StatementsFrom Wikipedia, the free encyclopedia

Chapter 1

Apophantic

Apophantic (Greek: ἀποφαντικός, “declaratory”, from ἀποφαίνειν apophainein, “to show, to make known”) is atermAristotle coined tomean a specific type of declaratory statement that can determine the truth or falsity of a logicalproposition or phenomenon. It was adopted by Edmund Husserl and Martin Heidegger as part of phenomenology.[1]Marcuse defines it as “the logic of judgment”.[2]

In Aristotle’s usage, the Greek term ἀποφαντικὸς λόγος (apophantic speech) describes a statement that, by examin-ing a proposition in itself, can determine what is true about a statement by establishing whether or not the predicate ofa sentence may logically be attributed to its subject. For example, logical propositions may be divided into ones thatare semantically determinate, as in the sentence “All penguins are birds,” and those that are semantically indetermi-nate, as in the sentence “All bachelors are unhappy.” In the first proposition, the subject is penguins and the predicateis birds, and the set of all birds is a category into which the subject of penguins should necessarily be put. In thesecond proposition, the subject is bachelors and the predicate is unhappy. This is a subjective, contingent connectionthat does not necessarily follow. An apophantic conclusion would, by examining the two statements—and not anyevidence supporting or denying them—make a judgment between them that identifies “All penguins are birds” asmore truthful than “All bachelors are unhappy.” One would reach this conclusion simply because of the propositions’nature, and not because any penguins or bachelors had been consulted.In phenomenology, Martin Heidegger argues that apophantic judgements are the most reliable means of obtainingtruth, as they do not rely upon subjective comparisons.[3] Before Heidegger, however, his former teacher Husserl hadalready centralized the role of apophantic judgment in his phenomenological 'transcendental logic', during the courselectures on passive synthesis in the mid 1920s.The concept appears in the Arabic Aristotelian tradition as jâzim, or 'truth-apt'.[4]

1.1 References[1] Roderick Munday, “Glossary of Terms in Being and Time", Retrieved 2012-05-27

[2] Herbert Marceuse, “One Dimensional Man: Part II, Chapter 5”, Retrieved 2012-05-27

[3] Roderick Munday, “Glossary of Terms in Being and Time", Retrieved 2012-05-27

[4] Stanford Encyclopedia of Philosophy, “Arabic and Islamic Philosophy of Language and Logic”, Retrieved 2012-05-27

1.2 External links• “Benedetto Croce, Aesthetic as Science of Expression and General Linguistic"

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Chapter 2

Artist’s statement

An artist’s statement (or artist statement) is an artist’s written description of their work. The brief verbal repre-sentation is for, and in support of, his or her own work to give the viewer understanding. As such it aims to inform,connect with an art context, and present the basis for the work; it is therefore didactic, descriptive, or reflective innature.

2.1 Description

The artist’s text intends to explain, justify, extend, and/or contextualize his or her body of work. It places or attemptsto place the work in relationship to art history and theory, the art world and the times. Further, the statement servesto show that the artist is conscious of their intentions, aware of their practice and its position within art parametersand of the discourse surrounding it. Therefore not only does it describe and place, but it indicates the level of theartist’s own comprehension of their field and making.Artists often write a short (50-100 word) and/or a long (500-1000 word) version of the same statement, and theymay maintain and revise these statements throughout their careers. [1]They may be edited to suit the requirements ofspecific funding bodies, galleries or call-outs as part of the application process.

2.2 History

The writing of artists’ statements is a comparatively recent phenomenon beginning in the 1990s.[2] In some respectsthe practice resembles the art manifesto and may derive in part from it. However, the artist’s statement generallyspeaks for an individual rather than a collective, and is not strongly associated with polemic. Rather, a contemporaryartist may be required to submit the statement in order to tender for commissions or apply for schools, residencies,jobs, awards, and other forms of institutional support, in justification of their submission.In their 2008 survey of North American art schools and university art programs, Garrett-Petts and Nash found thatnearly 90% teach the writing of artist statements as part of the curriculum; in addition they found that,

Like prefaces, forewords, prologues, and introductions in literary works, the artist statement performsa vital if complex rhetorical role: when included in an exhibition proposal and sent to a curator, the artiststatement usually provides a description of the work, some indication of the work’s art historical andtheoretical context, some background information about the artist and the artist’s intentions, technicalspecifications – and, at the same time, it aims to persuade the reader of the artwork’s value. Whenhung on a gallery wall, the statement (or “didactic”) becomes an invitation, an explanation, and, oftenindirectly, an element of the installation itself.[3]

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4 CHAPTER 2. ARTIST’S STATEMENT

Biography!!Monica Bonadies Hansel is a native of Palm Beach County, Florida, and learned to knit at the age of 9 from her paternal grandmother. After dabbling in architecture school, she studied mechanical engineering at the University of Central Florida and graduated with a bachelors degree shortly followed by a masters with a specialty in energy systems. She rediscovered her love for fiber art in 2008 while finishing her undergraduate degree and from there began picking apart patterns, spinning her own yarn, and knitting in the cafeteria on lunch breaks at work. Her day to day work focuses on power plants while her fiber art reflects her love of engineering and the sciences.!!Artist Statement!!I knit, crochet, and spin to ground myself. It is said that these techniques, used over hundreds of years, have become embedded in our DNA. So through the process of knitting a garment, I connect with my ancestors along with legions of women and men who turned heels, made hats, or elaborately cabled sweaters to identify their loved ones in a shipwreck. Thankfully, my creations identify the wearer as someone who perhaps has a low air conditioning temperature in their office rather than an ill-fated sailor. Along with a connection to crafters around the world, my work is a way for me to commemorate the memories of my two grandmothers and great-grandfather who knitted and continue their legacy.!!My designs are inspired by the laws of nature and the forms found in the sciences and engineering. Just as reality is bounded by physical laws, knitting and crocheting are bound by specific stitch forms. These forms can exact 45 degree angles, produce a circle, or something as simple as a rectangle. This is not even considering the inclusion of colors or texture. Using these basic forms, I work to represent the meeting of art and science.!!Description of the Work!“Wave” will represent a visual of several musical notes - G, E, C, and G. The sound wave is knitted into a scarf using a technique that shows the waveform when the scarf is tilted just so. Just as one looks closely at the Campanile and the bells within the tower are revealed, one must look closely to see the waveform. And, naturally, the colors must be Berkeley Blue and California Gold!!!

Artist’s statement by artist Monica Bonadies

2.3 As subject matter

On at least two occasions, artists’ statements have been the subject of gallery exhibitions. The first exhibition ofartists’ statements, The Art of the Artist’s Statement, was curated by Georgia Kotretsos and Maria Pashalidou at theHellenic Museum, Chicago, in the spring of 2005. It featured the work of 14 artists invited to create artwork offeringa visual commentary on the subject of artist statements. The second exhibition, Proximities: Artists’ Statements andTheir Works, was installed in the fall of 2005 at the Kamloops Art Gallery, Kamloops, British Columbia. Co-curatedby W.F. Garrett-Petts and Rachel Nash, the exhibition asked nine contributing artists to respond to the topic of

2.4. REFERENCES 5

artists’ statements by taking one or more of their own artist’s statements and working with the text(s) in a manner thatdocumented, represented, and annotated the original work, creating a new work in the process.

2.4 References[1] Tom Palin: Artist Statements 1992-2012. Workshop Press Leeds, 2013.

[2] Detterer, Gabriele. Ed. Art Recollection: Artists’ Interviews & Statements in the Nineties’]'. Florence: Danilo Montanari,Exit, and Zona Archives Editori, 1997.

[3] Garrett-Petts, W.F., and Rachel Nash. “Re-Visioning the Visual: Making Artistic Inquiry Visible.” Rhizomes 18 (Winter2008). Spec. issue on “Imaging Place”.

2.5 External links• Detterer, Gabriele. Ed. Art Recollection: Artists’ Interviews & Statements in the Nineties’ '. Florence: DaniloMontanari, Exit, and Zona Archives Editori, 1997.

• “Garrett-Petts, W.F. Literary Artists’ Statements”, Canadian Literature, No. 176 (Spring 2003): 111-114.

• Garrett-Petts, W.F., and Rachel Nash, eds. Proximities: Artists’ Statements and Their Works Kamloops, B.C.:Kamloops Art Gallery, 2005.

• “Nash, Rachel, and W.F. Garrett-Petts, eds. Artists’ Statements & the Nature of Artistic Inquiry, Open Letter.Thirteenth Series, No. 4, Strathroy, Canada, 2007.

• Garrett-Petts, W.F., and Rachel Nash. “Re-Visioning the Visual: Making Artistic Inquiry Visible.” Rhizomes18 (Winter 2008). Spec. issue on “Imaging Place”.

Chapter 3

Attending physician statement

An attending physician statement (APS) is a report by a physician, hospital or medical facility who has treated, orwho is currently treating, a person seeking insurance. In traditional underwriting, an APS is one of the most frequentlyordered additional sources of medical background information. The APS is one of the more expensive underwritingrequirements, as well as the most time consuming. It is usually completed only when a doctor has free time, as theirprimary focus is caring for patients. The underwriting cycle time is often severely hampered by the APS as it couldtake weeks or even months to obtain. Once obtained, it can be laborious to review and summarize as APS reportscan be large documents containing an indepth medical history information which may or may not be relevant.

3.1 Overview

In structured underwriting the data capture process with the proposed insured is asked very detailed questions toattempt to reduce the number of attending physician's statements necessary. However, many medical conditionsrequire supporting evidence from the physician. This is where an APS Summary can assist the underwriter in eval-uating the proposed insured’s medical risk(s). The APS summaries however, are only as good as the underwriter’sexperience which varies widely from person to person. Additionally, APS summaries when processed without a “tem-plate structure” guiding the information gathered from the APS often yield inconsistent or miss critical underwritinginformation. Either inconsistent or missing underwriting information will compromise the risk analysis process.By leveraging a seasoned underwriter’s knowledge and experiences, scripts can be built that ensure the appropriatemedical information is captured and recorded from the APS document. This is done by creating scripts that promptthe person summarizing the APS to enter all pertinent specific medical condition uncovered.

3.2 References

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Chapter 4

Co-premise

A co-premise is a premise in reasoning and informal logic which is not the main supporting reason for a contentionor a lemma, but is logically necessary to ensure the validity of an argument. One premise by itself, or a group ofco-premises can form a reason.

4.1 Logical structure

Every significant term or phrase appearing in a premise of a simple argument, should also appear in the con-tention/conclusion or in a co-premise. But this by itself does not guarantee a valid argument, see the fallacy ofthe undistributed middle for an example of this.Sometimes a co-premise will not be explicitly stated. This type of argument is known as an 'enthymematic' argument,and the co-premise may be referred to as a 'hidden' or an 'unstated' co-premise and will often be subject to an inferenceobjection. In this argument map of a simple argument the two reasons for the main contention are co-premises andnot separate reasons for believing the contention to be true. They are both necessary to ensure that the argument asa whole retains logical validity.

In this example, “What the Bible says is true” is a hidden co-premise.

4.2 See also

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Chapter 5

Conjecture

For text reconstruction, see Conjecture (textual criticism). For the annual science fiction convention held in SanDiego, see Conjecture (convention).In mathematics, a conjecture is a conclusion or proposition based on incomplete information, but for which no

The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivialzeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011. The Riemann hypothesis, a famous conjecture, says that all non-trivialzeros of the zeta function lie along the critical line.

proof has been found.[1][2] Conjectures such as the Riemann hypothesis (still a conjecture) or Fermat’s Last Theorem(which was a conjecture until proven in 1995) have shapedmuch of mathematical history as new areas of mathematicsare developed in order to prove them.

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5.1. IMPORTANT EXAMPLES 9

5.1 Important examples

5.1.1 Fermat’s Last Theorem

Main article: Fermat’s Last Theorem

In number theory, Fermat’s Last Theorem (sometimes called Fermat’s conjecture, especially in older texts) statesthat no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater thantwo.This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where heclaimed he had a proof that was too large to fit in the margin.[3] The first successful proof was released in 1994 byAndrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problemstimulated the development of algebraic number theory in the 19th century and the proof of the modularity theoremin the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it wasin the Guinness Book of World Records for “most difficult mathematical problems”.

5.1.2 Four color theorem

Main article: Four color theoremIn mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane

A four-coloring of a map of the states of the United States (ignoring lakes).

into contiguous regions, producing a figure called amap, no more than four colors are required to color the regions ofthe map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a commonboundary that is not a corner, where corners are the points shared by three or more regions.[4] For example, in themap of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share apoint that also belongs to Arizona and Colorado, are not.Möbius mentioned the problem in his lectures as early as 1840.[5] The conjecture was first proposed on October 23,1852[6] when Francis Guthrie, while trying to color the map of counties of England, noticed that only four differentcolors were needed. The five color theorem, which has a short elementary proof, states that five colors suffice to colora map and was proven in the late 19th century (Heawood 1890); however, proving that four colors suffice turned out

10 CHAPTER 5. CONJECTURE

to be significantly harder. A number of false proofs and false counterexamples have appeared since the first statementof the four color theorem in 1852.The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theoremto be proved using a computer. Appel and Haken’s approach started by showing that there is a particular set of 1,936maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. (If they did appear,you could make a smaller counter-example.) Appel and Haken used a special-purpose computer program to confirmthat each of these maps had this property. Additionally, any map that could potentially be a counterexample musthave a portion that looks like one of these 1,936 maps. Showing this required hundreds of pages of hand analysis.Appel and Haken concluded that no smallest counterexamples exists because any must contain, yet do not contain, oneof these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is thereforetrue. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasiblefor a human to check by hand (Swart 1980). Since then the proof has gained wider acceptance, although doubtsremain (Wilson 2002, 216–222).

5.1.3 Hauptvermutung

Main article: Hauptvermutung

TheHauptvermutung (German formain conjecture) of geometric topology is the conjecture that any two triangulationsof a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them. It wasoriginally formulated in 1908, by Steinitz and Tietze.This conjecture is now known to be false. The non-manifold version was disproved by John Milnor[7] in 1961 usingReidemeister torsion.The manifold version is true in dimensions m ≤ 3. The cases m = 2 and 3 were proved by Tibor Radó and Edwin E.Moise[8] in the 1920s and 1950s, respectively.

5.1.4 Weil conjectures

Main article: Weil conjectures

In mathematics, the Weil conjectures were some highly influential proposals by André Weil (1949) on the generatingfunctions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finitefields.A variety V over a finite field with q elements has a finite number of rational points, as well as points over every finitefield with qk elements containing that field. The generating function has coefficients derived from the numbers Nk ofpoints over the (essentially unique) field with qk elements.Weil conjectured that such zeta-functions should be rational functions, should satisfy a form of functional equation,and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the Riemannzeta function and Riemann hypothesis. The rationality was proved by Dwork (1960), the functional equation byGrothendieck (1965), and the analogue of the Riemann hypothesis was proved by Deligne (1974)

5.1.5 Poincaré conjecture

Main article: Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hyper-sphere that bounds the unit ball in four-dimensional space. The conjecture states:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopyequivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.

5.1. IMPORTANT EXAMPLES 11

Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaréconjecture claims that if such a space has the additional property that each loop in the space can be continuouslytightened to a point, then it is necessarily a three-dimensional sphere. An analogous result has been known in higherdimensions for some time.After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in threepapers made available in 2002 and 2003 on arXiv. The proof followed on from the program of Richard Hamilton touse the Ricci flow to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricciflow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way, butwas unable to prove this method “converged” in three dimensions.[9] Perelman completed this portion of the proof.Several teams of mathematicians have verified that Perelman’s proof is correct.The Poincaré conjecture, before being proven, was one of the most important open questions in topology.

5.1.6 Riemann hypothesis

Main article: Riemann hypothesis

In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the non-trivialzeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues,such as the Riemann hypothesis for curves over finite fields.The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations,some mathematicians consider it the most important unresolved problem in pure mathematics (Bombieri 2000). TheRiemann hypothesis, along with the Goldbach conjecture, is part of Hilbert’s eighth problem in David Hilbert's listof 23 unsolved problems; it is also one of the Clay Mathematics Institute Millennium Prize Problems.

5.1.7 P versus NP problem

Main article: P versus NP problem

The P versus NP problem is a major unsolved problem in computer science. Informally, it asks whether every problemwhose solution can be quickly verified by a computer can also be quickly solved by a computer; it is widely conjecturedthat the answer is no. It was essentially first mentioned in a 1956 letter written by Kurt Gödel to John von Neumann.Gödel asked whether a certain NP complete problem could be solved in quadratic or linear time.[10] The precisestatement of the P=NP problem was introduced in 1971 by Stephen Cook in his seminal paper “The complexity oftheorem proving procedures”[11] and is considered by many to be the most important open problem in the field.[12] Itis one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$1,000,000prize for the first correct solution.

5.1.8 Other conjectures

• Goldbach’s conjecture

• The twin prime conjecture

• The Collatz conjecture

• The Manin conjecture

• The Maldacena conjecture

• The Langlands program[13] is a far-reaching web of these ideas of 'unifying conjectures' that link different sub-fields of mathematics, e.g. number theory and representation theory of Lie groups; some of these conjectureshave since been proved.

12 CHAPTER 5. CONJECTURE

5.2 Resolution of conjectures

5.2.1 Proof

Formal mathematics is based on provable truth. In mathematics, any number of cases supporting a conjecture,no matter how large, is insufficient for establishing the conjecture’s veracity, since a single counterexample wouldimmediately bring down the conjecture. Mathematical journals sometimes publish theminor results of research teamshaving extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture,which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 ×1012 (over a trillion). However, the failure to find a counterexample after extensive search does not constitute a proofthat no counterexample exists nor that the conjecture is true, because the conjecture might be false but with a verylarge minimal counterexample.Instead, a conjecture is considered proven only when it has been shown that it is logically impossible for it to be false.There are various methods of doing so; see Mathematical proof#Methods for details.One method of proof, usable when there are only a finite number of cases that could lead to counterexamples, isknown as “brute force": in this approach, all possible cases are considered and shown not to give counterexamples.Sometimes the number of cases is quite large, in which situation a brute-force proof may require as a practical matterthe use of a computer algorithm to check all the cases: the validity of the 1976 and 1997 brute-force proofs of thefour color theorem by computer was initially doubted, but was eventually confirmed in 2005 by theorem-provingsoftware.When a conjecture has been proven, it is no longer a conjecture but a theorem. Many important theorems were onceconjectures, such as the Geometrization theorem (which resolved the Poincaré conjecture), Fermat’s Last Theorem,and others.

5.2.2 Disproof

Conjectures disproven through counterexample are sometimes referred to as false conjectures (cf. the Pólya conjectureand Euler’s sum of powers conjecture). In the case of the latter, the first counterexample found involved numbers inthe millions, although subsequently it has been found that the minimal counterexample is smaller than that.

5.2.3 Undecidable conjectures

Not every conjecture ends up being proven true or false. The continuum hypothesis, which tries to ascertain therelative cardinality of certain infinite sets, was eventually shown to be undecidable (or independent) from the generallyaccepted set of axioms of set theory. It is therefore possible to adopt this statement, or its negation, as a new axiomin a consistent manner (much as we can take Euclid's parallel postulate as either true or false).In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis(in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutralgeometry, i.e. no parallel postulate.) The one major exception to this in practice is the axiom of choice—unlessstudying this axiom in particular, the majority of researchers do not usually worry whether a result requires the axiomof choice.

5.3 Conditional proofs

Sometimes a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs ofother results. For example, the Riemann hypothesis is a conjecture from number theory that (amongst other things)makes predictions about the distribution of prime numbers. Few number theorists doubt that the Riemann hypothesisis true. In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent onthe truth of this conjecture. These are called conditional proofs: the conjectures assumed appear in the hypothesesof the theorem, for the time being.These “proofs”, however, would fall apart if it turned out that the hypothesis was false, so there is considerable interestin verifying the truth or falsity of conjectures of this type.

5.4. IN OTHER SCIENCES 13

5.4 In other sciences

Karl Popper pioneered the use of the term “conjecture” in scientific philosophy.[14] Conjecture is related to hypothesis,which in science refers to a testable conjecture.

5.5 See also• Hypotheticals

• List of conjectures

5.6 References[1] Oxford Dictionary of English (2010 ed.).

[2] Schwartz, JL (1995). Shuttling between the particular and the general: reflections on the role of conjecture and hypothesis inthe generation of knowledge in science and mathematics. p. 93.

[3] Ore, Oystein (1988) [1948], Number Theory and Its History, Dover, pp. 203–204, ISBN 978-0-486-65620-5

[4] Georges Gonthier (December 2008). “Formal Proof—The Four-Color Theorem”. Notices of the AMS 55 (11): 1382–1393.From this paper: Definitions: A planar map is a set of pairwise disjoint subsets of the plane, called regions. A simplemap is one whose regions are connected open sets. Two regions of a map are adjacent if their respective closures have acommon point that is not a corner of the map. A point is a corner of a map if and only if it belongs to the closures of atleast three regions. Theorem: The regions of any simple planar map can be colored with only four colors, in such a waythat any two adjacent regions have different colors.

[5] W. W. Rouse Ball (1960) The Four Color Theorem, in Mathematical Recreations and Essays, Macmillan, New York, pp222-232.

[6] Donald MacKenzie, Mechanizing Proof: Computing, Risk, and Trust (MIT Press, 2004) p103

[7] Milnor, John W. (1961). “Two complexes which are homeomorphic but combinatorially distinct”. Annals of Mathematics74 (2): 575–590. doi:10.2307/1970299. JSTOR 1970299. MR 133127.

[8] Moise, Edwin E. (1977). Geometric Topology in Dimensions 2 and 3. New York: New York : Springer-Verlag. ISBN978-0-387-90220-3.

[9] Hamilton, Richard S. (1997). “Four-manifolds with positive isotropic curvature”. Communications in Analysis and Geom-etry 5 (1): 1–92. MR 1456308. Zbl 0892.53018.

[10] Juris Hartmanis 1989, Gödel, von Neumann, and the P = NP problem, Bulletin of the European Association for TheoreticalComputer Science, vol. 38, pp. 101–107

[11] Cook, Stephen (1971). “The complexity of theorem proving procedures”. Proceedings of the Third Annual ACM Sympo-sium on Theory of Computing. pp. 151–158.

[12] Lance Fortnow, The status of the P versus NP problem, Communications of the ACM 52 (2009), no. 9, pp. 78–86.doi:10.1145/1562164.1562186

[13] Langlands, Robert (1967), Letter to Prof. Weil

[14] Popper, Karl (2004). Conjectures and refutations : the growth of scientific knowledge. London: Routledge. ISBN 0-415-28594-1.

5.7 External links• Open Problem Garden

• Unsolved Problems web site

Chapter 6

Corollary

A corollary (/ˈkɒrəlɛri/ KORR-əl-ur-ee or UK /kɒˈrɒləri/ ko-ROL-ər-ee) is a statement that follows readily from aprevious statement.In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition ortheorem, is intrinsically subjective. Proposition B is a corollary of proposition A if B can readily be deduced fromA or is self-evident from its proof, but the meaning of readily or self-evident varies depending upon the author andcontext. The importance of the corollary is often considered secondary to that of the initial theorem; B is unlikely tobe termed a corollary if its mathematical consequences are as significant as those of A. Sometimes a corollary has aproof that explains the derivation; sometimes the derivation is considered self-evident.

6.1 Peirce on corollarial and theorematic reasonings

Charles Sanders Peirce held that themost important division of kinds of deductive reasoning is that between corollarialand theorematic. He argued that, while finally all deduction depends in one way or another on mental experimen-tation on schemata or diagrams,[1] still in corollarial deduction “it is only necessary to imagine any case in whichthe premisses are true in order to perceive immediately that the conclusion holds in that case”, whereas theorematicdeduction “is deduction in which it is necessary to experiment in the imagination upon the image of the premiss inorder from the result of such experiment to make corollarial deductions to the truth of the conclusion.”[2] He held thatcorollarial deduction matches Aristotle’s conception of direct demonstration, which Aristotle regarded as the onlythoroughly satisfactory demonstration, while theorematic deduction (A) is the kind more prized by mathematicians,(B) is peculiar to mathematics,[1] and (C) involves in its course the introduction of a lemma or at least a defini-tion uncontemplated in the thesis (the proposition that is to be proved); in remarkable cases that definition is of anabstraction that “ought to be supported by a proper postulate.”[3]

6.2 See also

• Lemma (mathematics)

• Roosevelt Corollary to the Monroe Doctrine

6.3 References

• Weisstein, Eric W., “Corollary”, MathWorld.

• corollary at dictionary.com

• Chambers’s Encyclopaedia. Volume 3, Appleton 1864, p. 260 (online copy, p. 260, at Google Books)

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6.3. REFERENCES 15

[1] Peirce, C. S., from section dated 1902 by editors in the “Minute Logic” manuscript, Collected Papers v. 4, paragraph 233,quoted in part in "Corollarial Reasoning" in the Commens Dictionary of Peirce’s Terms, 2003–present, Mats Bergman andSami Paavola, editors, University of Helsinki.

[2] Peirce, C. S., the 1902 Carnegie Application, published in The New Elements of Mathematics, Carolyn Eisele, editor, alsotranscribed by Joseph M. Ransdell, see “From Draft A - MS L75.35-39” in Memoir 19 (once there, scroll down).

[3] Peirce, C. S., 1901 manuscript “On the Logic of Drawing History from Ancient Documents, Especially from Testimonies’,The Essential Peirce v. 2, see p. 96. See quote in "Corollarial Reasoning" in the Commens Dictionary of Peirce’s Terms.

Chapter 7

Corresponding conditional

This article is about the term “corresponding conditional” as it is used in logic

In logic, the corresponding conditional of an argument (or derivation) is a material conditional whose antecedentis the conjunction of the argument’s (or derivation’s) premises and whose consequent is the argument’s conclusion.An argument is valid if and only if its corresponding conditional is a logical truth. It follows that an argument is validif and only if the negation of its corresponding conditional is a contradiction. The construction of a correspondingconditional therefore provides a useful technique for determining the validity of argument

7.1 Example

Consider the argument A:

Either it is hot or it is coldIt is not hotTherefore it is cold

This argument is of the form:

Either P or QNot PTherefore Qor (using standard symbols of the propositional calculus):P Q¬P____________Q

The corresponding conditional C is:

IF ((P or Q) and not P) THEN Qor (using standard symbols):((P Q) & ¬P)→ Q

and the argument A is valid just in case the corresponding conditional C is a necessary truth.If C is a necessary truth then ¬C entails Falsity (The False).Thus, any argument is valid if and only if the denial of its corresponding conditional leads to a contradiction.

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7.2. APPLICATION 17

If we construct a truth table for C we will find that it comes out T (true) on every row (and of course if we constructa truth table for the negation of C it will come out F (false) in every row. These results confirm the validity of theargument ASome arguments need first-order predicate logic to reveal their forms and they cannot be tested properly by truthtables forms.Consider the argument A1:

Some mortals are not GreeksSome Greeks are not menNot every man is a logicianTherefore Some mortals are not logicians

To test this argument for validity, construct the corresponding conditional C1 (you will need first-order predicatelogic), negate it, and see if you can derive a contradiction from it. If you succeed then the argument is valid.

7.2 Application

Instead of attempting to derive the conclusion from the premises proceed as follows.To test the validity of an argument (a) translate, as necessary, each premise and the conclusion into sentential orpredicate logic sentences (b) construct from these the negation of the corresponding conditional (c) see if from it acontradiction can be derived (or if feasible construct a truth table for it and see if it comes out false on every row.)Alternatively construct a truth tree and see if every branch is closed. Success proves the validity of the originalargument.In case of difficulty trying to derive a contradiction proceed as follows. From the negation of the correspondingconditional derive a theorem in conjunctive normal form in the methodical fashions described in text books. If andonly if the original argument was valid will the theorem in conjunctive normal form be a contradiction, and if it isthen that it is will be apparent.

7.3 References

7.4 Further reading• First-order Logic: An Introduction

By Leigh S. Cauman Published by Walter de Gruyter, 1998 ISBN 3-11-015766-7, ISBN 978-3-11-015766-6, Page19

• The Cambridge Companion to Mill

By John Skorupski Published by Cambridge University Press, 1998 ISBN 0-521-42211-6, ISBN 978-0-521-42211-6,PAge 40

• The Languages of Logic: An Introduction to Formal Logic

By Samuel D. Guttenplan Published by Blackwell Publishing, 1997 ISBN 1-55786-988-X, 9781557869883, page90.

• The Value of Knowledge and the Pursuit of Understanding

By Jonathan L. Kvanvig Published by Cambridge University Press, 2003 ISBN 0-521-82713-2, ISBN 978-0-521-82713-3, page 175

18 CHAPTER 7. CORRESPONDING CONDITIONAL

• Logic

By Paul Tomassi Published by Routledge, 1999 ISBN 0-415-16696-9, ISBN 978-0-415-16696-6, page 153

7.5 External links• Corresponding conditional from the Free On-line Dictionary of Computing

• http://books.google.co.uk/books?id=TQlvJJgUiVoC&pg=PA19&lpg=PA19&dq=Corresponding+conditional&source=web&ots=V0GmWFcKsg&sig=JXjvWnQJpOKjU_-Nr-e3vE6s8PE&hl=en&sa=X&oi=book_result&resnum=3&ct=result

• http://books.google.co.uk/books?id=BVHwg_qNxosC&pg=PA40&lpg=PA40&dq=Corresponding+conditional&source=web&ots=MHRGHboBUd&sig=ha4gxQrKdKsINVcSOWBfrpvNQ00&hl=en&sa=X&oi=book_result&resnum=6&ct=result

• http://www.earlham.edu/~{}peters/courses/log/terms2.htm

• http://www.csus.edu/indiv/n/nogalesp/SymbolicLogicGustason/SymbolicLogicOverheads/Phil60GusCh2TruthTablesSemanticMethods/TTValidityCorrespondingConditional.doc

• http://books.google.co.uk/books?id=xfOdpyj1bSIC&pg=PA90&lpg=PA90&dq=Corresponding+conditional&source=web&ots=PNBSh6fukg&sig=7BEBKbCD5Qhq9TOIBri9Oa5Zah4&hl=en&sa=X&oi=book_result&resnum=6&ct=result

• http://books.google.co.uk/books?id=OxXopc5AjQ0C&pg=PA175&lpg=PA175&dq=Corresponding+conditional&source=web&ots=FCFY5L4_HB&sig=7pkTUrJ87AtojCVRzeej5eHgqnA&hl=en&sa=X&oi=book_result&resnum=2&ct=result

• http://books.google.co.uk/books?id=tb6bxjyrFJ4C&pg=PA153&dq=Corresponding+conditional+logic

Chapter 8

Elevator pitch

For other uses, see Pitch.

An elevator pitch, elevator speech or elevator statement is a short summary used to quickly and simply define aprocess, product, service, organization, or event and its value proposition.[1]

The name 'elevator pitch' reflects the idea that it should be possible to deliver the summary in the time span ofan elevator ride, or approximately thirty seconds to two minutes and is widely credited to Ilene Rosenzweig andMichael Caruso (while he was editor for Vanity Fair) for its origin.[2][3] The term itself comes from a scenario ofan accidental meeting with someone important in the elevator. If the conversation inside the elevator in those fewseconds is interesting and value adding, the conversation will either continue after the elevator ride, or end in exchangeof business cards or a scheduled meeting.[4]

A variety of people, including project managers, salespeople, evangelists, and policy-makers, commonly rehearse anduse elevator pitches to get their points across quickly.One idea behind an elevator pitch is to be able to actually not only say what you do but do it in a way that is interesting.For example, if you asked someone what they do and they answer “I am a financial planner” the conversation mayalmost end there. As the person who hears what they do may already feel they know what it is that person does. Soit is important to be able to explain what you do in more detail but without using terminology that will pigeon holeyou before you get started. [5]

8.1 See also

• Mission statement

• Vision statement

8.2 References

[1] Pincus, Aileen. “The Perfect (Elevator) Pitch”. Business Week.

[2] Hahn, Gerald J. (1989), “Statistics-Aided Manufacturing: A Look Into the Future”, The American Statistician 43 (2):74–79, doi:10.2307/2684502, JSTOR 2684502.

[3] Peters, Tom (1999), “The Wow Project” (PDF), Fast Company 24, p. 116.

[4] “What is an Elevator Pitch?". Mind Your Pitch. |first1= missing |last1= in Authors list (help)

[5] http://successclub.com.au/seven-ps-crafting-perfect-elevator-pitch/

19

20 CHAPTER 8. ELEVATOR PITCH

8.3 External links• Business Know-How - The Art of the Elevator Pitch

Chapter 9

Eternal statement

An eternal statement is a statement whose token instances all have the same truth value. For instance, every inscrip-tion or utterance of the sentence “On July 15, 2009 it rains in Boston.” has the same truth value, no matter when orwhere it is asserted. This type of statement is distinguished from others in that its context will not influence its truthvalue. Essentially, an eternal statement is a true statement, regardless of how it used.

9.1 References• W.V.O. Quine, Philosophy of Logic

21

Chapter 10

Fact

For other uses, see Fact (disambiguation).

A fact is something that has really occurred or is actually the case. The usual test for a statement of fact is verifiability—that is, whether it can be demonstrated to correspond to experience. Standard reference works are often used to checkfacts. Scientific facts are verified by repeatable careful observation or measurement (by experiments or other means).

10.1 Etymology and usage

The word fact derives from the Latin factum, and was first used in English with the same meaning: a thing doneor performed, a meaning now obsolete.[1] The common usage of “something that has really occurred or is the case”dates from the middle of the sixteenth century.[2]

Fact is sometimes used synonymously with truth, as distinct from opinions, falsehoods, or matters of taste. This useis found in such phrases as, "It is a fact that the cup is blue” or "Matter of fact”,[3] and "... not history, nor fact, butimagination.” Filmmaker Werner Herzog distinguishes clearly between the two, claiming that “Fact creates norms,and truth illumination.”[4]

Fact also indicates a matter under discussion deemed to be true or correct, such as to emphasize a point or prove adisputed issue; (e.g., "... the fact of the matter is ...”).[5][6]

Alternatively, fact may also indicate an allegation or stipulation of something that may or may not be a true fact,[7](e.g., “the author’s facts are not trustworthy”). This alternate usage, although contested by some, has a long historyin standard English.[8]

Fact may also indicate findings derived through a process of evaluation, including review of testimony, direct obser-vation, or otherwise; as distinguishable from matters of inference or speculation.[9] This use is reflected in the terms“fact-find” and “fact-finder” (e.g., “set up a fact-finding commission”).[10]

Facts may be checked by reason, experiment, personal experience, or may be argued from authority. Roger Baconwrote “If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to placethe foundations of knowledge in mathematics.”[11]

10.2 Fact in philosophy

In philosophy, the concept fact is considered in epistemology and ontology. Questions of objectivity and truth areclosely associated with questions of fact. A “fact” can be defined as something that is the case—that is, a state ofaffairs.[12][13]

Facts may be understood as information that makes a true sentence true.[14] Facts may also be understood as thosethings to which a true sentence refers. The statement “Jupiter is the largest planet in the solar system” is about thefact Jupiter is the largest planet in the solar system.[15]

22

10.3. FACT IN SCIENCE 23

10.2.1 Correspondence and the slingshot argument

Engel’s version of the correspondence theory of truth explains that what makes a sentence true is that it correspondsto a fact.[16] This theory presupposes the existence of an objective world.The Slingshot argument claims to show that all true statements stand for the same thing - the truth value true. Ifthis argument holds, and facts are taken to be what true statements stand for, then we reach the counter-intuitiveconclusion that there is only one fact - the truth.[17]

10.2.2 Compound facts

Any non-trivial true statement about reality is necessarily an abstraction composed of a complex of objects andproperties or relations.[18] For example, the fact described by the true statement "Paris is the capital city of France"implies that there is such a place as Paris, there is such a place as France, there are such things as capital cities, aswell as that France has a government, that the government of France has the power to define its capital city, and thatthe French government has chosen Paris to be the capital, that there is such a thing as a place or a government, andso on. The verifiable accuracy of all of these assertions, if facts themselves, may coincide to create the fact that Parisis the capital of France.Difficulties arise, however, in attempting to identify the constituent parts of negative, modal, disjunctive, or moralfacts.[19]

10.2.3 Fact–value distinction

Main article: Fact–value distinction

Moral philosophers since David Hume have debated whether values are objective, and thus factual. In A Treatise ofHuman Nature Hume pointed out there is no obvious way for a series of statements about what ought to be the caseto be derived from a series of statements of what is the case. Those who insist there is a logical gulf between facts andvalues, such that it is fallacious to attempt to derive values from facts, include G. E. Moore, who called attempting todo so the naturalistic fallacy.

10.2.4 Factual–counterfactual distinction

Main article: Counterfactual conditional

Factuality—what has occurred—can also be contrasted with counterfactuality: what might have occurred, but didnot. A counterfactual conditional or subjunctive conditional is a conditional (or “if-then”) statement indicating whatwould be the case if events had been other than they actually are. For example, “If Alexander had lived, his empirewould have been greater than Rome.” This contrasts with an indicative conditional, which indicates what is (in fact)the case if its antecedent is (in fact) true—for example, “If you drink this, it will make you well.”Such sentences are important to modal logic, especially since the development of possible world semantics.

10.3 Fact in science

Further information: scientific method and philosophy of science

In science, a fact is a repeatable careful observation or measurement (by experimentation or other means), also calledempirical evidence. Facts are central to building scientific theories. Various forms of observation and measurementlead to fundamental questions about the scientific method, and the scope and validity of scientific reasoning.In the most basic sense, a scientific fact is an objective and verifiable observation, in contrast with a hypothesis ortheory, which is intended to explain or interpret facts.[20]

24 CHAPTER 10. FACT

Various scholars have offered significant refinements to this basic formulation. Scientists are careful to distinguishbetween: 1) states of affairs in the external world and 2) assertions of fact that may be considered relevant in scientificanalysis. The term is used in both senses in the philosophy of science.[21]

Scholars and clinical researchers in both the social and natural sciences have written about numerous questions andtheories that arise in the attempt to clarify the fundamental nature of scientific fact.[22] Pertinent issues raised by thisinquiry include:

• the process by which “established fact” becomes recognized and accepted as such;[23]

• whether and to what extent “fact” and “theoretic explanation” can be considered truly independent and separablefrom one another;[24][25]

• to what extent “facts” are influenced by the mere act of observation;[25] and

• to what extent factual conclusions are influenced by history and consensus, rather than a strictly systematicmethodology.[26]

Consistent with the theory of confirmation holism, some scholars assert “fact” to be necessarily “theory-laden” tosome degree. Thomas Kuhn points out that knowing what facts to measure, and how to measure them, requires theuse of other theories. For example, the age of fossils is based on radiometric dating, which is justified by reasoningthat radioactive decay follows a Poisson process rather than a Bernoulli process. Similarly, Percy Williams Bridgmanis credited with the methodological position known as operationalism, which asserts that all observations are not onlyinfluenced, but necessarily defined by the means and assumptions used to measure them.

10.3.1 Fact and the scientific method

Apart from the fundamental inquiry into the nature of scientific fact, there remain the practical and social consid-erations of how fact is investigated, established, and substantiated through the proper application of the scientificmethod.[27] Scientific facts are generally believed independent of the observer: no matter who performs a scientificexperiment, all observers agree on the outcome.[28] In addition to these considerations, there are the social and insti-tutional measures, such as peer review and accreditation, that are intended to promote factual accuracy (among otherinterests) in scientific study.[29]

One researcher described the place of facts in science as follows:[30]

In case the difference between evidence and medical evidence or the difference between facts and sci-entific facts eludes you, let me explain. If I have a headache or a fever, that’s not a fact except to me. IfI tell a doctor about it, that’s what doctors call anecdotal evidence or testimonial. If the doctor takes mytemperature and writes it down, the headache becomes medical evidence. If another doctor copies it, itbecomes a scientific fact. Should I need proof that my fever was 101 last Tuesday and ask my doctorfor my chart, it will not be given to me. That plain garden variety fact has now become a scientific fact.It’s only available to another doctor. If I complain the doctor won’t give me the scientific facts about mypast condition, that’s anecdotal evidence again.

10.4 Fact in history

Further information: Historiography

A common rhetorical cliché states, "History is written by the winners.” This phrase suggests but does not examinethe use of facts in the writing of history.E. H. Carr in his 1961 volume, What is History?, argues that the inherent biases from the gathering of facts makesthe objective truth of any historical perspective idealistic and impossible. Facts are, “like fish in the Ocean,” of whichwe may only happen to catch a few, only an indication of what is below the surface. Even a dragnet cannot tell us forcertain what it would be like to live below the Ocean’s surface. Even if we do not discard any facts (or fish) presented,we will always miss the majority; the site of our fishing, the methods undertaken, the weather and even luck play avital role in what we will catch. Additionally, the composition of history is inevitably made up by the compilation

10.5. FACT IN LAW 25

of many different biases of fact finding - all compounded over time. He concludes that for a historian to attempt amore objective method, one must accept that history can only aspire to a conversation of the present with the past -and that one’s methods of fact gathering should be openly examined. Historical truth and facts therefore change overtime, and reflect only the present consensus (if that).

10.5 Fact in law

Further information: Evidence (law) and Trier of fact

In most common law jurisdictions,[31] the general concept and analysis of fact reflects fundamental principles ofjurisprudence, and is supported by several well-established standards.[32][33] Matters of fact have various formal def-initions under common law jurisdictions.These include:

• an element required in legal pleadings to demonstrate a cause of action;[34][35]

• the determinations of the finder of fact after evaluating admissible evidence produced in a trial or hearing;[36]

• a potential ground of reversible error forwarded on appeal in an appellate court;[37] and

• any of various matters subject to investigation by official authority to establish whether a crime has been per-petrated, and to establish culpability.[38]

10.5.1 Legal pleadings

Main article: Pleading

A party to a civil suit generally must clearly state all relevant allegations of fact that form the basis of a claim. Therequisite level of precision and particularity of these allegations varies, depending on the rules of civil procedureand jurisdiction. Parties who face uncertainties regarding facts and circumstances attendant to their side in a disputemay sometimes invoke alternative pleading.[39] In this situation, a party may plead separate sets of facts that (whenconsidered together) may be contradictory or mutually exclusive. This (seemingly) logically-inconsistent presentationof facts may be necessary as a safeguard against contingencies (such as res judicata) that would otherwise precludepresenting a claim or defense that depends on a particular interpretation of the underlying facts.[40]

10.5.2 Facts submitted by Amicus Curiae

The United States Supreme Court receives and often cites 'facts’ obtained from amicus briefs in its decisions. That isnot supposed to happen, according to Justice Scalia,:[41]

“Supreme Court briefs are an inappropriate place to develop the key facts in a case. We normally giveparties more robust protection, leaving important factual questions to district courts and juries aidedby expert witnesses and the procedural protections of discovery.” [Inadequate scrutiny can result in]“untested judicial fact-finding masquerading as statutory interpretation.”

Nonetheless, 'facts’ introduced in amicus briefs are cited in some Supreme Court decisions,[42] bringing up the needto distinguish between real facts and internet facts:[43]

“The Supreme Court has the same problem that the rest of us do: figuring out how to distinguish betweenreal facts and Internet facts...Amicus briefs from unreliable sources can contribute to that problem.”

26 CHAPTER 10. FACT

10.6 See also• Brute fact

• Consensus reality

• Counterfactual history

• De facto

• Factoid

• Lie

10.7 References[1] “Fact”. OED_2d_Ed_1989, (but note the conventional uses: after the fact and before the fact).

[2] “Fact” (1a). OED_2d_Ed_1989 Joye Exp. Dan. xi. Z vij b, Let emprours and kinges know this godly kynges fact. 1545

[3] “Fact” (4a) OED_2d_Ed_1989

[4] “Werner Herzog Film: Statements / Texts”. Wernerherzog.com. 1999-04-30. Retrieved 2014-07-14.

[5] “Fact” (6c). OED_2d_Ed_1989

[6] (See also “Matter” (2,6). Compact_OED)

[7] “Fact” (5). OED_2d_Ed_1989

[8] According to the American Heritage Dictionary of the English Language, “Fact has a long history of usage in the sense'allegation'" AHD_4th_Ed. The OED dates this use to 1729.

[9] “Fact” (6a). OED_2d_Ed_1989

[10] “Fact” (8). OED_2d_Ed_1989

[11] Roger Bacon, translated by Robert Burke Opus Majus, Book I, Chapter 2.

[12] “A fact, it might be said, is a state of affairs that is the case or obtains.” – Stanford Encyclopaedia of Philosophy. States ofAffairs

[13] See Wittgenstein, Tractatus Logico-Philosophicus, Proposition 2: What is the case -- a fact -- is the existence of states ofaffairs.

[14] “A fact is, traditionally, the worldly correlate of a true proposition, a state of affairs whose obtaining makes that propositiontrue.” – Fact in The Oxford Companion to Philosophy

[15] Alex Oliver, Fact, in Craig, Edward (2005). Shorter Routledge Encyclopedia of Philosophy. Routledge, Oxford. ISBN0-415-32495-5.

[16] Engel, Pascal (2002). Truth. McGill-Queen’s Press- MQUP. ISBN 0-7735-2462-2.

[17] The argument is presented in many places, but see for example Davidson, Truth and Meaning, in Davidson, Donald (1984).Truth and Interpretation. Clarendon Press, Oxford. ISBN 0-19-824617-X.

[18] “Facts possess internal structure, being complexes of objects and properties or relations” Oxford Companion to Philosophy

[19] Fact, in The Oxford Companion to Philosophy, Ted Honderich, editor. (Oxford, 1995) ISBN 0-19-866132-0

[20] Gower, Barry (1997). Scientific Method: A Historical and Philosophical Introduction. Routledge. ISBN 0-415-12282-1.

[21] Ravetz, Jerome Raymond (1996). Scientific Knowledge and Its Social Problems. Transaction Publishers. ISBN 1-56000-851-2.

[22] (Gower 1996)

[23] (see e.g., Ravetz, p. 182 fn. 1)

10.7. REFERENCES 27

[24] Ravetz, p. 185

[25] Gower, p. 138

[26] Gower, p. 7

[27] Ravetz p. 181 et. seq. (Chapter Six: “Facts and their evolution”)

[28] Cassell, Eric J. The Nature of Suffering and the Goals of Medicine Oxford University Press. Retrieved 16 May 2007.

[29] (Ravetz 1996)

[30] William Dufty (1975) Sugar Blues, page 93

[31] Ed. note: this section of the article emphasizes common law jurisprudence (as primarily represented in Anglo-American–based legal tradition). Nevertheless, the principles described herein have analogous treatment in other legal systems (suchas civil law systems) as well.

[32] Estrich, Willis Albert (1952). American Jurisprudence: A Comprehensive Text Statement of American Case Law. LawyersCo-operative Publishing Company.

[33] Elkouri, Frank (2003). How Arbitration Works. BNA Books. ISBN 1-57018-335-X.p. 305

[34] Bishin, William R. (1972). Law Language and Ethics: An Introduction to Law and Legal Method. Foundation Press.Original from the University of Michigan Digitized March 24, 2006.p. 277

[35] The Yale Law Journal: Volume 7. Yale Law Journal Co. 1898.

[36] Per Lord Shaw of Dunfermline, Clarke v. Edinburgh and District Tramways Co., 1919 S.C.(H.L.) 35, at p 36.

[37] Merrill, John Houston (1895). The American and English Encyclopedia of Law. E. Thompson. Original from HarvardUniversity Digitized April 26, 2007.

[38] Bennett, Wayne W. (2003). Criminal Investigation. Thomson Wadsworth. ISBN 0-534-61524-4.

[39] Roy W. McDonald, Alternative Pleading in the United States: I Columbia Law Review, Vol. 52, No. 4 (Apr., 1952), pp.443-478

[40] (McDonald 1952)

[41] Justice Antonin Scalia (June 9, 2011). “Justice Scalia, dissenting: On writ of certiorari to the United States Court of Appealsfor the Seventh Circuit, No. 09-11311” (PDF). p. 5.

[42] Examples are cited by Allison Orr Larsen, posted by Alan Meese (July 30, 2014). “Allison Orr Larsen on IntenselyEmpirical Amicus Briefs and Amicus Opportunism at the Supreme Court". Bishop Madison: Occasional commentary onpolitical economy and a free society.

[43] Kannon K. Shanmugam as quoted by Adam Liptak (September 1, 2014). “The dubious sources of some supreme court'facts’". New York Times (New York Times).

Chapter 11

False statement

A false statement is a statement that is not true. Although the word fallacy is sometimes used as a synonym for falsestatement, that is not how the word is used in philosophy, mathematics, logic and most formal contexts.A false statement need not be a lie. A lie is a statement that is known to be untrue and is used to mislead. A falsestatement is a statement that is untrue but not necessarily told to mislead, as a statement given by someone who doesnot know it is untrue.

11.1 Examples of false statements

Misleading statement (lie)

John told his little brother that sea otters aren't mammals, but fish, even though John himself was a marine biologistand knew otherwise. John simply wanted to see his little brother fail his class report, in order to teach him to beginprojects early, which help him develop skills necessary to succeed in life.

Statement made out of ignorance

James, John’s brother, stated in his class report that sea otters were fish. James got an F after his teacher pointed outwhy that statement was false. James did not know that sea otters were in fact mammals because he heard that seaotters were fish from his older brother John, a marine biologist.

11.2 In law

In some jurisdictions, false statement is a crime similar to perjury.

11.2.1 United States

Main article: Making false statements

In U.S. law, a “false statement” generally refers to United States federal false statements statute, contained in 18U.S.C.§ 1001. Most commonly, prosecutors use this statute to reach cover-up crimes such as perjury, false declarations, andobstruction of justice and government fraud cases.[1] Its earliest progenitor was the False Claims Act of 1863,[2] andin 1934 the requirement of an intent to defraud was eliminated to enforce the National Industrial Recovery Act of1933 (NIRA) against producers of “hot oil”, oil produced in violation of production restrictions established pursuantto the NIRA.[3]

The statute criminalizes a government official who “knowingly and willfully":[4]

28

11.3. SEE ALSO 29

(1) falsifies, conceals, or covers up by any trick, scheme, or device a material fact;(2) makes any materially false, fictitious, or fraudulent statement or representation; or

(3) makes or uses any false writing or document knowing the same to contain any materially false,fictitious, or fraudulent statement or entry.

11.3 See also• False accusation

• False statements of fact

• Jumping to conclusions

• Making false statements

11.4 References[1] Strader, Kelly J. Understanding White Collar Crime (2 ed.).

[2] Hubbard v. United States, 514 U.S. 695 (1995)

[3] United States v. Gilliland, 312 US 86, 93-94 (1941) (“Legislation had been sought by the Secretary of the Interior to aidthe enforcement of laws relating to the functions of the Department of the Interior and, in particular, to the enforcementof regulations under Sec. 9(c) of the [NIRA].”).

[4] 18 U.S.C. § 1001

Chapter 12

I Am a Man!

I Am aMan! is a declaration of civil rights, often used as a personal statement and as a declaration of independenceagainst oppression.

12.1 Am I Not a Man?

Historically, in countries such as the U.S. and South Africa, the term "boy" was used as a pejorative racist insulttowards men of color and slaves, indicating their subservient social status of being less than men.[1] In response,Am I Not A Man And A Brother? became a catchphrase used by British and American abolitionists. In 1787, JosiahWedgwood designed a medallion for the British anti-slavery campaign. He copied the original design from the Societyfor Effecting the Abolition of the Slave Trade as a cameo in black-and-white. It was widely reproduced and becamea popular fashion statement promoting justice, humanity and freedom.[2]

The question “Am I Not A Man?" was brought up again during the Dred Scott decision if the U.S. Supreme Court.[3]During the African-American Civil Rights Movement at the Memphis Sanitation Strike in 1968, “I AM A MAN!"signs were used to answer the same question.[4]

On trial for bringing his son back to Nebraska for burial, from a forced march to Oklahoma, in 1879 Ponca ChiefStanding Bear spoke to judge Dundy in his Omaha trial, “That hand is not the color of yours, but if I pierce it, I shallfeel pain. If you pierce your hand, you also feel pain. The blood that will flow from mine will be the same color asyours. I am a man. God made us both.” Standing Bear (and Native Americans) were granted habeas corpus meaningthat they had status in the court and were indeed human beings. “I Am a Man": Chief Standing Bear’s Journey forJustice, Joe Starita 2010

12.2 Modern use

“I Am a Man!" has been used as a title for books, plays and in film[5] to assert the rights of all people to be treatedwith dignity. “I Am a Man!" signs were used in Arabic language Ana Rajul during the Arab Spring.[6]

12.3 Other uses

• The Elephant Man declares, “I am not an elephant! I am not an animal! I am a human being! I ... am ... a ...man!"

12.4 See also

• Ain't I a Woman?

30

12.5. REFERENCES 31

Am I Not A Man And A Brother emblem used by abolitionists.

12.5 References[1] Andersen, Margaret L. (2008). Sociology With Infotrac: Understanding a Diverse Society. Thompson Learning. p. 61.

[2] Dabydeen, David (February 17, 2011). “The Black Figure in 18th-century Art”. BBC News. Retrieved December 18,2012.

[3] Am I Not a Man? by Mark L. Shurtleff

[4] Miami Herald

[5] Mining the Memphis Sound

[6] Dark Forebodings of the Arab Spring

32 CHAPTER 12. I AM A MAN!

“I AM A MAN!" diorama at the National Civil Rights Museum

Chapter 13

I-message

In interpersonal communication, an I-message or I-statement is an assertion about the feelings, beliefs, values etc.of the person speaking, generally expressed as a sentence beginning with the word “I”, and is contrasted with a "you-message" or "you-statement", which often begins with the word “you” and focuses on the person spoken to. ThomasGordon coined the term “I message” in the 1960s while doing play therapy with children. He added the concept tohis book for parents, P.E.T.: Parent Effectiveness Training (1970).[1][2]

I-messages are often used with the intent to be assertive without putting the listener on the defensive. They are alsoused to take ownership for one’s feelings rather than implying that they are caused by another person. An exampleof this would be to say: “I really am getting backed up on my work since I don't have the financial report yet,” ratherthan: “you didn't finish the financial report on time!" (The latter is an example of a “you-statement”).[3]

I-messages or I-statements can also be used in constructive criticism. For instance, one might say, “I had to read thatsection of your paper three times before I understood it,” rather than, “This section is worded in a really confusingway,” or “You need to learn how to word a paper more clearly.” The former comment leaves open the possibility thatthe fault lies with the giver of the criticism. According to the Conflict Resolution Network, I-statements are a disputeresolution conversation opener that can be used to state how one sees things and how one would like things to be,without using inflaming language[4]

13.1 I-message construction

While the underlying rationale and approach to I-messages is similar in various systems, there are both three-part andfour-part models for constructing I-messages. A three-part model is proposed by the University of Tennessee Family& Consumer Sciences for improving communication with children:

1. I feel... (Insert feeling word)

2. when... (tell what caused the feeling).

3. I would like... (tell what you want to happen instead).[5]

According to Hope E. Morrow, a common pitfall in I-statement construction is using phrases like “I feel that...” or “Ilike that...” which typically express an opinion or judgment. Morrow favors following “I feel...” with a feeling suchas “sad,” “angry,” etc.[6]

Gordon advises that to use an I-message successfully, there should be congruence between the words one is using andone’s affect, tone of voice, facial expression and body language. Gordon also describes a 3-part I-message, called a“confrontive” I-message, with the following parts:

• non-blameful description of the listener’s behavior

• the effect of that behavior on the speaker

• the speaker’s feelings about that effect

33

34 CHAPTER 13. I-MESSAGE

He describes the I-message as an appeal for help from the other person, and states that the other person is more likelyto respond positively when the message is presented in that way.[7]

13.1.1 Conflict resolution

If an “I” message contains “you-messages”, it can be problematic in conflict situations. For example: “I feel..., whenyou..., and I want you to...” This can put the receiver of the statement on the defensive. In a dispute, use of a phrasethat begins with “I want” may encourage the parties to engage in positional problem solving. This may make conflictsmore difficult to resolve. An “interest-based” approach to conflict resolution suggests using statements that reflectwhy the individual wants something.[8]

The goals of an “I” message in an interest-based approach:

• to avoid using “you” statements that will escalate the conflict

• to respond in a way that will de-escalate the conflict

• to identify feelings

• to identify behaviors that are causing the conflict

• to help individuals resolve the present conflict and/or prevent future conflicts.[8]

The Ohio Commission on Dispute Resolution and Conflict Management summarized this approach as follows: “Asender of a message can use a statement that begins with 'I' and expresses the sender’s feelings, identifies the unwantedbehavior, and indicates a willingness to resolve the dispute, without using 'you' statements or engaging in positionalproblem solving.[8]

The Commission proposed a four-part I-message:

1. “I feel like___ (taking responsibility for one’s own feelings)

2. “I don't like it when__ ” (stating the behavior that is a problem)

3. “because____” (what it is about the behavior or its consequences that one objects to)

4. “Can we work this out together?” (be open to working on the problem together).[8]

Marital stability and relationship analysis researcher John Gottman notes that although I-statements are less likelythan You-statements to be critical and to make the listener defensive, “you can also buck this general rule and comeup with 'I' statements like 'I think you are selfish' that are hardly gentle. So the point is not to start talking to yourspouse in some stilted psychobabble. Just keep in mind that if your words focus on how you're feeling rather than onaccusing your spouse, your discussion will be far more successful.”[9]

13.1.2 The Benefits of I-Statements, Self-Talk

I-statements have been found to offer a tremendous benefit to clients - patients. I-statements encourage growthand maturation. They are beneficial when employed with an individual struggling with self-defeating thoughts andmindset.The therapeutic model varies on its take on the use of I-statements. I-statements are designed to rid the myths fromthe reality of life. I-statements are further productive in challenging one’s innermost feelings. Dr. Asa Don Brown,author with the Canadian Counseling and Psychotherapy Association stated that “Self-talk reflects your innermostfeelings.” If they reflect your “inner most” feelings, then understanding those feelings are necessary in overcomingone’s negative perceptions and worldviews, according to Dr. Brown.I-statements are capable of influencing one’s path and design in life. According to Girlshealth.gov, “An I-statementis a sentence that begins with the word “I.” It helps the... (individual) take responsibility for their feelings instead ofsaying they are caused by the other person. This can help keep relationships open and honest between people whenthere is a conflict.” [10]

13.2. USE OF THE CONCEPT 35

I statements are important for clarifying one’s position, contribution, and desires around a situation, event, and/orlife perspective. According to family psychology movement, I-statements are necessary for establishing a healthyrelationship and an appropriate level of intimacy.I-statements help the individual avoid blame, turning blame into personal responsibility. Personal responsibility iskey to learning to use I-statements. Without personal responsibility, I-statements are null in their intention.

13.1.3 Shifting gears

Gordon states, “Although I-messages are more likely to influence others to change than You-messages, still it is afact that being confronted with the prospect of having to change is often disturbing to the changee.” A quick shiftby the sender of the I-message to an active listening posture can achieve several important functions in this situation,according to Gordon. He states that in Leader Effectiveness Training courses, this is called “shifting gears”, and statesthat the person might shift back to an I-message later in the conversation.[11]

13.2 Use of the concept

A book about mentoring states that communications specialists find that I-messages are a less threatening way toconfront someone one wants to influence, and suggests a three-part I-message: a neutral description of plannedbehaviour, consequences of the behaviour, and the emotions of the speaker about the situation.[12]

Amanual for health care workers calls I-messages an “important skill”, but emphasizes that use of an I-message doesnot guarantee that the other person will respond in a helpful way. It presents an I-message as a way that one can takeresponsibility for one’s own feelings and express them without blaming someone else.[13] Amanual for social workerspresents I-messages as a technique with the purpose of improving the effectiveness of communication.[14]

13.3 Research

A study in Hong Kong of children’s reactions to messages from their mothers found that children are most receptiveto I-messages that reveal distress, and most antagonistic towards critical you-messages.[15] A study with universitystudents as subjects did not find differences in emotional reactions to I-messages and you-messages for negativeemotions, but did find differences in reactions for positive emotions.[16]

A study of self-reported emotional reactions to I-statements and you-statements by adolescents found that accusatoryyou-statements evoked greater anger and a greater inclination for antagonistic response than assertive I-statements.[17]

13.4 See also

• Conflict resolution

• Face saving

• Flaming (Internet)

• Nonviolent Communication

13.5 Notes[1] Gordon 1995 p. xiii

[2] Gordon, Thomas. Origins of the Gordon Model. Gordon Training International. Retrieved on: 2012-01-17.

[3] “I” Statements not “You” Statements, International Online Training Program On Intractable Conflict, Conflict ResearchConsortium, University of Colorado, USA

36 CHAPTER 13. I-MESSAGE

[4] When to Use “I” Statements from 12 Skills: 4. Appropriate Assertiveness. Conflict Resolution Network. Retrieved 2007-11-25.

[5] Brandon, Denise [fcs.tennessee.edu/humandev/kidsmart/ks_c2a.pdf “I” Message Worksheet]. University of Tennessee,Extension Family and Consumer Sciences. Retrieved on: 2012-01-17.

[6] Constructing I-Statements, Hope E. Morrow, MA, MFT, CTS, 1998-2009.

[7] Gordon 1995 p. 112

[8] Rethinking “I” Statements, from Communication Skills, Skills and Concepts of Conflict Management. Ohio Commissionon Dispute Resolution & Conflict Management. Retrieved 2011-02-12.

[9] Gottman, John and Silver, Nan (1999). “Solve Your Solvable Problems”. The Seven Principles for Making Marriage Work.Three Rivers Press. pp. 164–165. ISBN 0-609-80579-7.

[10] Changing statements from “you” to “I” – Relationships

[11] Gordon, Thomas (2001). Leader Effectiveness Training (L.E.T.): The Foundation for Participative Management and Em-ployee Involvement. Perigee. pp. 113–115. ISBN 9780399527135.

[12] Shea 2001 p. 50

[13] Davis 1996 p. 100

[14] Sheafor 1996 p. 166

[15] Cheung 2003 pp. 3–14

[16] Bippus 2005 pp. 26–45

[17] Kubany, E.S. et. al., “Verbalized Anger and Accusatory “You” Messages as Cues for Anger and Antagonism amongAdolescents”, Adolescence, Vol. 27, No. 107, pp. 505-16, Fall 1992.

13.6 References• Gordon, Thomas; W. Sterling Edwards (1995). Making the patient your partner: Communication Skills for

Doctors and Other Caregivers (Edition of 1997). Greenwood Publishing Group. ISBN 9780865692558.

• Cheung, Siu-Kau; Sylvia Y.C. Kwok (2003). “How do Hong Kong children react to maternal I-messages andinductive reasoning?". The Hong Kong Journal of Social Work 37 (1). Retrieved 2008-08-23.

• Bippus, Amy M.; Stacy L. Young (2005). “Owning Your Emotions: Reactions to Expressions of Self- versusOther-Attributed Positive and Negative Emotions”. Journal of Applied Communication Research 33 (1): 26.doi:10.1080/0090988042000318503. Retrieved 2008-08-23.

• Shea, Gordon (2001). How to Develop Successful Mentor Behaviors. Thomas Crisp Learning. ISBN 1-56052-642-4.

• Davis, Carol M. (2006). Patient Practitioner Interaction: An Experiential Manual for Developing the Art ofHealth Care (4th ed.). SLACK Incorporated. ISBN 9781556427206.

• Sheafor, Bradford W.; Charles R. Horejsi; Gloria A. Horejsi (1996). Techniques and Guidelines for SocialWork Practice. Allyn and Bacon (Original from the University of Michigan). ISBN 9780205191772.

Chapter 14

Illocutionary act

Illocutionary act is a term in linguistics introduced by the philosopher John L. Austin in his investigation of thevarious aspects of speech acts. In Austin’s framework, locution is what was said, illocution is what was meant, andperlocution is what happened as a result. For example, when somebody says “Is there any salt?" at the dinner table,the illocutionary act (the meaning conveyed) is effectively “please give me some salt” even though the locutionary act(the literal sentence) was to ask a question about the presence of salt. The perlocutionary act (the actual effect), wasto cause somebody to hand over the salt.The notion of an illocutionary act is closely connected with Austin’s doctrine of the so-called 'performative' and'constative utterances’: an utterance is “performative” if, and only if it is issued in the course of the “doing of anaction” (1975, 5), by which, again, Austin means the performance of an illocutionary act (Austin 1975, 6 n2, 133).According to Austin’s original exposition in How to Do Things With Words, an illocutionary act is an act (1) for theperformance of which I must make it clear to some other person that the act is performed (Austin speaks of the'securing of uptake'), and (2) the performance of which involves the production of what Austin calls 'conventionalconsequences’ as, e.g., rights, commitments, or obligations (Austin 1975, 116f., 121, 139). Thus, for example, inorder to make a promise I must make clear to my audience that the act I am performing is the making of a promise,and in the performance of the act I will be undertaking an obligation to do the promised thing: so promising is anillocutionary act in the present sense. Since Austin’s death, the term has been defined differently by various authors.One way to think about the difference between an illocutionary act (e.g., a declaration, command, or a promise),and a perlocutionary act (e.g., an insult or a persuasion attempt) is to note how in the former case, by uttering theobject—for example, “I hereby declare,” or “I command,” or “I hereby promise you”—the act has taken place. Thatis to say, in each case a declaration, command, or promise has necessarily taken place in virtue of the utterance itself,whether the hearer believes in the declaration, command, or promise or not. On the other hand, with a perlocutionaryact, the object of the utterance has not taken place unless the hearer deems it so—for example, if one utters, “I herebyinsult you,” or “I hereby persuade you,” one would not assume an insult has necessarily occurred, nor persuasionhas necessarily taken place, unless the hearer were suitably offended or persuaded by the utterance. In a sense aperlocutionary act demands the cooperation of the hearer, whereas an illocutionary act does not.

14.1 Approaches to defining “illocutionary act”

Many define the term “illocutionary act” with reference to examples, saying for example that any speech act (likestating, asking, commanding, promising, and so on) is an illocutionary act. This approach has generally failed togive any useful hints about what traits and elements make up an illocutionary act; that is, what defines such an act.It is also often emphasised that Austin introduced the illocutionary act by means of a contrast with other kinds ofacts or aspects of acting: the illocutionary act, he says, is an act performed in saying something, as contrasted witha locutionary act, the act of saying something, and also contrasted with a perlocutionary act, an act performed bysaying something. Austin, however, eventually abandoned the “in saying” / “by saying” test (1975, 123).According to the conception adopted by Bach and Harnish in 'Linguistic Communication and Speech Acts’ (1979),an illocutionary act is an attempt to communicate, which they analyse as the expression of an attitude. Anotherconception of the illocutionary act goes back to Schiffer’s book 'Meaning' (1972, 103), in which the illocutionary actis represented as just the act of meaning something.

37

38 CHAPTER 14. ILLOCUTIONARY ACT

According to a widespread opinion, an adequate and useful account of “illocutionary acts” has been provided byJohn Searle (e.g., 1969, 1975, 1979). In recent years, however, it has been doubted whether Searle’s account iswell-founded. A wide ranging critique is in FC Doerge 2006. Collections of articles examining Searle’s account are:Burkhardt 1990 and Lepore / van Gulick 1991.

14.2 Classes of illocutionary acts

Searle (1975) set up the following classification of illocutionary speech acts:

• assertives = speech acts that commit a speaker to the truth of the expressed proposition

• directives = speech acts that are to cause the hearer to take a particular action, e.g. requests, commands andadvice

• commissives = speech acts that commit a speaker to some future action, e.g. promises and oaths

• expressives = speech acts that express on the speaker’s attitudes and emotions towards the proposition, e.g.congratulations, excuses and thanks

• declarations = speech acts that change the reality in accord with the proposition of the declaration, e.g. bap-tisms, pronouncing someone guilty or pronouncing someone husband and wife

14.3 Illocutionary force

Several speech act theorists, including Austin himself, make use of the notion of an illocutionary force. In Austin’soriginal account, the notion remains rather unclear. Some followers of Austin, such as David Holdcroft, view il-locutionary force as the property of an utterance to be made with the intention to perform a certain illocutionaryact—rather than as the successful performance of the act (which is supposed to further require the appropriatenessof certain circumstances). According to this conception, the utterance of “I bet you five pounds that it will rain”may well have an illocutionary force even if the addressee doesn't hear it. However, Bach and Harnish assume il-locutionary force if, and only if this or that illocutionary act is actually (successfully) performed. According to thisconception, the addressee must have heard and understood that the speaker intends to make a bet with them in orderfor the utterance to have 'illocutionary force'.If we adopt the notion of illocutionary force as an aspect of meaning, then it appears that the (intended) 'force' ofcertain sentences, or utterances, is not quite obvious. If someone says, “It sure is cold in here”, there are severaldifferent illocutionary acts that might be aimed at by the utterance. The utterer might intend to describe the room,in which case the illocutionary force would be that of 'describing'. But she might also intend to criticise someonewho should have kept the room warm. Or it might be meant as a request to someone to close the window. Theseforces may be interrelated: it may be by way of stating that the temperature is too cold that one criticises someoneelse. Such a performance of an illocutionary act by means of the performance of another is referred to as an indirectspeech act.

14.4 Illocutionary force indicating devices (IFIDs)

Searle andVanderveken (1985) often speak about what they call 'illocutionary force indicating devices’ (IFIDs). Theseare supposed to be elements, or aspects of linguistic devices which indicate either (dependent on which conceptions of“illocutionary force” and “illocutionary act” are adopted) that the utterance is made with a certain illocutionary force,or else that it constitutes the performance of a certain illocutionary act. In English, for example, the interrogativemood is supposed to indicate that the utterance is (intended as) a question; the directive mood indicates that theutterance is (intended as) a directive illocutionary act (an order, a request, etc.); the words “I promise” are supposedto indicate that the utterance is (intended as) a promise. Possible IFIDs in English include: word order, stress,intonation contour, punctuation, the mood of the verb, and performative verbs.

14.5. ILLOCUTIONARY NEGATIONS 39

14.5 Illocutionary negations

Another notion Searle and Vanderveken use is that of an 'illocutionary negation'. The difference of such an 'illo-cutionary negation' to a 'propositional negation' can be explained by reference to the difference between “I do notpromise to come” and “I promise not to come”. The first is an illocutionary negation—the 'not' negates the promise.The second is a propositional negation. In the view of Searle and Vanderveken, illocutionary negations change thetype of illocutionary act.

14.6 See also• Direction of fit

• J. L. Austin

• John Searle

• Linguistics

• Performative utterance

• Perlocutionary act

• Pragmatics

• Semantics

• Speech act

14.7 References• Alston, William P.. Illocutionary Acts and Sentence Meaning. Ithaca: Cornell University Press. 2000

• Austin, John L.. How To Do Things with Words. Oxford: Oxford University Press. 1975[1962] ISBN 0-19-281205-X

• Burkhardt, Armin (ed.). Speech Acts, Meaning and Intentions: Critical Approaches to the Philosophy of JohnR. Searle. Berlin / New York 1990 ISBN 0-89925-357-1

• Doerge, Friedrich Christoph. Illocutionary Acts – Austin’s Account and What Searle Made Out of It. Tuebingen2006.

• Lepore, Ernest / van Gulick, Robert (eds). John Searle and his Critics. Oxford: Basil Blackwell 1991. ISBN0-631-15636-4

• Searle, John R. Speech Acts. Cambridge University Press. 1969 ISBN 0-521-07184-4

• Searle, John R. “A Taxonomy of Illocutionary Acts”, in: Günderson, K. (ed.), Language, Mind, and Knowledge,Minneapolis, vol. 7. 1975

• Searle, John R. Expression and Meaning. Cambridge University Press. 1979 ISBN 0-521-22901-4

• Searle, John R. and Daniel Vanderveken. Foundations of Illocutionary Logic. Cambridge University Press.1985. ISBN 0-521-26324-7

14.8 Further reading• Discussion of illocutionary acts in sec. 1 of Stanford Encycolopedia of Philosophy, "Assertion”.

• Kuhi, D & Almasi, K. (2013). "The Study of Impact of Learner’s Personal Constructs in Illocutionary ActsInduction,” International Journal of Enhanced Research in Educational Development.Er Publications. Vol. 1Issue 6, Sept–Oct 2013, pp 4–37.

Chapter 15

Leave and Earnings Statement

Example of an LES

A Leave and Earnings Statement, generally referred to as an LES, is a document given on a monthly basis tomembers of the United States military which documents their pay and leave status on a monthly basis.Employees in the civil service receive a similar document each pay period, called a Civilian Leave and EarningsStatement, a link to which the Defense Finance and Accounting Service emails two days prior to the scheduled pay

40

15.1. DELIVERY 41

day.

15.1 Delivery

While paper LES documents were originally mailed or handed out in person, it is now almost always retrieved bythe member from an online system called MyPay from the Defense Finance and Accounting Service. For U.S. CoastGuard personnel LES can be retrieved by the member from an online system called Direct Access that is controlledby U.S. Coast Guard Personnel Services Command.

15.2 Sections

There are multiple sections to an LES.

15.2.1 Pay

The first section lists the monetary entitlements that month. For all members it consists of Basic Pay, and for manymembers it also includes Basic Allowance for Housing and Basic Allowance for Subsistence. Other kinds of payincluding Cost of Living Allowance, Overseas Housing Allowance, incentive pay, bonus pay, or hazardous duty paymay be included.Per diem and TDY money are usually not included in the LES.

42 CHAPTER 15. LEAVE AND EARNINGS STATEMENT

15.2.2 Deductions

The second section describes the money that has been deducted. Common deductions include:

• Federal and state taxes

• Social Security

• Servicemembers’ Group Life Insurance (SGLI) (if the member is a participant)

• Montgomery GI Bill deduction for the first year (if the member is a participant)

• The service retirement center

• Thrift Savings Plan (TSP) (if the member is a participant)

• Mid month pay

• Most members receive their money two times per month, on the 15th of the month (known as mid monthpay) and on the 1st of the following month (known as end of month pay). The mid month pay is alsolisted in the deductions section.

15.2.3 Allotments

Common allotments include:

• Bank account

• Savings• Checking• Loan payment (car, etc.)

• TRICARE premiums

• Combined Federal Campaign (CFC) contributions

• Service relief societies (i.e.: Navy-Marine Corps Relief Society, Air Force Aid Society, etc.)

15.2.4 End of Month Pay

After all the pay is added, all deductions & allotments are subtracted, and loose ends from previous months areaccounted for, what is left is the end of month pay, which is indicated on the far right.

15.2.5 Leave

Main article: Leave (U.S. military)

Military members accumulate 2.5 days of leave per month or 30 days per year. The maximum amount of leave thatcan accrue is 60 days (this can be more if a member was deployed within the year). The fiscal year ends on September30, unless Congress decides to make a change.

• BF Bal - Brought forward leave balance. This is the unused leave rolled over from the last fiscal year.

• Ernd - The cumulative amount of leave earned in the current fiscal year, or current term of service if the servicemember re-enlisted or extended since the start of the fiscal year.

• Used - The cumulative amount of leave used during the current fiscal year, or term of enlistment.

• Cr Bal - The current leave balance as of the end of the period covered by the LES. It is calculated by BF BAL+ ERNED - USED.

15.3. SEE ALSO 43

• ETS Bal - The projected leave available through the current Expiration Term of Service. This helps makeappropriate plans if the member does not plan to re-enlist.

• LV LOST - The amount of leave that has been lost, usually because of having too high a balance at the end ofthe fiscal year.

• LV Paid - The number of days of leave for which you have been paid to date.

• Use/Lose - The projected number of days of leave that will be lost if not taken on in the current fiscal year ona monthly basis. The number of days of leave in this block will decrease with any leave usage.

15.2.6 TSP

Main article: Thrift Savings Plan

TSP information is listed near the bottom, along with the YTD (calendar year) total tax withholdings.

15.3 See also• United States Military Pay

15.4 External links

Chapter 16

Locutionary act

In linguistics and the philosophy of mind, a locutionary act is the performance of an utterance, and hence of a speechact. The term equally refers to the surface meaning of an utterance because, according to J. L. Austin's posthumous“How To Do Things With Words”, a speech act should be analysed as a locutionary act (i.e. the actual utteranceand its ostensible meaning, comprising phonetic, phatic and rhetic acts corresponding to the verbal, syntactic andsemantic aspects of any meaningful utterance), as well as an illocutionary act (the semantic 'illocutionary force' ofthe utterance, thus its real, intended meaning), and in certain cases a further perlocutionary act (i.e. its actual effect,whether intended or not).

16.1 Example

For example, my saying to you “Don't go into the water” (a locutionary act with distinct phonetic, syntactic andsemantic features) counts as warning you not to go into the water (an illocutionary act), and if you heed my warningI have thereby succeeded in persuading you not to go into the water (a perlocutionary act). This taxonomy of speechacts was inherited by John R. Searle, Austin’s pupil at Oxford and subsequently an influential exponent of speech acttheory.

16.2 References

44

Chapter 17

Loosely associated statements

A loosely associated statement is a type of simple non-inferential passage wherein statements about a general subjectare juxtaposed but make no inferential claim.[1] As a rhetorical device, loosely associated statements may be intendedby the speaker to infer a claim or conclusion, but because they lack a coherent logical structure any such interpretationis subjective as loosely associated statements prove nothing and attempt no obvious conclusion.[2] Loosely associatedstatements can be said to serve no obvious purpose, such as illustration or explanation.[3]

Included statements can be premises, conclusions or both, and both true or false, but missing from the passage is aclaim that any one statement supports another.

17.1 Examples

In A concise introduction to logic, Hurley demonstrates the concept with a quote by Lao-Tzu:

Not to honor men of worth will keep the people from contention; not to value goods which are hardto come by will keep them from theft; not to display what is desirable will keep them from being unset-tled of mind.— Lao-Tzu

While each clause in the quote may seem related to the others, each provides no reason to believe another.

17.2 See also

17.3 References[1] Hurley, Patrick J. (2008). A Concise Introduction to Logic 10th ed. Thompson Wadsworth. p. 17. ISBN 0-495-50383-5.

[2] “The logic of arguments”. Retrieved April 28, 2012.

[3] “NONargument - Loosely associated statements”. Retrieved April 28, 2012.

45

Chapter 18

Maxim (philosophy)

A maxim is a ground rule or subjective principle of action; in that sense, a maxim is a thought that can motivateindividuals.

18.1 Deontological ethics

In deontological ethics, mainly in Kantian ethics, maxims are understood as subjective principles of action. A maximis thought to be part of an agent’s thought process for every rational action, indicating in its standard form: (1) theaction, or type of action; (2) the conditions under which it is to be done; and (3) the end or purpose to be achieved bythe action, or the motive. The maxim of an action is often referred to as the agent’s intention. In Kantian ethics, thecategorical imperative provides a test on maxims for determining whether the actions they refer to are right, wrong,or permissible.The categorical imperative is stated canonically as:"Act only according to that maxim whereby you can, at the sametime, will that it should become a universal law.”[2]

in his Critique of Practical Reason, Kant provided the following example of a maxim and of how to apply the test ofthe categorical imperative:

“I have, for example, made it my maxim to increase my wealth by any safe means. Now I have adeposit in my hands, the owner of which has died and left no record of it. . . . I therefore apply themaxim to the present case and ask whether it could indeed take the form of a law, and consequentlywhether I could through my maxim at the same time give such a law as this: that everyone may deny adeposit which no one can prove has been made. I at once become aware that such a principle, as a law,would annihilate itself since it would bring it about that there would be no deposits at all.”[3]

Also, an action is said to have “moral worth” if the maxim upon which the agent acts cites the purpose of conformingto a moral requirement. That is, a person’s action has moral worth when she does her duty purely for the sake ofduty, or does the right thing for the right reason. Kant himself believed that it is impossible to know whether anyone’saction has ever had moral worth. It might appear to someone that he has acted entirely “from duty,” but this couldalways be an illusion of self-interest: of wanting to see oneself in the best, most noble light. This indicates that agentsare not always the best judges of their own maxims or motives.

18.2 Personal knowledge

Michael Polanyi in his account of tacit knowledge stressed the importance of the maxim in focusing both explicitand implicit modes of understanding. “Maxims are rules, the correct application of which is part of the art theygovern....Maxims can only function within a framework of personal (i.e., experiential) knowledge”.[4]

46

18.3. SEE ALSO 47

18.3 See also• Aphorism

• Categorical imperative

• Immanuel Kant

18.4 References[1] Oxford Dictionary of Philosophy, Maxim (Oxford University Press 2008) p. 226

[2] Kant, Immanuel; translated by JamesW. Ellington [1785] (1993). Grounding for theMetaphysics ofMorals 3rd ed. Hackett.p. 30. ISBN 0-87220-166-X. Cite uses deprecated parameter |coauthors= (help)

[3] Kant, Immanuel; translated by Mary Gregor [1788] (1997). Critique of Practical Reason. Cambridge University Press. pp.25/5:27. ISBN 0-521-59051-5. Cite uses deprecated parameter |coauthors= (help)

[4] Quoted in Guy Claxton, Live and learn (1992) p. 116

18.5 External links• Locksley Hall - Alfred Lord Tennyson

• Online works of Immanuel Kant at Gutenberg

• Maxims of Action

Chapter 19

Meaningless statement

This article is about meaningless rhetoric. For the literary use, see Nonsense verse.

A meaningless statement posits nothing of substance with which one could agree or disagree. In the context oflogical arguments, the inclusion of ameaningless statement in the premises will undermine the validity of the argumentsince that premise can neither be true nor false.There are many classes of meaningless statement:

• A statement may be considered meaningless if it asserts that two categories are disjoint without proposing acriterion to distinguish between them. For example, the claim, “I wouldn't know how pornography differs fromerotica" is a distinction without a difference.

• A statement may be meaningless if its terms are undefined, or if it contains unbound variables. For instance,the sentence “All X have Y" is meaningless unless the terms X and Y are defined (or bound).

• A grammatically correct sentence may be meaningless if it ascribes properties to particulars which admit ofno such properties. For example, the famous sentence "Colorless green ideas sleep furiously" cannot be takenliterally.

• An ungrammatical sentence admits of no meaning. For instance, the string of words “deities Olympus Greekreside The. upon” offers no information regarding the Greek deities, or their whereabouts.

• Lewis Carrol's poem Jabberwocky is a famous example of the nonsensical as literature.

19.1 See also• Non sequitur (logic)

• Nonsense

48

Chapter 20

Mission statement

“Statement of purpose” redirects here. For use in the university and college admissions, see admissions essay.

A mission statement is a statement which is used as a way of communicating the purpose of the organization.Although most of the time it will remain the same for a long period of time, it is not uncommon for organizations toupdate their mission statement and generally happens when an organization evolves. Mission statements are normallyshort and simple statements which outline what the organization’s purpose is and are related to the specific sector anorganization operates in.Properly crafted mission statements (1) serve as filters to separate what is important from what is not, (2) clearly statewhich markets will be served and how, and (3) communicate a sense of intended direction to the entire organization.A mission is different from a vision in that the former is the cause and the latter is the effect; a mission is something tobe accomplished whereas a vision is something to be pursued for that accomplishment. Also called company mission,corporate mission, or corporate purpose.[1]

The mission statement should guide the actions of the organization, spell out its overall goal, provide a path, and guidedecision-making. It provides “the framework or context within which the company’s strategies are formulated.” It islike a goal for what the company wants to do for the world.[2]

According to Dr. Christopher Bart,[3] the commercial mission statement consists of three essential components:

1. Key market: Who is your target client or customer (generalize if needed)?

2. Contribution: What product or service do you provide to that client?

3. Distinction: What makes your product or service unique, so that the client would choose you?

A personalmission statement is developed inmuch the sameway that an organizationalmission statement is created.A personal mission statement is a brief description of what an individual wants to focus on, wants to accomplish andwants to become. It is a way to focus energy, actions, behaviors and decisions towards the things that are mostimportant to the individual.

20.1 Purpose of a mission statement

The sole purpose of a mission statement is to serve as your company’s goal/agenda, it outlines clearly what the goalof the company is.[4] Some generic examples of mission statements would be, “To provide the best service possiblewithin the banking sector for our customers.” or “To provide the best experience for all of our customers.” The reasonwhy businesses make use of mission statements is to make it clear what they look to achieve as an organisation, notonly to themselves and their employees but to the customers and other people who are a part of the business, such asshareholders. As a company evolves, so will their mission statement, this is to make sure that the company remainson track and to ensure that the mission statement does not lose its touch and become boring or stale.An article which can be found here explains the purpose of a mission statement as the following:

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50 CHAPTER 20. MISSION STATEMENT

“The mission statement reflects every facet of your business: the range and nature of the products you offer, pricing,quality, service, marketplace position, growth potential, use of technology, and your relationships with your cus-tomers, employees, suppliers, competitors and the community.”[5]

It is important that a mission statement is not confused with a vision statement. As discussed earlier, the main purposeof a mission statement is to get across the ambitions of an organisation in a short and simple fashion, it is not necessaryto go into detail for the mission statement which is evident in examples given. The reason why it is important thata mission statement and vision statement are not confused is because they both serve different purposes. Visionstatements tend to be more related to strategic planning and lean more towards discussing where a company aims tobe in the future.

20.2 Mission statement vs vision statement

The definition of a vision statement according to BusinessDictionary is “An aspirational description of what an or-ganisation would like to achieve or accomplish in the mid-term or long-term future. It is intended to serve as a clearguide for choosing current and future courses of action.”[6]

It is not hard to see why a lot of people confuse a mission statement and a vision statement, although both statementsserve a different purpose for a company.A mission statement is all about how an organisation will get to where they want to be and makes the purposes andobjectives clear, whereas a vision statement is outlining where the organisation wants to be in the future. Missionstatements are more concerned about the current times and tend to answer questions about what the business does orwhat makes them stand out compared to the competition, whilst vision statements are solely focused on where theorganisation sees themselves in the future and where they aim to be. Both statements may be adapted later into theorganisation’s life, however it is important to keep the core of the statement there such as core values, customer needsand vision.[7]

Although it may not seem very important to know the difference between the two types of statements, it is veryimportant to businesses. This is because it is common for businesses to base their strategic plans around clear visionand mission statements. Both statements play a big factor in the strategic planning of a business. A study carried outby Bain & Company showed that companies which had clearly outlined vision and mission statements outperformedother businesses that did not have clear vision and mission statements.[8]

20.3 Advantages of a mission statement

Provides direction: Mission statements are a great way to direct a business into the right path, it plays a part inhelping the business make better decisions which can be beneficial to them. Without the mission statement providingdirection, businesses may struggle when it comes tomaking decisions and planning for the future, this is why providingdirection could be considered one of the most advantageous points of a mission statement.Clear purpose: Having a clear purpose can remove any potential ambiguities that can surround the existence of abusiness. People who are interested in the progression of the business, such as stakeholders, will want to know that thebusiness is making the right choices and progressing more towards achieving their goals, which will help to removeany doubt the stakeholders may have in the business.[9]

The benefit of having a simple and clear mission statement is that it can be beneficial in many different ways. Amissionstatement can help to play as a motivational tool within an organisation, it can allow employees to all work towardsone common goal that benefits both the organisation and themselves. This can help with factors such as employeesatisfaction and productivity. It is important that employees feel a sense of purpose, by giving them this sense ofpurpose it will allow them to focus more on their daily tasks and help them to realise the goals of the organisationand their role.[10][11]

20.4 Disadvantages of a mission statement

Although it is mostly beneficial for a business to craft a good mission statement, there are some situations where amission statement can be considered pointless or not useful to a business.

20.5. DESIGNING A MISSION STATEMENT 51

Unrealistic: In most cases, mission statements turn out to be unrealistic and far too optimistic.[9] An unrealisticmission statement can also affect the performance and morale of the employees within the workplace. This is becausean unrealistic mission statement would reduce the likelihood of employees being able to meet this standard whichcould demotivate employees in the long term. Unrealistic mission statements also serve no purpose and can beconsidered a waste of management’s time. Another issue which could arise from an unrealistic mission statement isthat poor decisions could be made in an attempt to achieve this goal which has the potential to harm the business andbe seen as a waste of both time and resources.Waste of time and resources: Mission statements require planning, this takes time and effort for those who areresponsible for creating the mission statement. If the mission statement is not achieved, then the process of creatingthe mission statement could be seen as a waste of time for all of the people involved. A lot of thought and time isspent in designing a good mission statement, and to have all of that time wasted is not what businesses can affordto be doing. The wasted time could have been spent on much more important tasks within the organisation such asdecision-making for the business.

20.5 Designing a mission statement

According to Forbes, the following questions must be answered in the mission statement. "What we do?" Themission statement should clearly outline the main purpose of the organisation, and what they do. "How do we doit?" It should alsomention how you plan on achieving themission statement. "Whomdowe do it for?"The audienceof the mission statement should be clearly stated within the mission statement. "What value are we bringing?" Thebenefits and values of the mission statement should be clearly outlined.[12] When designing your mission statement,it should be very clear to the audience what the purpose of it is. It is ideal for a business to be able to communicatetheir mission, goals and objectives to the reader without including any unnecessary information through the missionstatement.[13] “Your mission is the soul of your brand.”.[14]

FEMA’s Mission Statement Poster

Richard Branson has commented on ways of crafting a good mission statement; he explains the importance of havinga mission statement that is clear and straight to the point and does not contain unnecessary baffling. He went on toanalyse a mission statement, using Yahoo's mission statement at the time (2013) as an example, in his evaluationof the mission statement he seemed to suggest that while the statement sounded interesting most people will not beable to understand the message it is putting across, in other words the message of the mission statement potentially

52 CHAPTER 20. MISSION STATEMENT

meant nothing to the audience.[15] This further backs up the idea that a good mission statement is one that is clear andanswers the right questions in a simple manner, and does not over complicate things. An example of a good missionstatement would be Google’s, which is “to organise the world’s information and make it universally accessible anduseful.”[16][17]

20.6 See also• Strategic planning

20.7 References[1] “What is a mission statement? definition and meaning”. BusinessDictionary.com. Retrieved 27 October 2015.

[2] Hill, Charles; Jones, Gareth (2008). Strategic Management: An Integrated Approach (8th Revised edition). Mason, OH:South-Western Educational Publishing. p. 11. ISBN 978-0-618-89469-7.

[3] Bart, Christopher (2015). “Sex, Lies and Mission Statements”. papers.ssrn.com. Retrieved June 16, 2015.

[4] “What is a Mission Statement?". Business News Daily. Retrieved 2015-11-01.

[5] “Mission Statement”. Entrepreneur. Retrieved 2015-11-01.

[6] “What is a vision statement? definition and meaning”. BusinessDictionary.com. Retrieved 2015-11-01.

[7] “Mission Statement vs Vision Statement - Difference and Comparison | Diffen”. www.diffen.com. Retrieved 2015-11-01.

[8] “Vision and Mission - What’s the difference and why does it matter?". Psychology Today. Retrieved 2015-11-01.

[9] “Advantages and Disadvantages of a Mission Statement”. Investment & Small Business Accountants. Retrieved 2015-11-01.

[10] “Benefits of Vision and Mission Statements | Clearlogic Consulting Professionals”. clearlogic.ca. Retrieved 2015-11-01.

[11] “How organizations benefit from having a clearly defined mission - Smart Business Magazine”. Smart Business Magazine.Retrieved 2015-11-02.

[12] “Answer 4 Questions to Get a Great Mission Statement”. Forbes. Retrieved 2015-11-02.

[13] “How to Write Your Mission Statement”. Entrepreneur. Retrieved 2015-11-02.

[14] “How to Write a Powerful Mission Statement”. www.kinesisinc.com. Retrieved 2015-11-02.

[15] “Richard Branson on Crafting Your Mission Statement”. Entrepreneur. Retrieved 2015-11-02.

[16] “Google’s Vision Statement & Mission Statement - Panmore Institute”. Panmore Institute. Retrieved 2015-11-02.

[17] “Company – Google”. www.google.co.uk. Retrieved 2015-11-02.

20.8 External links• NORC Blueprint: A Guide to Community Action’s Developing a Mission Statement Guide

• Personal Mission Statement Guidelines

Chapter 21

Normative statement

In economics and philosophy, a normative statement expresses a value judgment about whether a situation is sub-jectively desirable or undesirable. It looks at the world as it “should” be.[1] “The world would be a better place if themoon were made of green cheese" is a normative statement because it expresses a judgment about what ought to be.Notice that there is no way of testing the veracity of the statement; even if you disagree with it, you have no sureway of proving to someone who believes the statement that he or she is wrong by mere appeal to facts. Normativestatements are characterised by the modal verbs “should”, “would” or “could”. They form the basis of normativeeconomics, and are the opposite of positive statements. For further information see normative science.

21.1 References[1] “Business Dictionary”. http://www.businessdictionary.com/definition/normative-economics.html''.

• Lipsey, Richard G. (1975). An introduction to positive economics (fourth ed.). Weidenfeld & Nicolson. pp.4–6. ISBN 0-297-76899-9.

21.2 External links• Economae: An Encyclopedia

53

Chapter 22

Objection (argument)

“Refute” redirects here. For the Transformer, see Refute (Transformers).In informal logic an objection (also called expostulation or refutation), is a reason arguing against a premise,

Refutation on Graham's Hierarchy of Disagreement

lemma, or main contention. An objection to an objection is known as a rebuttal.

22.1 See also• Argument map• Argumentation theory• Inference objection

54

Chapter 23

Opening statement

A legalman making an opening statement for the prosecution to a jury during a mock trial.

An opening statement is generally the first occasion that the trier of fact (jury or judge) has to hear from a lawyer ina trial, aside possibly from questioning during voir dire. The opening statement is generally constructed to serve as a“roadmap” for the fact-finder. This is especially essential, in many jury trials, since jurors (at least theoretically) knownothing at all about the case before the trial, (or if they do, they are strictly instructed by the judge to put preconceivednotions aside). Though such statements may be dramatic and vivid, they must be limited to the evidence reasonablyexpected to be presented during the trial. Attorneys generally conclude opening statements with a reminder that atthe conclusion of evidence, the attorney will return to ask the fact-finder to find in his or her client’s favor.Opening statements are, in theory, not allowed to be argumentative, or suggest the inferences that fact-finders shoulddraw from the evidence they will hear. In actual practice, the line between statement and argument is often unclear andmany attorneys will infuse at least a little argumentation into their opening (often prefacing borderline arguments withsome variation on the phrase, “As we will show you...”). Objections, though permissible during opening statements,are very unusual, and by professional courtesy are usually reserved only for egregious conduct.Generally, the prosecution in a criminal case and plaintiff in a civil case is the first to offer an opening statement, and

55

56 CHAPTER 23. OPENING STATEMENT

defendants go second. Defendants are also allowed the option of delaying their opening statement until after the closeof the prosecution or plaintiff’s case. Few take this option, however, so as not to allow the other party’s argument tostand uncontradicted for so long.The techniques of opening statements are taught in courses on trial advocacy.[1] The opening statement is integratedwith the overall case strategy through either a theme and theory or, with more advanced strategies, a line of effort.[2]

23.1 See also• Closing argument

23.2 References[1] Lubet, Steven; Modern Trial Advocacy, NITA, New York, NY 2004 pp. 415 et. seq., ISBN 1556818866

[2] Dreier, A.S.; Strategy, Planning & Litigating to Win; Conatus, Boston, MA, 2012 pp. 46–73, ISBN 0615676952

23.3 External links

Differences Between Opening Statements and Closing Arguments

Chapter 24

Positive statement

In economics and philosophy, a positive statement concerns what “is”, “was”, or “will be”, and contains no indicationof approval or disapproval (what should be). Positive statement is based on empirical evidence. For examples, “Anincrease in taxation will result in less consumption” and “A fall in supply of petrol will lead to an increase in its price”.However, positive statement can be factually incorrect: “The moon is made of green cheese” is empirically false, butis still a positive statement, as it is a statement about what is, not what should be.[1]

24.1 Positive statements and normative statements

Positive statements are distinct from normative statements. Positive statements are based on empirical evidence, canbe tested, and involve no value judgements. Positive statements refer towhat is and contain no indication of approvalor disapproval. When values or opinions come into the analysis, then it is in the realm of normative economics. Anormative statement expresses a subjective judgment about whether a situation is desirable or undesirable, whichcan carry value judgements. These refer to what ought to be.

24.2 Use of positive statement

Positive statements are widely used to describe something measurable, like the rate of inflation in an economy. Andit is mainly used in explanations of theories and concepts. Using positive statement does not mean you can't haveyour own opinions on issues. However, when you are writing academic essays it’s better to use positive statement,since it can be verified by evidence .

24.3 How to transform positive statements to normative statements

24.4 See also

• Falsifiability

• Positivism

• Normative statement

24.5 References

[1] “Positive and normative economics”. http://www.soas.ac.uk/cedep-demos/000_P570_IEEP_K3736-Demo/unit1/page_16.htm. External link in |website= (help);

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58 CHAPTER 24. POSITIVE STATEMENT

• Lipsey, Richard G. (1975). An introduction to positive economics (fourth ed.). Weidenfeld & Nicolson. pp.4–6. ISBN 0-297-76899-9.

• http://www.unc.edu/depts/econ/byrns_web/Economicae/Figures/Positive-Normative.htmEconomae: AnEn-cyclopedia

Chapter 25

Precept

For the community in the United States, see Precept, Nebraska.

A precept (from the Latin: præcipere, to teach) is a commandment, instruction, or order intended as an authoritativerule of action.

25.1 Religion

In religion, precepts are usually commands respecting moral conduct.

25.1.1 Christianity

The term is encountered frequently in the Jewish and Christian Scriptures:

Thou hast commanded thy precepts to be kept diligently. O that my ways may be steadfast in keepingthy statutes!— Psalm 119(118):4–5, RSV

The usage of precepts in the Revised Standard Version of the Bible corresponds with that of the Hebrew Bible. TheSeptuagint (Samuel Rengster edition) has Greek entolas, which, too, may be rendered with precepts.Roman Catholic Canon law, which is based on Roman Law, makes a distinction between precept and law in Canon49:

A singular precept is a decree which directly and legitimately enjoins a specific person or persons todo or omit something, especially in order to urge the observance of law.

In Catholicism, the "Commandments of the Church" may also be called “Precepts of the Church”.

25.1.2 Buddhism

Main article: Buddhist precepts

In Buddhism, the fundamental code of ethics is known as the Five Precepts (Pañcaśīla in Sanskrit, or Pañcasīla inPāli), practiced by laypeople, either for a given period of time or for a lifetime. There are other levels of precepts,varying amongst traditions. In Theravadan tradition there are Eight Precepts, Ten Precepts, and the Patimokkha.Eight Precepts are a more rigorous practice for laypeople. Ten Precepts are the training rules for samaneras andsamaneris, novice monks and nuns, respectively. The Patimokkha is the basic Theravada code of monastic discipline,consisting of 227 rules for monks, (bhikkhus) and 311 rules for nuns (bhikkhunis).[1]

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60 CHAPTER 25. PRECEPT

25.2 Secular law

In secular law, a precept is a command in writing; a species of writ issued from a court or other legal authority. It isnow chiefly used of an order demanding payment. The Latin form praecipe (i.e., to enjoin, command) is used of thenote of instructions delivered by a plaintiff or his lawyer to be filed by the officer of the court, giving the names ofthe plaintiff and defendant.

25.3 Higher education

Princeton University uses the term precept to describe what many other universities refer to as recitations: large classesare often divided into several smaller discussion sections called precepts, which are led by the professor or graduateteaching assistants. Precepts or recitations usually meet once a week to supplement the lectures and provide a venuefor discussion of the course material.[2]

25.4 References[1] Roshi, Robert Aitken. “The Second Paramita (Buddhist Precepts)". Kaohsiung, Taiwan Expat Community Forum. Re-

trieved 28 August 2012.

[2] Aaron Sommers, The Nature of Time. Preceptorial University of New Hampshire.

25.5 Bibliography• Article entolē in Exegetical Dictionary of the New Testament, H. Balz and G. Schneider (ed.), Edinburgh 1990,Vol. I, pp. 459–60, which also cites sources for a discussion of the term’s distinction from Greek nomos/"law”.

• The Code of Canon Law, 1983, in the English translation prepared by the Canon Law Society of Great Britainand Ireland

• The Oxford English Dictionary lists the origin of precept as from the Latin roots of pre-septum. Thus preceptis a pre coming-together/closure.

• This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911).Encyclopædia Britannica (11th ed.). Cambridge University Press.

• This article incorporates text from a publication now in the public domain: Porter, Noah, ed. (1913),Webster’sDictionary, Springfield, Massachusetts: C. & G. Merriam Co.

Chapter 26

Presidential Statement

A Presidential Statement is often created when the United Nations Security Council cannot reach consensus or areprevented from passing a resolution by a permanent member’s veto, or threat thereof. Such statements are similar incontent, format, and tone to resolutions, but are not legally binding.[1]

The adoption of a Presidential Statement requires consensus, although Security Council members may abstain. TheStatement is signed by the sitting Security Council President.

26.1 References[1] Matam Farrall, Jeremy (2007). United Nations sanctions and the rule of law. Cambridge University Press. p. 21. ISBN

978-0-521-87802-9.

26.2 External links• Presidential Statements of the Security Council

61

Chapter 27

Prior consistent statements and priorinconsistent statements

Prior consistent statements and prior inconsistent statements, in the law of evidence, occur where a witness,testifying at trial, makes a statement that is either consistent or inconsistent, respectively, with a previous statementgiven at an earlier time such as during a discovery, interview, or interrogation. The examiner can impeach the witnesswhen an inconsistent statement is found, and may conversely bolster the credibility of an impeached witness with aprior consistent statement.

27.1 Impeachment with a prior inconsistent statement

Before the witness can be impeached the examiner must have extrinsic evidence of the prior statement. The examinermust also provide the witness with the opportunity to adopt or reject the previous statement. [1]

In the majority of U.S. jurisdictions, prior inconsistent statements may not be introduced to prove the truth of theprior statement itself, as this constitutes hearsay, but only to impeach the credibility of the witness.However, under Federal Rule of Evidence 801 and the minority of U.S. jurisdictions that have adopted this rule, aprior inconsistent statement may be introduced as evidence of the truth of the statement itself if the prior statementwas given in live testimony and under oath as part of a formal hearing, proceeding, trial, or deposition.[2]

• Note that under California Evidence Code (“CEC”) §§769, 770, and 1235, prior inconsistent statements maybe used for both impeachment and as substantive evidence, even if they were not originally made under oathat a formal proceeding, as long as “the witness was so examined while testifying as to give him an opportunityto explain or to deny the statement.”[3]

27.2 Bolstering with a prior consistent statement

27.2.1 Prior consistent statements

Aprior consistent statement is not a hearsay exception, the FRE specifically define it as non hearsay. A prior consistentstatement is admissible:

1. to rebut an express or implied charge that the declarant recently fabricated a statement, for instance, during hertestimony at trial;

2. the witness testifies at the present trial; and

3. the witness is subject to cross-examination about the prior statement [4]

There is no requirement that the prior consistent statement have been made under oath at a prior trial or hearing.

62

27.3. REFERENCES 63

A form of prior consistent statement excepted from this rule is that of prior identification by the witness of anotherperson in a lineup.

27.3 References[1] Federal Rules of Evidence, Rule 613

[2] In some U.S. jurisdictions, a prior inconsistent statement that is signed or adopted by a witness is admissible both forimpeachment and substantive purposes. See, e.g., Commonwealth v. Brady, 507 A.2d 66 (Pa. 1986). This approach hasbeen rejected in the federal system.

[3] California Evidence Code §770

[4] Federal Rules of Evidence, Rule 801 (d)(1)

Chapter 28

Proposition

This article is about the term in logic and philosophy. For other uses, see Proposition (disambiguation).Not to be confused with preposition.

The term proposition has a broad use in contemporary philosophy. It is used to refer to some or all of the following:the primary bearers of truth-value, the objects of belief and other "propositional attitudes" (i.e., what is believed,doubted, etc.), the referents of that-clauses and the meanings of declarative sentences. Propositions are the sharableobjects of attitudes and the primary bearers of truth and falsity. This stipulation rules out certain candidates forpropositions, including thought- and utterance-tokens which are not sharable, and concrete events or facts, whichcannot be false.[1]

28.1 Historical usage

28.1.1 By Aristotle

Aristotelian logic identifies a proposition as a sentence which affirms or denies a predicate of a subject. AnAristotelianproposition may take the form “All men are mortal” or “Socrates is a man.” In the first example the subject is “Allmen” and the predicate “are mortal.” In the second example the subject is “Socrates” and the predicate is “is a man.”

28.1.2 By the logical positivists

Often propositions are related to closed sentences to distinguish them from what is expressed by an open sentence.In this sense, propositions are “statements” that are truth-bearers. This conception of a proposition was supported bythe philosophical school of logical positivism.Some philosophers argue that some (or all) kinds of speech or actions besides the declarative ones also have proposi-tional content. For example, yes–no questions present propositions, being inquiries into the truth value of them. Onthe other hand, some signs can be declarative assertions of propositions without forming a sentence nor even beinglinguistic, e.g. traffic signs convey definite meaning which is either true or false.Propositions are also spoken of as the content of beliefs and similar intentional attitudes such as desires, preferences,and hopes. For example, “I desire that I have a new car,” or “I wonder whether it will snow" (or, whether it is thecase that “it will snow”). Desire, belief, and so on, are thus called propositional attitudes when they take this sort ofcontent.

28.1.3 By Russell

Bertrand Russell held that propositions were structured entities with objects and properties as constituents. Wittgen-stein held that a proposition is the set of possible worlds/states of affairs in which it is true. One important differencebetween these views is that on the Russellian account, two propositions that are true in all the same states of affairs

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28.2. RELATION TO THE MIND 65

can still be differentiated. For instance, the proposition that two plus two equals four is distinct on a Russellian ac-count from three plus three equals six. If propositions are sets of possible worlds, however, then all mathematicaltruths (and all other necessary truths) are the same set (the set of all possible worlds).

28.2 Relation to the mind

In relation to the mind, propositions are discussed primarily as they fit into propositional attitudes. Propositionalattitudes are simply attitudes characteristic of folk psychology (belief, desire, etc.) that one can take toward a propo-sition (e.g. 'it is raining,' 'snow is white,' etc.). In English, propositions usually follow folk psychological attitudes bya “that clause” (e.g. “Jane believes that it is raining”). In philosophy of mind and psychology, mental states are oftentaken to primarily consist in propositional attitudes. The propositions are usually said to be the “mental content” ofthe attitude. For example, if Jane has a mental state of believing that it is raining, her mental content is the proposi-tion 'it is raining.' Furthermore, since such mental states are about something (namely propositions), they are said tobe intentional mental states. Philosophical debates surrounding propositions as they relate to propositional attitudeshave also recently centered on whether they are internal or external to the agent or whether they are mind-dependentor mind-independent entities (see the entry on internalism and externalism in philosophy of mind).

28.3 Treatment in logic

As noted above, in Aristotelian logic a proposition is a particular kind of sentence, one which affirms or denies apredicate of a subject. Aristotelian propositions take forms like “All men are mortal” and “Socrates is a man.”Propositions show up in formal logic as objects of a formal language. A formal language begins with different types ofsymbols. These types can include variables, operators, function symbols, predicate (or relation) symbols, quantifiers,and propositional constants. (Grouping symbols are often added for convenience in using the language but do not playa logical role.) Symbols are concatenated together according to recursive rules in order to construct strings to whichtruth-values will be assigned. The rules specify how the operators, function and predicate symbols, and quantifiers areto be concatenated with other strings. A proposition is then a string with a specific form. The form that a propositiontakes depends on the type of logic.The type of logic called propositional, sentential, or statement logic includes only operators and propositional constantsas symbols in its language. The propositions in this language are propositional constants, which are considered atomicpropositions, and composite propositions, which are composed by recursively applying operators to propositions.Application here is simply a short way of saying that the corresponding concatenation rule has been applied.The types of logics called predicate, quantificational, or n-order logic include variables, operators, predicate andfunction symbols, and quantifiers as symbols in their languages. The propositions in these logics are more complex.First, terms must be defined. A term is (i) a variable or (ii) a function symbol applied to the number of terms requiredby the function symbol’s arity. For example, if + is a binary function symbol and x, y, and z are variables, thenx+(y+z) is a term, which might be written with the symbols in various orders. A proposition is (i) a predicate symbolapplied to the number of terms required by its arity, (ii) an operator applied to the number of propositions requiredby its arity, or (iii) a quantifier applied to a proposition. For example, if = is a binary predicate symbol and ∀ is aquantifier, then ∀x,y,z [(x = y) → (x+z = y+z)] is a proposition. This more complex structure of propositions allowsthese logics to make finer distinctions between inferences, i.e., to have greater expressive power.In this context, propositions are also called sentences, statements, statement forms, formulas, and well-formed formu-las, though these terms are usually not synonymous within a single text. This definition treats propositions as syntacticobjects, as opposed to semantic or mental objects. That is, propositions in this sense are meaningless, formal, abstractobjects. They are assigned meaning and truth-values by mappings called interpretations and valuations, respectively.

28.4 Objections to propositions

Attempts to provide a workable definition of proposition include

Two meaningful declarative sentences express the same proposition if and only if they mean thesame thing.

66 CHAPTER 28. PROPOSITION

thus defining proposition in terms of synonymity. For example, “Snow is white” (in English) and “Schnee ist weiß"(in German) are different sentences, but they say the same thing, so they express the same proposition.

Two meaningful declarative sentence-tokens express the same proposition if and only if they meanthe same thing.

Unfortunately, the above definition has the result that two sentences/sentence-tokens which have the same meaningand thus express the same proposition, could have different truth-values, e.g. “I am Spartacus” said by Spartacus andsaid by John Smith; and e.g. “It is Wednesday” said on a Wednesday and on a Thursday.A number of philosophers and linguists claim that all definitions of a proposition are too vague to be useful. Forthem, it is just a misleading concept that should be removed from philosophy and semantics. W.V. Quine maintainedthat the indeterminacy of translation prevented any meaningful discussion of propositions, and that they should bediscarded in favor of sentences.[2] Strawson advocated the use of the term “statement”.

28.5 See also• Main contention

28.6 References[1] “Propositions (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved 2014-06-23.

[2] Quine W.V. Philosophy of Logic, Prentice-Hall NJ USA: 1970, pp 1-14

28.7 External links• Stanford Encyclopedia of Philosophy articles on:

• Propositions, by Matthew McGrath• Singular Propositions, by Greg Fitch• Structured Propositions, by Jeffrey C. King

Chapter 29

Propositional formula

In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truthvalue. If the values of all variables in a propositional formula are given, it determines a unique truth value. Apropositional formula may also be called a propositional expression, a sentence, or a sentential formula.A propositional formula is constructed from simple propositions, such as “five is greater than three” or propositionalvariables such as P and Q, using connectives such as NOT, AND, OR, and IMPLIES; for example:

(P AND NOT Q) IMPLIES (P OR Q).

In mathematics, a propositional formula is often more briefly referred to as a "proposition", but, more precisely, apropositional formula is not a proposition but a formal expression that denotes a proposition, a formal object underdiscussion, just like an expression such as "x + y" is not a value, but denotes a value. In some contexts, maintainingthe distinction may be of importance.

29.1 Propositions

For the purposes of the propositional calculus, propositions (utterances, sentences, assertions) are considered to beeither simple or compound.[1] Compound propositions are considered to be linked by sentential connectives, someof the most common of which are “AND”, “OR”, “IF ... THEN ...”, “NEITHER ... NOR...”, "... IS EQUIVALENTTO ...” . The linking semicolon ";", and connective “BUT” are considered to be expressions of “AND”. A sequenceof discrete sentences are considered to be linked by “AND"s, and formal analysis applies a recursive “parenthesisrule” with respect to sequences of simple propositions (see more below about well-formed formulas).

For example: The assertion: “This cow is blue. That horse is orange but this horse here is purple.” isactually a compound proposition linked by “AND"s: ( (“This cow is blue” AND “that horse is orange”)AND “this horse here is purple” ) .

Simple propositions are declarative in nature, that is, they make assertions about the condition or nature of a particularobject of sensation e.g. “This cow is blue”, “There’s a coyote!" (“That coyote IS there, behind the rocks.”).[2] Thusthe simple “primitive” assertions must be about specific objects or specific states of mind. Each must have at least asubject (an immediate object of thought or observation), a verb (in the active voice and present tense preferred), andperhaps an adjective or adverb. “Dog!" probably implies “I see a dog” but should be rejected as too ambiguous.

Example: “That purple dog is running”, “This cow is blue”, “Switch M31 is closed”, “This cap is off”,“Tomorrow is Friday”.

For the purposes of the propositional calculus a compound proposition can usually be reworded into a series of simplesentences, although the result will probably sound stilted.

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68 CHAPTER 29. PROPOSITIONAL FORMULA

29.1.1 Relationship between propositional and predicate formulas

The predicate calculus goes a step further than the propositional calculus to an “analysis of the inner structure ofpropositions”[3] It breaks a simple sentence down into two parts (i) its subject (the object (singular or plural) ofdiscourse) and (ii) a predicate—a verb or possibly verb-clause that asserts a quality or attribute of the object(s)).The predicate calculus then generalizes the “subject|predicate” form (where | symbolizes concatenation (stringingtogether) of symbols) into a form with the following blank-subject structure " ___|predicate”, and the predicate inturn generalized to all things with that property.

Example: “This blue pig has wings” becomes two sentences in the propositional calculus: “This pig haswings” AND “This pig is blue”, whose internal structure is not considered. In contrast, in the predicatecalculus, the first sentence breaks into “this pig” as the subject, and “has wings” as the predicate. Thusit asserts that object “this pig” is a member of the class (set, collection) of “winged things”. The secondsentence asserts that object “this pig” has an attribute “blue” and thus is a member of the class of “bluethings”. One might choose to write the two sentences connected with AND as:

p|W AND p|B

The generalization of “this pig” to a (potential) member of two classes “winged things” and “blue things” means thatit has a truth-relationship with both of these classes. In other words, given a domain of discourse “winged things”,either we find p to be a member of this domain or not. Thus we have a relationship W (wingedness) between p (pig)and { T, F }, W(p) evaluates to { T, F }. Likewise for B (blueness) and p (pig) and { T, F }: B(p) evaluates to { T, F}. So we now can analyze the connected assertions “B(p) AND W(p)" for its overall truth-value, i.e.:

( B(p) AND W(p) ) evaluates to { T, F }

In particular, simple sentences that employ notions of “all”, “some”, “a few”, “one of”, etc. are treated by the predicatecalculus. Along with the new function symbolism “F(x)" two new symbols are introduced: ∀ (For all), and ∃ (Thereexists ..., At least one of ... exists, etc.). The predicate calculus, but not the propositional calculus, can establish theformal validity of the following statement:

“All blue pigs have wings but some pigs have no wings, hence some pigs are not blue”.

29.1.2 Identity

Tarski asserts that the notion of IDENTITY (as distinguished from LOGICAL EQUIVALENCE) lies outside thepropositional calculus; however, he notes that if a logic is to be of use for mathematics and the sciences it must containa “theory” of IDENTITY.[4] Some authors refer to “predicate logic with identity” to emphasize this extension. Seemore about this below.

29.2 An algebra of propositions, the propositional calculus

An algebra (and there are many different ones), loosely defined, is a method by which a collection of symbols calledvariables together with some other symbols such as parentheses (, ) and some sub-set of symbols such as *, +, ~, &,V, =, ≡, ⋀, ¬ are manipulated within a system of rules. These symbols, and well-formed strings of them, are saidto represent objects, but in a specific algebraic system these objects do not have meanings. Thus work inside thealgebra becomes an exercise in obeying certain laws (rules) of the algebra’s syntax (symbol-formation) rather thanin semantics (meaning) of the symbols. The meanings are to be found outside the algebra.For a well-formed sequence of symbols in the algebra—a formula -- to have some usefulness outside the algebra thesymbols are assigned meanings and eventually the variables are assigned values; then by a series of rules the formulais evaluated.When the values are restricted to just two and applied to the notion of simple sentences (e.g. spoken utterancesor written assertions) linked by propositional connectives this whole algebraic system of symbols and rules andevaluation-methods is usually called the propositional calculus or the sentential calculus.

29.2. AN ALGEBRA OF PROPOSITIONS, THE PROPOSITIONAL CALCULUS 69

While some of the familiar rules of arithmetic algebra continue to hold in the algebra of propositions (e.g. thecommutative and associative laws for AND and OR), some do not (e.g. the distributive laws for AND, OR andNOT).

29.2.1 Usefulness of propositional formulas

Analysis: In deductive reasoning, philosophers, rhetoricians and mathematicians reduce arguments to formulas andthen study them (usually with truth tables) for correctness (soundness). For example: Is the following argumentsound?

“Given that consciousness is sufficient for an artificial intelligence and only conscious entities can passthe Turing test, before we can conclude that a robot is an artificial intelligence the robot must pass theTuring test.”

Engineers analyze the logic circuits they have designed using synthesis techniques and then apply various reductionand minimization techniques to simplify their designs.Synthesis: Engineers in particular synthesize propositional formulas (that eventually end up as circuits of symbols)from truth tables. For example, one might write down a truth table for how binary addition should behave given theaddition of variables “b” and “a” and “carry_in” “ci”, and the results “carry_out” “co” and “sum” Σ:

Example: in row 5, ( (b+a) + ci ) = ( (1+0) + 1 ) = the number “2”. written as a binary number this is102, where “co"=1 and Σ=0 as shown in the right-most columns.

29.2.2 Propositional variables

The simplest type of propositional formula is a propositional variable. Propositions that are simple (atomic), sym-bolic expressions are often denoted by variables named a, b, or A, B, etc. A propositional variable is intended torepresent an atomic proposition (assertion), such as “It is Saturday” = a (here the symbol = means " ... is assignedthe variable named ...”) or “I only go to the movies on Monday” = b.

29.2.3 Truth-value assignments, formula evaluations

Evaluation of a propositional formula begins with assignment of a truth value to each variable. Because eachvariable represents a simple sentence, the truth values are being applied to the “truth” or “falsity” of these simplesentences.Truth values in rhetoric, philosophy and mathematics: The truth values are only two: { TRUTH “T”, FALSITY“F” }. An empiricist puts all propositions into two broad classes: analytic—true no matter what (e.g. tautology), andsynthetic—derived from experience and thereby susceptible to confirmation by third parties (the verification theory ofmeaning).[5] Empiricits hold that, in general, to arrive at the truth-value of a synthetic proposition, meanings (pattern-matching templates) must first be applied to the words, and then these meaning-templates must be matched againstwhatever it is that is being asserted. For example, my utterance “That cow is blue!" Is this statement a TRUTH? TrulyI said it. And maybe I am seeing a blue cow—unless I am lying my statement is a TRUTH relative to the object ofmy (perhaps flawed) perception. But is the blue cow “really there"? What do you see when you look out the samewindow? In order to proceed with a verification, you will need a prior notion (a template) of both “cow” and “blue”,and an ability to match the templates against the object of sensation (if indeed there is one).Truth values in engineering: Engineers try to avoid notions of truth and falsity that bedevil philosophers, but in thefinal analysis engineers must trust their measuring instruments. In their quest for robustness, engineers prefer to pullknown objects from a small library—objects that have well-defined, predictable behaviors even in large combinations,(hence their name for the propositional calculus: “combinatorial logic”). The fewest behaviors of a single object aretwo (e.g. { OFF, ON }, { open, shut }, { UP, DOWN } etc.), and these are put in correspondence with { 0, 1 }.Such elements are called digital; those with a continuous range of behaviors are called analog. Whenever decisionsmust be made in an analog system, quite often an engineer will convert an analog behavior (the door is 45.32146%UP) to digital (e.g. DOWN=0 ) by use of a comparator.[6]

70 CHAPTER 29. PROPOSITIONAL FORMULA

Thus an assignment ofmeaning of the variables and the two value-symbols { 0, 1 } comes from “outside” the formulathat represents the behavior of the (usually) compound object. An example is a garage door with two “limit switches”,one for UP labelled SW_U and one for DOWN labelled SW_D, and whatever else is in the door’s circuitry. Inspectionof the circuit (either the diagram or the actual objects themselves—door, switches, wires, circuit board, etc.) mightreveal that, on the circuit board “node 22” goes to +0 volts when the contacts of switch “SW_D” are mechanically incontact (“closed”) and the door is in the “down” position (95% down), and “node 29” goes to +0 volts when the dooris 95% UP and the contacts of switch SW_U are in mechanical contact (“closed”).[7] The engineer must define themeanings of these voltages and all possible combinations (all 4 of them), including the “bad” ones (e.g. both nodes22 and 29 at 0 volts, meaning that the door is open and closed at the same time). The circuit mindlessly responds towhatever voltages it experiences without any awareness of TRUTH or FALSEHOOD, RIGHT or WRONG, SAFEor DANGEROUS.

29.3 Propositional connectives

Arbitrary propositional formulas are built from propositional variables and other propositional formulas using propositionalconnectives. Examples of connectives include:

• The unary negation connective. If α is a formula, then ¬α is a formula.

• The classical binary connectives ∧,∨,→,↔ . Thus, for example, if α and β are formulas, so is (α → β) .

• Other binary connectives, such as NAND, NOR, and XOR

• The ternary connective IF ... THEN ... ELSE ...

• Constant 0-ary connectives ⊤ and ⊥ (alternately, constants { T, F }, { 1, 0 } etc. )

• The “theory-extension” connective EQUALS (alternately, IDENTITY, or the sign " = " as distinguished fromthe “logical connective”↔ )

29.3.1 Connectives of rhetoric, philosophy and mathematics

The following are the connectives common to rhetoric, philosophy and mathematics together with their truth tables.The symbols used will vary from author to author and between fields of endeavor. In general the abbreviations “T”and “F” stand for the evaluations TRUTH and FALSITY applied to the variables in the propositional formula (e.g.the assertion: “That cow is blue” will have the truth-value “T” for Truth or “F” for Falsity, as the case may be.).The connectives go by a number of different word-usages, e.g. “a IMPLIES b” is also said “IF a THEN b”. Some ofthese are shown in the table.

29.3.2 Engineering connectives

In general, the engineering connectives are just the same as the mathematics connectives excepting they tend toevaluate with “1” = “T” and “0” = “F”. This is done for the purposes of analysis/minimization and synthesis offormulas by use of the notion of minterms and Karnaugh maps (see below). Engineers also use the words logicalproduct from Boole's notion (a*a = a) and logical sum from Jevons' notion (a+a = a).[8]

29.3.3 CASE connective: IF ... THEN ... ELSE ...

The IF ... THEN ... ELSE ... connective appears as the simplest form of CASE operator of recursion theoryand computation theory and is the connective responsible for conditional goto’s (jumps, branches). From this oneconnective all other connectives can be constructed (see more below). Although " IF c THEN b ELSE a " soundslike an implication it is, in its most reduced form, a switch that makes a decision and offers as outcome only one oftwo alternatives “a” or “b” (hence the name switch statement in the C programming language).[9]

The following three propositions are equivalent (as indicated by the logical equivalence sign ≡ ):

29.3. PROPOSITIONAL CONNECTIVES 71

Engineering symbols have varied over the years, but these are commonplace. Sometimes they appear simply as boxes with symbolsin them. “a” and “b” are called “the inputs” and “c” is called “the output”. An output will typical “connect to” an input (unless it is thefinal connective); this accomplishes the mathematical notion of substitution.

• (1) ( IF 'counter is zero' THEN 'go to instruction b ' ELSE 'go to instruction a ') ≡• (2) ( (c → b) & (~c → a) ) ≡ ( ( IF 'counter is zero' THEN 'go to instruction b ' ) AND ( IF 'It isNOT the case that counter is zero' THEN 'go to instruction a ) " ≡

• (3) ( (c & b) V (~c & a) ) = " ( 'Counter is zero' AND 'go to instruction b ) OR ( 'It is NOT thecase that 'counter is zero' AND 'go to instruction a ) "

Thus IF ... THEN ... ELSE—unlike implication—does not evaluate to an ambiguous “TRUTH” when the firstproposition is false i.e. c = F in (c → b). For example, most people would reject the following compound propositionas a nonsensical non sequitur because the second sentence is not connected in meaning to the first.[10]

Example: The proposition " IF 'Winston Churchill was Chinese' THEN 'The sun rises in the east' "evaluates as a TRUTH given that 'Winston Church was Chinese' is a FALSEHOOD and 'The sun risesin the east' evaluates as a TRUTH.

In recognition of this problem, the sign → of formal implication in the propositional calculus is called materialimplication to distinguish it from the everyday, intuitive implication.[11]

The use of the IF ... THEN ... ELSE construction avoids controversy because it offers a completely deterministicchoice between two stated alternatives; it offers two “objects” (the two alternatives b and a), and it selects betweenthem exhaustively and unabiguously.[12] In the truth table below, d1 is the formula: ( (IF c THEN b) AND (IF NOT-cTHEN a) ). Its fully reduced form d2 is the formula: ( (c AND b) OR (NOT-c AND a). The two formulas areequivalent as shown by the columns "=d1” and "=d2”. Electrical engineers call the fully reduced formula the AND-OR-SELECT operator. The CASE (or SWITCH) operator is an extension of the same idea to n possible, but mutuallyexclusive outcomes. Electrical engineers call the CASE operator a multiplexer.

29.3.4 IDENTITY and evaluation

The first table of this section stars *** the entry logical equivalence to note the fact that "Logical equivalence" is notthe same thing as “identity”. For example, most would agree that the assertion “That cow is blue” is identical to theassertion “That cow is blue”. On the other hand, logical equivalence sometimes appears in speech as in this example:" 'The sun is shining' means 'I'm biking' " Translated into a propositional formula the words become: “IF 'the sun isshining' THEN 'I'm biking', AND IF 'I'm biking' THEN 'the sun is shining'":[13]

“IF 's’ THEN 'b' AND IF 'b' THEN 's’ " is written as ((s → b) & (b → s)) or in an abbreviated form as(s ↔ b). As the rightmost symbol string is a definition for a new symbol in terms of the symbols on theleft, the use of the IDENTITY sign = is appropriate:

72 CHAPTER 29. PROPOSITIONAL FORMULA

((s → b) & (b → s)) = (s ↔ b)

Different authors use different signs for logical equivalence: ↔ (e.g. Suppes, Goodstein, Hamilton), ≡ (e.g. Robbin),⇔ (e.g. Bender andWilliamson). Typically identity is written as the equals sign =. One exception to this rule is foundin Principia Mathematica. For more about the philosophy of the notion of IDENTITY see Leibniz’s law.As noted above, Tarski considers IDENTITY to lie outside the propositional calculus, but he asserts that without thenotion, “logic” is insufficient for mathematics and the deductive sciences. In fact the sign comes into the propositionalcalculus when a formula is to be evaluated.[14]

In some systems there are no truth tables, but rather just formal axioms (e.g. strings of symbols from a set { ~, →, (,), variables p1, p2, p3, ... } and formula-formation rules (rules about how to make more symbol strings from previousstrings by use of e.g. substitution and modus ponens). the result of such a calculus will be another formula (i.e. awell-formed symbol string). Eventually, however, if one wants to use the calculus to study notions of validity andtruth, one must add axioms that define the behavior of the symbols called “the truth values” {T, F} ( or {1, 0}, etc.)relative to the other symbols.For example, Hamilton uses two symbols = and ≠ when he defines the notion of a valuation v of any wffs A and Bin his “formal statement calculus” L. A valuation v is a function from the wffs of his system L to the range (output){ T, F }, given that each variable p1, p2, p3 in a wff is assigned an arbitrary truth value { T, F }.

• (i) v(A) ≠ v(~A)

• (ii) v(A→ B) = F if and only if v(A) = T and v(B) = F

The two definitions (i) and (ii) define the equivalent of the truth tables for the ~ (NOT) and → (IMPLICATION)connectives of his system. The first one derives F ≠ T and T ≠ F, in other words " v(A) does not mean v(~A)".Definition (ii) specifies the third row in the truth table, and the other three rows then come from an application ofdefinition (i). In particular (ii) assigns the value F (or a meaning of “F”) to the entire expression. The definitions alsoserve as formation rules that allow substitution of a value previously derived into a formula:Some formal systems specify these valuation axioms at the outset in the form of certain formulas such as the law ofcontradiction or laws of identity and nullity. The choice of which ones to use, together with laws such as commutationand distribution, is up to the system’s designer as long as the set of axioms is complete (i.e. sufficient to form and toevaluate any well-formed formula created in the system).

29.4 More complex formulas

As shown above, the CASE (IF c THEN b ELSE a ) connective is constructed either from the 2-argument connectivesIF...THEN... and AND or from OR and AND and the 1-argument NOT. Connectives such as the n-argument AND(a & b & c & ... & n), OR (a V b V c V ... V n) are constructed from strings of two-argument AND and OR andwritten in abbreviated form without the parentheses. These, and other connectives as well, can then used as buildingblocks for yet further connectives. Rhetoricians, philosophers, and mathematicians use truth tables and the varioustheorems to analyze and simplify their formulas.Electrical engineering uses drawn symbols and connect them with lines that stand for the mathematicals act of sub-stitution and replacement. They then verify their drawings with truth tables and simplify the expressions as shownbelow by use of Karnaugh maps or the theorems. In this way engineers have created a host of “combinatorial logic”(i.e. connectives without feedback) such as “decoders”, “encoders”, “mutifunction gates”, “majority logic”, “binaryadders”, “arithmetic logic units”, etc.

29.4.1 Definitions

A definition creates a new symbol and its behavior, often for the purposes of abbreviation. Once the definition ispresented, either form of the equivalent symbol or formula can be used. The following symbolism =D is followingthe convention of Reichenbach.[15] Some examples of convenient definitions drawn from the symbol set { ~, &, (,) } and variables. Each definition is producing a logically equivalent formula that can be used for substitution orreplacement.

29.5. INDUCTIVE DEFINITION 73

• definition of a new variable: (c & d) =D s• OR: ~(~a & ~b) =D (a V b)• IMPLICATION: (~a V b) =D (a → b)• XOR: (~a & b) V (a & ~b) =D (a ⊕ b)• LOGICAL EQUIVALENCE: ( (a → b) & (b → a) ) =D ( a ≡ b )

29.4.2 Axiom and definition schemas

The definitions above for OR, IMPLICATION, XOR, and logical equivalence are actually schemas (or “schemata”),that is, they are models (demonstrations, examples) for a general formula format but shown (for illustrative purposes)with specific letters a, b, c for the variables, whereas any variable letters can go in their places as long as the lettersubstitutions follow the rule of substitution below.

Example: In the definition (~a V b) =D (a → b), other variable-symbols such as “SW2” and “CON1”might be used, i.e. formally:

a =D SW2, b =D CON1, so we would have as an instance of the definition schema (~SW2V CON1) =D (SW2 → CON1)

29.4.3 Substitution versus replacement

Substitution: The variable or sub-formula to be substituted with another variable, constant, or sub-formula must bereplaced in all instances throughout the overall formula.

Example: (c & d) V (p & ~(c & ~d)), but (q1 & ~q2) ≡ d. Now wherever variable “d” occurs, substitute(q1 & ~q2):

(c & (q1 & ~q2)) V (p & ~(c & ~(q1 & ~q2)))

Replacement: (i) the formula to be replaced must be within a tautology, i.e. logically equivalent ( connected by ≡or ↔) to the formula that replaces it, and (ii) unlike substitution its permissible for the replacement to occur only inone place (i.e. for one formula).

Example: Use this set of formula schemas/equivalences: 1: ( (a V 0) ≡ a ). 2: ( (a & ~a) ≡ 0 ). 3: ( (~aV b) =D (a → b) ). 6. ( ~(~a) ≡ a )

• start with “a": a• Use 1 to replace “a” with (a V 0): (a V 0)• Use the notion of “schema” to substitute b for a in 2: ( (a & ~a) ≡ 0 )• Use 2 to replace 0 with (b & ~b): ( a V (b & ~b) )• (see below for how to distribute “a V” over (b & ~b), etc

29.5 Inductive definition

The classical presentation of propositional logic (see Enderton 2002) uses the connectives ¬,∧,∨,→,↔ . The setof formulas over a given set of propositional variables is inductively defined to be the smallest set of expressions suchthat:

• Each propositional variable in the set is a formula,

• (¬α) is a formula whenever α is, and

• (α□β) is a formula whenever α and β are formulas and □ is one of the binary connectives ∧,∨,→,↔ .

74 CHAPTER 29. PROPOSITIONAL FORMULA

This inductive definition can be easily extended to cover additional connectives.The inductive definition can also be rephrased in terms of a closure operation (Enderton 2002). Let V denote a set ofpropositional variables and let XV denote the set of all strings from an alphabet including symbols in V, left and rightparentheses, and all the logical connectives under consideration. Each logical connective corresponds to a formulabuilding operation, a function from XXV to XXV:

• Given a string z, the operation E¬(z) returns (¬z) .

• Given strings y and z, the operation E∧(y, z) returns (y ∧ x) . There are similar operations E∨ , E→ , and E↔corresponding to the other binary connectives.

The set of formulas over V is defined to be the smallest subset of XXV containing V and closed under all the formulabuilding operations.

29.6 Parsing formulas

The following “laws” of the propositional calculus are used to “reduce” complex formulas. The “laws” can be easilyverified with truth tables. For each law, the principal (outermost) connective is associated with logical equivalence≡ or identity =. A complete analysis of all 2n combinations of truth-values for its n distinct variables will result ina column of 1’s (T’s) underneath this connective. This finding makes each law, by definition, a tautology. And, fora given law, because its formula on the left and right are equivalent (or identical) they can be substituted for oneanother.

Example: The following truth table is De Morgan’s law for the behavior of NOT over OR: ~(a V b) ≡(~a & ~b). To the left of the principal connective ≡ (yellow column labelled “taut”) the formula ~(b Va) evaluates to (1, 0, 0, 0) under the label “P”. On the right of “taut” the formula (~(b) V ~(a)) alsoevaluates to (1, 0, 0, 0) under the label “Q”. As the two columns have equivalent evaluations, the logicalequivalence ≡ under “taut” evaluates to (1, 1, 1, 1), i.e. P ≡ Q. Thus either formula can be substitutedfor the other if it appears in a larger formula.

Enterprising readers might challenge themselves to invent an “axiomatic system” that uses the symbols { V, &, ~, (,), variables a, b, c }, the formation rules specified above, and as few as possible of the laws listed below, and thenderive as theorems the others as well as the truth-table valuations for V, &, and ~. One set attributed to Huntington(1904) (Suppes:204) uses eight of the laws defined below.Note that that if used in an axiomatic system, the symbols 1 and 0 (or T and F) are considered to be wffs and thusobey all the same rules as the variables. Thus the laws listed below are actually axiom schemas, that is, they stand inplace of an infinite number of instances. Thus ( x V y ) ≡ ( y V x ) might be used in one instance, ( p V 0 ) ≡ ( 0 V p) and in another instance ( 1 V q ) ≡ ( q V 1 ), etc.

29.6.1 Connective seniority (symbol rank)

In general, to avoid confusion during analysis and evaluation of propositional formulas make liberal use parentheses.However, quite often authors leave them out. To parse a complicated formula one first needs to know the seniority, orrank, that each of the connectives (excepting *) has over the other connectives. To “well-form” a formula, start withthe connective with the highest rank and add parentheses around its components, then move down in rank (payingclose attention to the connective’s scope over which the it is working). From most- to least-senior, with the predicatesigns ∀x and ∃x, the IDENTITY = and arithmetic signs added for completeness:[16]

≡ (LOGICAL EQUIVALENCE), → (IMPLICATION), & (AND), V (OR), ~ (NOT), ∀x(FORALLx),∃x (THEREEXISTSANx),= (IDENTITY),+ (arithmetic sum), *(arithmeticmultiply), ' (s, arithmetic successor).

Thus the formula can be parsed—but note that, because NOT does not obey the distributive law, the parenthesesaround the inner formula (~c & ~d) is mandatory:

29.6. PARSING FORMULAS 75

Example: " d & c V w " rewritten is ( (d & c) V w )Example: " a & a → b ≡ a & ~a V b " rewritten (rigorously) is

• ≡ has seniority: ( ( a & a → b ) ≡ ( a & ~a V b ) )• → has seniority: ( ( a & (a → b) ) ≡ ( a & ~a V b ) )• & has seniority both sides: ( ( ( (a) & (a → b) ) ) ≡ ( ( (a) & (~a V b) ) )• ~ has seniority: ( ( ( (a) & (a → b) ) ) ≡ ( ( (a) & (~(a) V b) ) )• check 9 ( -parenthesis and 9 ) -parenthesis: ( ( ( (a) & (a → b) ) ) ≡ ( ( (a) & (~(a) V b)) )

Example:

d & c V p & ~(c & ~d) ≡ c & d V p & c V p & ~d rewritten is ( ( (d & c) V ( p & ~((c &~(d)) ) ) ) ≡ ( (c & d) V (p & c) V (p & ~(d)) ) )

29.6.2 Commutative and associative laws

Both AND and OR obey the commutative law and associative law:

• Commutative law for OR: ( a V b ) ≡ ( b V a )

• Commutative law for AND: ( a & b ) ≡ ( b & a )

• Associative law for OR: (( a V b ) V c ) ≡ ( a V (b V c) )

• Associative law for AND: (( a & b ) & c ) ≡ ( a & (b & c) )

Omitting parentheses in strings of AND and OR: The connectives are considered to be unary (one-variable, e.g.NOT) and binary (i.e. two-variable AND, OR, IMPLIES). For example:

( (c & d) V (p & c) V (p & ~d) ) above should be written ( ((c & d) V (p & c)) V (p & ~(d) ) ) or possibly( (c & d) V ( (p & c) V (p & ~(d)) ) )

However, a truth-table demonstration shows that the form without the extra parentheses is perfectly adequate.Omitting parentheses with regards to a single-variable NOT: While ~(a) where a is a single variable is perfectlyclear, ~a is adequate and is the usual way this literal would appear. When the NOT is over a formula with more thanone symbol, then the parentheses are mandatory, e.g. ~(a V b).

29.6.3 Distributive laws

OR distributes over AND and AND distributes over OR. NOT does not distribute over AND or OR. See below aboutDe Morgan’s law:

• Distributive law for OR: ( c V ( a & b) ) ≡ ( (c V a) & (c V b) )

• Distributive law for AND: ( c & ( a V b) ) ≡ ( (c & a) V (c & b) )

29.6.4 De Morgan’s laws

NOT, when distributed over OR or AND, does something peculiar (again, these can be verified with a truth-table):

• De Morgan’s law for OR: ~(a V b) ≡ (~a & ~b)

• De Morgan’s law for AND: ~(a & b) ≡ (~a V ~b)

76 CHAPTER 29. PROPOSITIONAL FORMULA

29.6.5 Laws of absorption

Absorption, in particular the first one, causes the “laws” of logic to differ from the “laws” of arithmetic:

• Absorption (idempotency) for OR: (a V a) ≡ a

• Absorption (idempotency) for AND: (a & a) ≡ a

29.6.6 Laws of evaluation: Identity, nullity, and complement

The sign " = " (as distinguished from logical equivalence ≡, alternately ↔ or⇔) symbolizes the assignment of value ormeaning. Thus the string (a & ~(a)) symbolizes “1”, i.e. itmeans the same thing as symbol “1” ". In some “systems”this will be an axiom (definition) perhaps shown as ( (a & ~(a)) =D 1 ); in other systems, it may be derived in thetruth table below:

• Commutation of equality: (a = b) ≡ (b = a)

• Identity for OR: (a V 0) = a or (a V F) = a

• Identity for AND: (a & 1) = a or (a & T) = a

• Nullity for OR: (a V 1) = 1 or (a V T) = T

• Nullity for AND: (a & 0) = 0 or (a & F) = F

• Complement for OR: (a V ~a) = 1 or (a V ~a) = T, law of excluded middle

• Complement for AND: (a & ~a) = 0 or (a & ~a) = F, law of contradiction

29.6.7 Double negative (involution)

• ~(~a) = a

29.7 Well-formed formulas (wffs)

A key property of formulas is that they can be uniquely parsed to determine the structure of the formula in termsof its propositional variables and logical connectives. When formulas are written in infix notation, as above, uniquereadability is ensured through an appropriate use of parentheses in the definition of formulas. Alternatively, formulascan be written in Polish notation or reverse Polish notation, eliminating the need for parentheses altogether.The inductive definition of infix formulas in the previous section can be converted to a formal grammar in Backus-Naur form:

<formula> ::= <propositional variable>| ( ¬ <formula> )| ( <formula> ∧ <formula>)| ( <formula> ∨ <formula> )| ( <formula>→ <formula> )| ( <formula>↔ <formula> )

It can be shown that any expression matched by the grammar has a balanced number of left and right parentheses, andany nonempty initial segment of a formula has more left than right parentheses.[17] This fact can be used to give analgorithm for parsing formulas. For example, suppose that an expression x begins with (¬ . Starting after the secondsymbol, match the shortest subexpression y of x that has balanced parentheses. If x is a formula, there is exactly onesymbol left after this expression, this symbol is a closing parenthesis, and y itself is a formula. This idea can be usedto generate a recursive descent parser for formulas.

29.8. REDUCED SETS OF CONNECTIVES 77

Example of parenthesis counting:This method locates as “1” the principal connective -- the connective under which the overall evaluation of theformula occurs for the outer-most parentheses (which are often omitted).[18] It also locates the inner-most connectivewhere one would begin evaluatation of the formula without the use of a truth table, e.g. at “level 6”.

29.7.1 Wffs versus valid formulas in inferences

The notion of valid argument is usually applied to inferences in arguments, but arguments reduce to propositionalformulas and can be evaluated the same as any other propositional formula. Here a valid inference means: “Theformula that represents the inference evaluates to “truth” beneath its principal connective, no matter what truth-values are assigned to its variables”, i.e. the formula is a tautology.[19] Quite possibly a formula will be well-formedbut not valid. Another way of saying this is: “Being well-formed is necessary for a formula to be valid but it is notsufficient.” The only way to find out if it is both well-formed and valid is to submit it to verification with a truth tableor by use of the “laws":

Example 1: What does one make of the following difficult-to-follow assertion? Is it valid? “If it’s sunny,but if the frog is croaking then it’s not sunny, then it’s the same as saying that the frog isn't croaking.”Convert this to a propositional formula as follows:

" IF (a AND (IF b THEN NOT-a) THEN NOT-a” where " a " represents “its sunny” and " b" represents “the frog is croaking":( ( (a) & ( (b) → ~(a) ) ≡ ~(b) )

This is well-formed, but is it valid? In other words, when evaluated will this yield a tautology (all T)beneath the logical-equivalence symbol ≡ ? The answer is NO, it is not valid. However, if reconstructedas an implication then the argument is valid.“Saying it’s sunny, but if the frog is croaking then it’s not sunny, implies that the frog isn't croaking.”

Other circumstances may be preventing the frog from croaking: perhaps a crane ate it.

Example 2 (from Reichenbach via Bertrand Russell):

“If pigs have wings, some winged animals are good to eat. Some winged animals are good toeat, so pigs have wings.”( ((a) → (b)) & (b) → (a) ) is well formed, but an invalid argument as shown by the redevaluation under the principal implication:

29.8 Reduced sets of connectives

A set of logical connectives is called complete if every propositional formula is tautologically equivalent to a formulawith just the connectives in that set. There are many complete sets of connectives, including {∧,¬} , {∨,¬} , and{→,¬} . There are two binary connectives that are complete on their own, corresponding to NAND and NOR,respectively.[20] Some pairs are not complete, for example {∧,∨} .

29.8.1 The stroke (NAND)

The binary connective corresponding to NAND is called the Sheffer stroke, and written with a vertical bar | or verticalarrow ↑. The completeness of this connective was noted in Principia Mathematica (1927:xvii). Since it is complete onits own, all other connectives can be expressed using only the stroke. For example, where the symbol " ≡ " representslogical equivalence:

~p ≡ p|pp → q ≡ p|~qp V q ≡ ~p|~qp & q ≡ ~(p|q)

78 CHAPTER 29. PROPOSITIONAL FORMULA

The engineering symbol for the NAND connective (the 'stroke') can be used to build any propositional formula. The notion that truth(1) and falsity (0) can be defined in terms of this connective is shown in the sequence of NANDs on the left, and the derivations ofthe four evaluations of a NAND b are shown along the bottom. The more common method is to use the definition of the NAND fromthe truth table.

In particular, the zero-ary connectives ⊤ (representing truth) and ⊥ (representing falsity) can be expressed using thestroke:

⊤ ≡ (a|(a|a))

⊥ ≡ (⊤|⊤)

29.8.2 IF ... THEN ... ELSE

This connective together with { 0, 1 }, ( or { F, T } or { ⊥ , ⊤ } ) forms a complete set. In the following theIF...THEN...ELSE relation (c, b, a) = d represents ( (c → b) V (~c → a) ) ≡ ( (c & b) V (~c & a) ) = d

(c, b, a):(c, 0, 1) ≡ ~c(c, b, 1) ≡ (c → b)(c, c, a) ≡ (c V a)(c, b, c) ≡ (c & b)

29.9. NORMAL FORMS 79

Example: The following shows how a theorem-based proof of "(c, b, 1) ≡ (c → b)" would proceed, below the proofis its truth-table verification. ( Note: (c → b) is defined to be (~c V b) ):

• Begin with the reduced form: ( (c & b) V (~c & a) )• Substitute “1” for a: ( (c & b) V (~c & 1) )• Identity (~c & 1) = ~c: ( (c & b) V (~c) )• Law of commutation for V: ( (~c) V (c & b) )• Distribute "~c V” over (c & b): ( ((~c) V c ) & ((~c) V b )• Law of excluded middle (((~c) V c ) = 1 ): ( (1) & ((~c) V b ) )• Distribute "(1) &" over ((~c) V b): ( ((1) & (~c)) V ((1) & b )) )• Commutivity and Identity (( 1 & ~c) = (~c & 1) = ~c, and (( 1 & b) ≡ (b & 1) ≡ b: ( ~c V b )• ( ~c V b ) is defined as c → b Q. E. D.

In the following truth table the column labelled “taut” for tautology evaluates logical equivalence (symbolized here by≡) between the two columns labelled d. Because all four rows under “taut” are 1’s, the equivalence indeed representsa tautology.

29.9 Normal forms

An arbitrary propositional formulamay have a very complicated structure. It is often convenient to workwith formulasthat have simpler forms, known as normal forms. Some common normal forms include conjunctive normal formand disjunctive normal form. Any propositional formula can be reduced to its conjunctive or disjunctive normal form.

29.9.1 Reduction to normal form

Reduction to normal form is relatively simple once a truth table for the formula is prepared. But further attempts tominimize the number of literals (see below) requires some tools: reduction by De Morgan’s laws and truth tablescan be unwieldy, but Karnaugh maps are very suitable a small number of variables (5 or less). Some sophisticatedtabular methods exist for more complex circuits with multiple outputs but these are beyond the scope of this article;for more see Quine–McCluskey algorithm.

Literal, term and alterm

In electrical engineering a variable x or its negation ~(x) is lumped together into a single notion called a literal. Astring of literals connected by ANDs is called a term. A string of literals connected by OR is called an alterm.Typically the literal ~(x) is abbreviated ~x. Sometimes the &-symbol is omitted altogether in the manner of algebraicmultiplication.

Example: a, b, c, d are variables. ((( a & ~(b) ) & ~(c)) & d) is a term. This can be abbreviated as (a &~b & ~c & d), or a~b~cd.Example: p, q, r, s are variables. (((p & ~(q) ) & r) & ~(s) ) is an alterm. This can be abbreviated as (pV ~q V r V ~s).

Minterms

In the same way that a 2n-row truth table displays the evaluation of a propositional formula for all 2n possible valuesof its variables, n variables produces a 2n-square Karnaugh map (even though we cannot draw it in its full-dimensionalrealization). For example, 3 variables produces 23 = 8 rows and 8 Karnaugh squares; 4 variables produces 16 truth-table rows and 16 squares and therefore 16 minterms. Each Karnaugh-map square and its corresponding truth-tableevaluation represents one minterm.

80 CHAPTER 29. PROPOSITIONAL FORMULA

Any propositional formula can be reduced to the “logical sum” (OR) of the active (i.e. “1"- or “T"-valued) minterms.When in this form the formula is said to be in disjunctive normal form. But even though it is in this form, it is notnecessarily minimized with respect to either the number of terms or the number of literals.In the following table, observe the peculiar numbering of the rows: (0, 1, 3, 2, 6, 7, 5, 4, 0). The first column is thedecimal equivalent of the binary equivalent of the digits “cba”, in other words:

Example: cba2 = c*22 + b*21 + a*20:

cba = (c=1, b=0, a=0) = 1012 = 1*22 + 0*21 + 1*20 = 510

This numbering comes about because as one moves down the table from row to row only one variable at a timechanges its value. Gray code is derived from this notion. This notion can be extended to three and four-dimensionalhypercubes called Hasse diagrams where each corner’s variables change only one at a time as one moves around theedges of the cube. Hasse diagrams (hypercubes) flattened into two dimensions are either Veitch diagrams or Karnaughmaps (these are virtually the same thing).When working with Karnaugh maps one must always keep in mind that the top edge “wrap arounds” to the bottomedge, and the left edge wraps around to the right edge—the Karnaugh diagram is really a three- or four- or n-dimensional flattened object.

29.9.2 Reduction by use of the map method (Veitch, Karnaugh)

Veitch improved the notion of Venn diagrams by converting the circles to abutting squares, and Karnaugh simplifiedthe Veitch diagram by converting the minterms, written in their literal-form (e.g. ~abc~d) into numbers.[21] Themethod proceeds as follows:

(1) Produce the formula’s truth table

Produce the formula’s truth table. Number its rows using the binary-equivalents of the variables (usually just sequen-tially 0 through n-1) for n variables.

Technically, the propositional function has been reduced to its (unminimized) conjunctive normal form:each row has its minterm expression and these can be OR'd to produce the formula in its (unminimized)conjunctive normal form.

Example: ((c & d) V (p & ~(c & (~d)))) = q in conjunctive normal form is:

( (~p & d & c ) V (p & d & c) V (p & d & ~c) V (p & ~d & ~c) ) = q

However, this formula be reduced both in the number of terms (from 4 to 3) and in the total count of its literals (12to 6).

(2) Create the formula’s Karnaugh map

Use the values of the formula (e.g. “p”) found by the truth-table method and place them in their into their respective(associated) Karnaugh squares (these are numbered per the Gray code convention). If values of “d” for “don't care”appear in the table, this adds flexibility during the reduction phase.

(3) Reduce minterms

Minterms of adjacent (abutting) 1-squares (T-squares) can be reduced with respect to the number of their literals,and the number terms also will be reduced in the process. Two abutting squares (2 x 1 horizontal or 1 x 2 vertical,even the edges represent abutting squares) lose one literal, four squares in a 4 x 1 rectangle (horizontal or vertical)or 2 x 2 square (even the four corners represent abutting squares) lose two literals, eight squares in a rectangle lose 3literals, etc. (One seeks out the largest square or rectangles and ignores the smaller squares or rectangles contained

29.10. IMPREDICATIVE PROPOSITIONS 81

totally within it. ) This process continues until all abutting squares are accounted for, at which point the propositionalformula is minimized.For example, squares #3 and #7 abut. These two abutting squares can lose one literal (e.g. “p” from squares #3 and#7), four squares in a rectangle or square lose two literals, eight squares in a rectangle lose 3 literals, etc. (One seeksout the largest square or rectangles.) This process continues until all abutting squares are accounted for, at whichpoint the propositional formula is said to be minimized.Example: The map method usually is done by inspection. The following example expands the algebraic method toshow the “trick” behind the combining of terms on a Karnaugh map:

Minterms #3 and #7 abut, #7 and #6 abut, and #4 and #6 abut (because the table’s edges wrap around).So each of these pairs can be reduced.

Observe that by the Idempotency law (A V A) = A, we can create more terms. Then by association and distributivelaws the variables to disappear can be paired, and then “disappeared” with the Law of contradiction (x & ~x)=0. Thefollowing uses brackets [ and ] only to keep track of the terms; they have no special significance:

• Put the formula in conjunctive normal form with the formula to be reduced:

q = ( (~p & d & c ) V (p & d & c) V (p & d & ~c) V (p & ~d & ~c) ) = ( #3 V#7 V #6 V #4 )

• Idempotency (absorption) [ A V A) = A:

( #3 V [ #7 V #7 ] V [ #6 V #6 ] V #4 )

• Associative law (x V (y V z)) = ( (x V y) V z )

( [ #3 V #7 ] V [ #7 V #6 ] V [ #6 V #4] )[ (~p & d & c ) V (p & d & c) ] V [ (p & d & c) V (p & d & ~c) ] V [ (p & d & ~c)V (p & ~d & ~c) ].

• Distributive law ( x & (y V z) ) = ( (x & y) V (x & z) ) :

( [ (d & c) V (~p & p) ] V [ (p & d) V (~c & c) ] V [ (p & ~c) V (c & ~c) ] )

• Commutative law and law of contradiction (x & ~x) = (~x & x) = 0:

( [ (d & c) V (0) ] V [ (p & d) V (0) ] V [ (p & ~c) V (0) ] )

• Law of identity ( x V 0 ) = x leading to the reduced form of the formula:

q = ( (d & c) V (p & d) V (p & ~c) )

(4) Verify reduction with a truth table

29.10 Impredicative propositions

Given the following examples-as-definitions, what does one make of the subsequent reasoning:

(1) “This sentence is simple.” (2) “This sentence is complex, and it is conjoined by AND.”

Then assign the variable “s” to the left-most sentence “This sentence is simple”. Define “compound” c = “not simple”~s, and assign c = ~s to “This sentence is compound"; assign “j” to “It [this sentence] is conjoined by AND”. Thesecond sentence can be expressed as:

82 CHAPTER 29. PROPOSITIONAL FORMULA

( NOT(s) AND j )

If truth values are to be placed on the sentences c = ~s and j, then all are clearly FALSEHOODS: e.g. “This sentenceis complex” is a FALSEHOOD (it is simple, by definition). So their conjunction (AND) is a falsehood. But whentaken in its assembed form, the sentence a TRUTH.This is an example of the paradoxes that result from an impredicative definition—that is, when an object m has aproperty P, but the object m is defined in terms of property P.[22] The best advice for a rhetorician or one involvedin deductive analysis is avoid impredicative definitions but at the same time be on the lookout for them because theycan indeed create paradoxes. Engineers, on the other hand, put them to work in the form of propositional formulaswith feedback.

29.11 Propositional formula with “feedback”

The notion of a propositional formula appearing as one of its own variables requires a formation rule that allows theassignment of the formula to a variable. In general there is no stipulation (either axiomatic or truth-table systems ofobjects and relations) that forbids this from happening.[23]

The simplest case occurs when an OR formula becomes one its own inputs e.g. p = q. Begin with (p V s) = q, thenlet p = q. Observe that q’s “definition” depends on itself “q” as well as on “s” and the OR connective; this definitionof q is thus impredicative. Either of two conditions can result:[24] oscillation or memory.It helps to think of the formula as a black box. Without knowledge of what is going on “inside” the formula-"box”from the outside it would appear that the output is no longer a function of the inputs alone. That is, sometimesone looks at q and sees 0 and other times 1. To avoid this problem one has to know the state (condition) of the“hidden” variable p inside the box (i.e. the value of q fed back and assigned to p). When this is known the apparentinconsistency goes away.To understand [predict] the behavior of formulas with feedback requires the more sophisticated analysis of sequentialcircuits. Propositional formulas with feedback lead, in their simplest form, to state machines; they also lead tomemories in the form of Turing tapes and counter-machine counters. From combinations of these elements onecan build any sort of bounded computational model (e.g. Turing machines, counter machines, register machines,Macintosh computers, etc.).

29.11.1 Oscillation

In the abstract (ideal) case the simplest oscillating formula is a NOT fed back to itself: ~(~(p=q)) = q. Analysis ofan abstract (ideal) propositional formula in a truth-table reveals an inconsistency for both p=1 and p=0 cases: Whenp=1, q=0, this cannot be because p=q; ditto for when p=0 and q=1.Oscillation with delay: If an delay[25] (ideal or non-ideal) is inserted in the abstract formula between p and q thenp will oscillate between 1 and 0: 101010...101... ad infinitum. If either of the delay and NOT are not abstract (i.e.not ideal), the type of analysis to be used will be dependent upon the exact nature of the objects that make up theoscillator; such things fall outside mathematics and into engineering.Analysis requires a delay to be inserted and then the loop cut between the delay and the input “p”. The delay mustbe viewed as a kind of proposition that has “qd” (q-delayed) as output for “q” as input. This new proposition addsanother column to the truth table. The inconsistency is now between “qd” and “p” as shown in red; two stable statesresulting:

29.11.2 Memory

Without delay, inconsistencies must be eliminated from a truth table analysis. With the notion of “delay”, this con-dition presents itself as a momentary inconsistency between the fed-back output variable q and p = q ₑ ₐ ₑ .A truth table reveals the rows where inconsistencies occur between p = q ₑ ₐ ₑ at the input and q at the output. After“breaking” the feed-back,[26] the truth table construction proceeds in the conventional manner. But afterwards, inevery row the output q is compared to the now-independent input p and any inconsistencies between p and q are noted(i.e. p=0 together with q=1, or p=1 and q=0); when the “line” is “remade” both are rendered impossible by the Law

29.12. HISTORICAL DEVELOPMENT 83

of contradiction ~(p & ~p)). Rows revealing inconsistencies are either considered transient states or just eliminatedas inconsistent and hence “impossible”.

Once-flip memory

About the simplest memory results when the output of an OR feeds back to one of its inputs, in this case output“q” feeds back into “p”. Given that the formula is first evaluated (initialized) with p=0 & q=0, it will “flip” oncewhen “set” by s=1. Thereafter, output “q” will sustain “q” in the “flipped” condition (state q=1). This behavior, nowtime-dependent, is shown by the state diagram to the right of the once-flip.

Flip-flop memory

The next simplest case is the “set-reset” flip-flop shown below the once-flip. Given that r=0 & s=0 and q=0 at theoutset, it is “set” (s=1) in a manner similar to the once-flip. It however has a provision to “reset” q=0 when “r"=1.And additional complication occurs if both set=1 and reset=1. In this formula, the set=1 forces the output q=1 sowhen and if (s=0 & r=1) the flip-flop will be reset. Or, if (s=1 & r=0) the flip-flop will be set. In the abstract (ideal)instance in which s=1 => s=0 & r=1 => r=0 simultaneously, the formula q will be indeterminate (undecidable). Dueto delays in “real” OR, AND and NOT the result will be unknown at the outset but thereafter predicable.

Clocked flip-flop memory

The formula known as “clocked flip-flop” memory (“c” is the “clock” and “d” is the “data”) is given below. It worksas follows: When c = 0 the data d (either 0 or 1) cannot “get through” to affect output q. When c = 1 the data d “getsthrough” and output q “follows” d’s value. When c goes from 1 to 0 the last value of the data remains “trapped” atoutput “q”. As long as c=0, d can change value without causing q to change.Example: ( ( c & d ) V ( p & ( ~( c & ~( d ) ) ) ) = q, but now let p = q:

Example: ( ( c & d ) V ( q & ( ~( c & ~( d ) ) ) ) = q

The state diagram is similar in shape to the flip-flop’s state diagram, but with different labelling on the transitions.

29.12 Historical development

Bertrand Russell (1912:74) lists three laws of thought that derive from Aristotle: (1) The law of identity: “Whateveris, is.”, (2) The law of contradiction: “Nothing cannot both be and not be”, and (3) The law of excluded middle:“Everything must be or not be.”

Example: Here O is an expression about an objects BEING or QUALITY:

(1) Law of Identity: O = O(2) Law of contradiction: ~(O & ~(O))(3) Law of excluded middle: (O V ~(O))

The use of the word “everything” in the law of excluded middle renders Russell’s expression of this law open todebate. If restricted to an expression about BEING or QUALITY with reference to a finite collection of objects (afinite “universe of discourse”) -- the members of which can be investigated one after another for the presence orabsence of the assertion—then the law is considered intuitionistically appropriate. Thus an assertion such as: “Thisobject must either BE or NOT BE (in the collection)", or “This object must either have this QUALITY or NOT havethis QUALITY (relative to the objects in the collection)" is acceptable. See more at Venn diagram.Although a propositional calculus originated with Aristotle, the notion of an algebra applied to propositions had towait until the early 19th century. In an (adverse) reaction to the 2000 year tradition of Aristotle’s syllogisms, JohnLocke's Essay concerning human understanding (1690) used the word semiotics (theory of the use of symbols). By1826 Richard Whately had critically analyzed the syllogistic logic with a sympathy toward Locke’s semiotics. George

84 CHAPTER 29. PROPOSITIONAL FORMULA

Bentham's work (1827) resulted in the notion of “quantification of the predicate” (1827) (nowadays symbolized as∀ ≡ “for all”). A “row” instigated by William Hamilton over a priority dispute with Augustus De Morgan “inspiredGeorge Boole to write up his ideas on logic, and to publish them as MAL [Mathematical Analysis of Logic] in 1847”(Grattin-Guinness and Bornet 1997:xxviii).About his contribution Grattin-Guinness and Bornet comment:

“Boole’s principal single innovation was [the] law [ xn = x ] for logic: it stated that the mental acts ofchoosing the property x and choosing x again and again is the same as choosing x once... As consequenceof it he formed the equations x•(1-x)=0 and x+(1-x)=1 which for him expressed respectively the law ofcontradiction and the law of excluded middle” (p. xxviiff). For Boole “1” was the universe of discourseand “0” was nothing.

Gottlob Frege's massive undertaking (1879) resulted in a formal calculus of propositions, but his symbolism is sodaunting that it had little influence excepting on one person: Bertrand Russell. First as the student of Alfred NorthWhitehead he studied Frege’s work and suggested a (famous and notorious) emendation with respect to it (1904)around the problem of an antinomy that he discovered in Frege’s treatment ( cf Russell’s paradox ). Russell’s workled to a collatoration with Whitehead that, in the year 1912, produced the first volume of Principia Mathematica(PM). It is here that what we consider “modern” propositional logic first appeared. In particular, PM introduces NOTand OR and the assertion symbol ⊦ as primitives. In terms of these notions they define IMPLICATION → ( def.*1.01: ~p V q ), then AND (def. *3.01: ~(~p V ~q) ), then EQUIVALENCE p ←→ q (*4.01: (p → q) & ( q → p )).

• Henry M. Sheffer (1921) and Jean Nicod demonstrate that only one connective, the “stroke” | is sufficient toexpress all propositional formulas.

• Emil Post (1921) develops the truth-table method of analysis in his “Introduction to a general theory of ele-mentary propositions”. He notes Nicod’s stroke | .

• Whitehead and Russell add an introduction to their 1927 re-publication of PM adding, in part, a favorabletreatment of the “stroke”.

Computation and switching logic:

• William Eccles and F. W. Jordan (1919) describe a “trigger relay” made from a vacuum tube.

• George Stibitz (1937) invents the binary adder using mechanical relays. He builds this on his kitchen table.

Example: Given binary bits aᵢ and bᵢ and carry-in ( c_inᵢ), their summation Σᵢ and carry-out (c_outᵢ) are:

• ( ( aᵢ XOR bᵢ ) XOR c_inᵢ )= Σᵢ• ( aᵢ & bᵢ ) V c_inᵢ ) = c_outᵢ;

• Alan Turing builds a multiplier using relays (1937–1938). He has to hand-wind his own relay coils to do this.

• Textbooks about “switching circuits” appear in early 1950s.

• Willard Quine 1952 and 1955, E. W. Veitch 1952, and M. Karnaugh (1953) develop map-methods for simpli-fying propositional functions.

• George H. Mealy (1955) and Edward F. Moore (1956) address the theory of sequential (i.e. switching-circuit)“machines”.

• E. J. McCluskey and H. Shorr develop a method for simplifying propositional (switching) circuits (1962).

29.13. FOOTNOTES 85

29.13 Footnotes[1] Hamilton 1978:1

[2] PM p. 91 eschews “the” because they require a clear-cut “object of sensation"; they stipulate the use of “this”

[3] (italics added) Reichenbach p.80.

[4] Tarski p.54-68. Suppes calls IDENTITY a “further rule of inference” and has a brief development around it; Robbin,Bender and Williamson, and Goodstein introduce the sign and its usage without comment or explanation. Hamilton p. 37employs two signs ≠ and = with respect to the valuation of a formula in a formal calculus. Kleene p. 70 and Hamilton p.52 place it in the predicate calculus, in particular with regards to the arithmetic of natural numbers.

[5] Empiricits eschew the notion of a priori (built-in, born-with) knowledge. “Radical reductionists” such as John Lockeand David Hume “held that every idea must either originate directly in sense experience or else be compounded of ideasthus originating"; quoted from Quine reprinted in 1996 The Emergence of Logical Empriricism, Garland Publishing Inc.http://www.marxists.org/reference/subject/philosophy/works/us/quine.htm

[6] Neural net modelling offers a good mathematical model for a comparator as follows: Given a signal S and a threshold “thr”,subtract “thr” from S and substitute this difference d to a sigmoid function: For large “gains” k, e.g. k=100, 1/( 1 + e-k*(d)) = 1/( 1 + e-k*(S-thr) ) = { ≃0, ≃1 }. For example, if “The door is DOWN” means “The door is less than 50% of the wayup”, then a threshold thr=0.5 corresponding to 0.5*5.0 = +2.50 volts could be applied to a “linear” measuring-device withan output of 0 volts when fully closed and +5.0 volts when fully open.

[7] In actuality the digital 1 and 0 are defined over non-overlapping ranges e.g. { “1” = +5/+0.2/−1.0 volts, 0 = +0.5/−0.2volts }. When a value falls outside the defined range(s) the value becomes “u” -- unknown; e.g. +2.3 would be “u”.

[8] While the notion of logical product is not so peculiar (e.g. 0*0=0, 0*1=0, 1*0=0, 1*1=1), the notion of (1+1=1 is peculiar;in fact (a "+" b) = (a + (b - a*b)) where "+" is the “logical sum” but + and - are the true arithmetic counterparts. Occasionallyall four notions do appear in a formula: A AND B = 1/2*( A plus B minus ( A XOR B ) ] (cf p. 146 in John Wakerly 1978,Error Detecting Codes, Self-Checking Circuits and Applications, North-Holland, New York, ISBN 0-444-00259-6 pbk.)

[9] A careful look at its Karnaugh map shows that IF...THEN...ELSE can also be expressed, in a rather round-about way, interms of two exclusive-ORs: ( (b AND (c XOR a)) OR (a AND (c XOR b)) ) = d.

[10] Robbin p. 3.

[11] Rosenbloom p. 30 and p. 54ff discusses this problem of implication at some length. Most philosophers and mathematiciansjust accept the material definition as given above. But some do not, including the intuitionists; they consider it a form ofthe law of excluded middle misapplied.

[12] Indeed, exhaustive selection between alternatives --mutual exclusion -- is required by the definition that Kleene gives theCASE operator (Kleene 1952229)

[13] The use of quote marks around the expressions is not accidental. Tarski comments on the use of quotes in his “18. Identityof things and identity of their designations; use of quotation marks” p. 58ff.

[14] Hamilton p. 37. Bender and Williamson p. 29 state “In what follows, we'll replace “equals” with the symbol " ⇔ "(equivalence) which is usually used in logic. We use the more familiar " = " for assigning meaning and values.”

[15] Reichenbach p. 20-22 and follows the conventions of PM. The symbol =D is in the metalanguage and is not a formalsymbol with the following meaning: “by symbol ' s ' is to have the same meaning as the formula '(c & d)' ".

[16] Rosenbloom 1950:32. Kleene 1952:73-74 ranks all 11 symbols.

[17] cf Minsky 1967:75, section 4.2.3 “The method of parenthesis counting”. Minsky presents a state machine that will do thejob, and by use of induction (recursive definition) Minsky proves the “method” and presents a theorem as the result. Afully generalized “parenthesis grammar” requires an infinite state machine (e.g. a Turing machine) to do the counting.

[18] Robbin p. 7

[19] cf Reichenbach p. 68 for a more involved discussion: “If the inference is valid and the premises are true, the inference iscalled conclusive.

[20] As well as the first three, Hamilton pp.19-22 discusses logics built from only | (NAND), and ↓ (NOR).

[21] Wickes 1967:36ff. Wickes offers a good example of 8 of the 2 x 4 (3-variable maps) and 16 of the 4 x 4 (4-variable)maps. As an arbitrary 3-variable map could represent any one of 28=256 2x4 maps, and an arbitrary 4-variable map couldrepresent any one of 216 = 65,536 different formula-evaluations, writing down every one is infeasible.

86 CHAPTER 29. PROPOSITIONAL FORMULA

[22] This definition is given by Stephen Kleene. Both Kurt Gödel and Kleene believed that the classical paradoxes are uniformlyexamples of this sort of definition. But Kleene went on to assert that the problem has not been solved satisfactorily andimpredicative definitions can be found in analysis. He gives as example the definition of the least upper bound (l.u.b) u ofM. Given a Dedekind cut of the number line C and the two parts into which the number line is cut, i.e. M and (C -M),l.u.b. = u is defined in terms of the notion M, whereas M is defined in terms of C. Thus the definition of u, an elementof C, is defined in terms of the totality C and this makes its definition impredicative. Kleene asserts that attempts to arguethis away can be used to uphold the impredicative definitions in the paradoxes.(Kleene 1952:43).

[23] McCluskey comments that “it could be argued that the analysis is still incomplete because the word statement “The outputsare equal to the previous values of the inputs” has not been obtained"; he goes on to dismiss such worries because “Englishis not a formal language in a mathematical sense, [and] it is not really possible to have a formal procedure for obtainingword statements” (p. 185).

[24] More precisely, given enough “loop gain”, either oscillation or memory will occur (cf McCluskey p. 191-2). In abstract(idealized) mathematical systems adequate loop gain is not a problem.

[25] The notion of delay and the principle of local causation as caused ultimately by the speed of light appears in Robin Gandy(1980), “Church’s thesis and Principles for Mechanisms”, in J. Barwise, H. J. Keisler and K. Kunen, eds., The KleeneSymposium, North-Holland Publishing Company (1980) 123-148. Gandy considered this to be the most important of hisprinciples: “Contemporary physics rejects the possibility of instantaneous action at a distance” (p. 135). Gandy was AlanTuring's student and close friend.

[26] McKlusky p. 194-5 discusses “breaking the loop” and inserts “amplifiers” to do this; Wickes (p. 118-121) discuss insertingdelays. McCluskey p. 195ff discusses the problem of “races” caused by delays.

29.14 References

• Bender, Edward A. andWilliamson, S. Gill, 2005, A Short Course in Discrete Mathematics, Dover Publications,Mineola NY, ISBN 0-486-43946-1. This text is used in a “lower division two-quarter [computer science]course” at UC San Diego.

• Enderton, H. B., 2002, AMathematical Introduction to Logic. Harcourt/Academic Press. ISBN 0-12-238452-0

• Goodstein, R. L., (Pergamon Press 1963), 1966, (Dover edition 2007), Boolean Algebra, Dover Publications,Inc. Minola, New York, ISBN 0-486-45894-6. Emphasis on the notion of “algebra of classes” with set-theoretic symbols such as ∩, ∪, ' (NOT), ⊂ (IMPLIES). Later Goldstein replaces these with &, ∨, ¬, → (re-spectively) in his treatment of “Sentence Logic” pp. 76–93.

• Ivor Grattan-Guinness and Gérard Bornet 1997, George Boole: Selected Manuscripts on Logic and its Philoso-phy, Birkhäuser Verlag, Basil, ISBN 978-0-8176-5456-6 (Boston).

• A. G. Hamilton 1978, Logic for Mathematicians, Cambridge University Press, Cambridge UK, ISBN 0-521-21838-1.

• E. J. McCluskey 1965, Introduction to the Theory of Switching Circuits, McGraw-Hill Book Company, NewYork. No ISBN. Library of Congress Catalog Card Number 65-17394. McCluskey was a student of WillardQuine and developed some notable theorems with Quine and on his own. For those interested in the history,the book contains a wealth of references.

• Marvin L. Minsky 1967, Computation: Finite and Infinite Machines, Prentice-Hall, Inc, Englewood Cliffs, N.J..No ISBN. Library of Congress Catalog Card Number 67-12342. Useful especially for computability, plus goodsources.

• Paul C. Rosenbloom 1950, Dover edition 2005, The Elements of Mathematical Logic, Dover Publications, Inc.,Mineola, New York, ISBN 0-486-44617-4.

• Joel W. Robbin 1969, 1997, Mathematical Logic: A First Course, Dover Publications, Inc., Mineola, NewYork, ISBN 0-486-45018-X (pbk.).

• Patrick Suppes 1957 (1999Dover edition), Introduction to Logic, Dover Publications, Inc., Mineola, NewYork.ISBN 0-486-40687-3 (pbk.). This book is in print and readily available.

29.14. REFERENCES 87

• On his page 204 in a footnote he references his set of axioms to E. V. Huntington, “Sets of IndependentPostulates for the Algebra of Logic”, Transactions of the American Mathematical Society, Vol. 5 91904) pp.288-309.

• Alfred Tarski 1941 (1995 Dover edition), Introduction to Logic and to the Methodology of Deductive Sciences,Dover Publications, Inc., Mineola, New York. ISBN 0-486-28462-X (pbk.). This book is in print and readilyavailable.

• Jean van Heijenoort 1967, 3rd printing with emendations 1976, From Frege to Gödel: A Source Book in Mathe-matical Logic, 1879-1931, Harvard University Press, Cambridge, Massachusetts. ISBN 0-674-32449-8 (pbk.)Translation/reprints of Frege (1879), Russell’s letter to Frege (1902) and Frege’s letter to Russell (1902),Richard’s paradox (1905), Post (1921) can be found here.

• Alfred North Whitehead and Bertrand Russell 1927 2nd edition, paperback edition to *53 1962, PrincipiaMathematica, Cambridge University Press, no ISBN. In the years between the first edition of 1912 and the 2ndedition of 1927, H. M. Sheffer 1921 and M. Jean Nicod (no year cited) brought to Russell’s and Whitehead’sattention that what they considered their primitive propositions (connectives) could be reduced to a single |,nowadays known as the “stroke” or NAND (NOT-AND, NEITHER ... NOR...). Russell-Whitehead discussthis in their “Introduction to the Second Edition” and makes the definitions as discussed above.

• William E. Wickes 1968, Logic Design with Integrated Circuits, John Wiley & Sons, Inc., New York. NoISBN. Library of Congress Catalog Card Number: 68-21185. Tight presentation of engineering’s analysis andsynthesis methods, references McCluskey 1965. Unlike Suppes, Wickes’ presentation of “Boolean algebra”starts with a set of postulates of a truth-table nature and then derives the customary theorems of them (p.18ff).

88 CHAPTER 29. PROPOSITIONAL FORMULA

A truth table will contain 2n rows, where n is the number of variables (e.g. three variables “p”, “d”, “c” produce 23 rows). Eachrow represents a minterm. Each minterm can be found on the Hasse diagram, on the Veitch diagram, and on the Karnaugh map.(The evaluations of “p” shown in the truth table are not shown in the Hasse, Veitch and Karnaugh diagrams; these are shown in theKarnaugh map of the following section.)

29.14. REFERENCES 89

Steps in the reduction using a Karnaugh map. The final result is the OR (logical “sum”) of the three reduced terms.

90 CHAPTER 29. PROPOSITIONAL FORMULA

29.14. REFERENCES 91

About the simplest memory results when the output of an OR feeds back to one of its inputs, in this case output “q” feeding backinto “p”. The next simplest is the “flip-flop” shown below the once-flip. Analysis of these sorts of formulas can be done by eithercutting the feedback path(s) or inserting (ideal) delay in the path. A cut path and an assumption that no delay occurs anywhere in the“circuit” results in inconsistencies for some of the total states (combination of inputs and outputs, e.g. (p=0, s=1, r=1) results in aninconsistency). When delay is present these inconsistencies are merely transient and expire when the delay(s) expire. The drawingson the right are called state diagrams.

92 CHAPTER 29. PROPOSITIONAL FORMULA

A “clocked flip-flop” memory (“c” is the “clock” and “d” is the “data”). The data can change at any time when clock c=0; when clockc=1 the output q “tracks” the value of data d. When c goes from 1 to 0 it “traps” d = q’s value and this continues to appear at q nomatter what d does (as long as c remains 0).

Chapter 30

Proxy statement

A proxy statement is a statement required of a firm when soliciting shareholder votes. This statement is filed inadvance of the annual meeting. The firm needs to file a proxy statement, otherwise known as a Form DEF 14A(Definitive Proxy Statement), with the U.S. Securities and Exchange Commission. This statement is useful in assess-ing how management is paid and potential conflict-of-interest issues with auditors. The statement includes:

• Voting procedure and information.

• Background information about the company’s nominated directors including relevant history in the companyor industry, positions on other corporate boards, and potential conflicts in interest.

• Board compensation.

• Executive compensation, including salary, bonus, non-equity compensation, stock awards, options, and deferredcompensation. Also, information is included about perks such as personal use of company aircraft, travel, andtax gross-ups. Many companies will also include pre-determined payout packages for if an executive leavesthe company.

• Who is on the audit committee, as well as a breakdown of audit and non-audit fees paid to the auditor.

SEC proxy rules: The term “proxy statement” means the statement required by Section 240.14a-3(a) whether or notcontained in a single document.In many cases, shareholder votes - particularly institutional shareholder votes - are determined by proxy firms whichadvise the shareholders...Traditionally, broker-dealers have been permitted to vote for “routine” proposals on behalf of their shareholders if theshareholders do not return the proxy statement. This has been controversial, and in 2006 the NYSE Proxy WorkingGroup recommended that the rules be modified so that uncontested director elections were not considered routine.[1]The SEC approved the rule on July 1, 2009.[2]

In July 2010, the SEC announced that it was seeking public comment on the efficiency of the proxy system.[3]

There has been some controversy over “proxy access” which is a method to allow shareholders to nominate candidateswhich appear on the proxy statement. Currently, only the nominating board can place candidates on the proxystatement. The United States Dodd–Frank Wall Street Reform and Consumer Protection Act specifically allowed theSEC to rule on this issue. In 2010, the SEC passed a rule which allowed certain shareholders to place candidates onthe proxy statement,;[4] however, the rule was struck down by the United States Court of Appeals for the District ofColumbia Circuit in 2011.[5]

30.1 References

[1] NYSEWorkingGroup. (June 5, 2006). REPORTANDRECOMMENDATIONSOFTHEPROXYWORKINGGROUPTO THE NEW YORK STOCK EXCHANGE.

93

94 CHAPTER 30. PROXY STATEMENT

[2] Order Approving Proposed Rule Change, as modified byAmendment No. 4, to AmendNYSERule 452 and CorrespondingListed CompanyManual Section 402.08 to Eliminate Broker Discretionary Voting for the Election of Directors, Except forCompanies Registered under the Investment CompanyAct of 1940, and to Codify Two Previously Published Interpretationsthat Do Not Permit Broker Discretionary Voting for Material Amendments to Investment Advisory Contracts with anInvestment Company. Release No. 34-60215; File No. SR-NYSE-2006-92. Discussed by the NY Times Dealbook at:Curbs on Broker Voting Are Approved.

[3] SEC. SEC Votes to Seek Public Comment on U.S. Proxy System.

[4] SEC Adopts New Measures to Facilitate Director Nominations by Shareholders.

[5] What Would Proxy Access Look Like if Done Right?. Corpgov.net.

30.2 External links• Recently filed proxy statements

• List of items required in proxy statements in Schedule 14A (SEC) (PDF) - note: the SEC published its finalrules governing disclosure on August 11, 2006. This Schedule does not reflect the additions and changes.

• Wall Street Journal primer on how to read a proxy statement

Chapter 31

Risk and Safety Statements

Risk and Safety Statements, also known as R/S statements, R/S numbers, R/S phrases, and R/S sentences,is a system of hazard codes and phrases for labeling dangerous chemicals and compounds. The R/S statement ofa compound consists of a risk part (R) and a safety part (S), each followed by a combination of numbers. Eachnumber corresponds to a phrase. The phrase corresponding to the letter/number combination has the same meaningin different languages—see 'languages’ in the menu on the left.In 2015, the risk and safety statements will be replaced by hazard statements and precautionary statements in thecourse of harmonising classification, labelling and packaging of chemicals by introduction of the UN Globally Har-monized System of Classification and Labelling of Chemicals (GHS).

31.1 Example

The R/S statement code for fuming hydrochloric acid (37%):

R: 34-37 S: 26-36-45.

The corresponding English language phrases:

Risks

R: 34 Causes burnsR: 37 Irritating to the respiratory system.

Safety

S: 26 In case of contact with eyes, rinse immediately with plenty of water and seek medicaladvice.S: 36Wear suitable protective clothing.S: 45 In case of accident or if you feel unwell, seek medical advice immediately (show labelwhere possible).

Dashes separate the phrase numbers. They are not to be confused with range indicators.

Example: R: 34-37 Causes burns, irritating to the respiratory system.

Slashes indicate fixed combinations of single phrases.

Example: R: 36/37/38 Irritating to eyes, respiratory system and skin.

More detailed hazard and safety information can be found in the material safety data sheets (MSDS) of a compound.

95

96 CHAPTER 31. RISK AND SAFETY STATEMENTS

31.2 See also• MSDS

• List of R-phrases

• List of S-phrases

31.3 External links• European Commission Directive 2001/59/EC of 6August 2001 adapting to technical progress for the 28th timeCouncil Directive 67/548/EEC on the approximation of the laws, regulations and administrative provisionsrelating to the classification, packaging and labelling of dangerous substances

• Chemical Risk & Safety Phrases in 23 European Languages

• List of R/S statements at Sigma-Aldrich

• Australian hazardous substances information system

• The EU Joint Research Centre’s Institute for Health & Consumer Protection (IHCP)

Chapter 32

Safety statement

Safety statement is the name given to the document that outlines how a company manages their health and safety inthe Republic of Ireland, based upon the Safety, Health and Welfare at Work Act, 2005. The requirement to have awritten safety statement is outlined in Section 20 of the above Act, although it was also a requirement in the original(but now revoked) Act of 1989. The document is primarily based on the risk assessment of workplace hazards.

32.1 Summary

The safety statement document details how the organisation identifies, assesses and controls risks in the workplace andwhat arrangements are in place to achieve this. It is a legal requirement for all employers, regardless of the number ofemployees. The exceptions are if there are three or fewer employed by a company or self-employed person and thesector or work activity carried out is covered under a Code of Practice prepared by the Health and Safety Authority,then that is considered to be sufficient (although in this case, there is still a requirement to carry out risk assessmentsfor the work activity).

32.2 Background

It is a legal requirement in nearly all countries for employers to ensure the health and safety of employees and otherswho may be affected by an organisations work activities (such as members of the public, contractors, site visitorsor guests etc.). This requirement is normally explicitly stated in the health and safety legislation that applies toeach country. There are usually a number of basic criteria that need to be met by employers in order to achievethis goal (such as adequate training, suitable instruction and supervision, the provision of information, provision ofemergency response plans and the assessment of risk amongst others). In addition, there is normally a requirementfor an organisation to document in writing how it will ensure that it meets the minimum legal health and safetyrequirements.

32.3 References

32.4 External links• Irish Health and Safety Authority

97

Chapter 33

Scope statement

Scope statements may take many forms depending on the type of project being implemented and the nature of theorganization. The scope statement details the project deliverables and describes the major objectives. The objectivesshould include measurable success criteria for the project.A scope statement should be written before the statement of work and it should capture, in very broad terms, theproduct of the project, for example, “developing a software based system to capture and track orders for software.” Ascope statement should also include the list of users using the product, as well as the features in the resulting product.[1]

As a baseline scope statements should contain:

• The project charter

• The project owner, sponsors, and stakeholders

• The problem statement

• The project goals and objectives

• The project requirements

• The project deliverables

• The project non-goals (what is out of scope)

• Milestones

• Cost estimates

In more project oriented organizations the scope statement may also contain these and other sections:

• Project scope management plan

• Approved change requests

• Project assumptions and risks

• Project acceptance criteria

33.1 References[1] Nielsen, David How to Write the Project Statement of Work (SOW) – Retrieved March 22, 2010

98

Chapter 34

Sentence (linguistics)

A sentence is a linguistic unit consisting of one or more words that are grammatically linked. A sentence can includewords grouped meaningfully to express a statement, question, exclamation, request, command or suggestion.[1] Asentence is a set of words that in principle tells a complete thought (although it may make little sense taken in isolationout of context); thus it may be a simple phrase, but it conveys enough meaning to imply a clause, even if it is notexplicit. For example, “Two” as a sentence (in answer to the question “How many were there?") implies the clause“There were two”. Typically a sentence contains a subject and predicate. A sentence can also be defined purelyin orthographic terms, as a group of words starting with a capital letter and ending in a full stop.[2] (However, thisdefinition is useless for unwritten languages, or languages written in a system that does not employ both devices,or precise analogues thereof.) For instance, the opening of Charles Dickens's novel Bleak House begins with thefollowing three sentences:

London. Michaelmas term lately over, and the Lord Chancellor sitting in Lincoln’s Inn Hall. ImplacableNovember weather.

The first sentence involves one word, a proper noun. The second sentence has only a non-finite verb (although usingthe definition given above, e.g. “Chancellor sitting in Lincoln’s Inn Hall.” would be a sentence by itself). The third isa single nominal group. Only an orthographic definition encompasses this variation.In the teaching of writing skills (composition skills), students are generally required to express (rather than imply)the elements of a sentence, leading to the schoolbook definition of a sentence as one that must [explicitly] includea subject and a verb. For example, in second-language acquisition, teachers often reject one-word answers that onlyimply a clause, commanding the student to “give me a complete sentence”, by which they mean an explicit one.As with all language expressions, sentences might contain function and content words and contain properties such ascharacteristic intonation and timing patterns.Sentences are generally characterized in most languages by the inclusion of a finite verb, e.g. "The quick brown foxjumps over the lazy dog".

34.1 Components

34.1.1 Clauses

A clause typically contains at least a subject noun phrase and a finite verb. Although the subject is usually a nounphrase, other kinds of phrases (such as gerund phrases) work as well, and some languages allow subjects to be omitted.There are two types of clauses: independent and subordinate (dependent). An independent clause is a completesentence in itself, although it may not express a complete thought: for example, They did it.. A subordinate clause isnot a complete sentence: for example, because I have no friends. See also copula for the consequences of the verb tobe on the theory of sentence structure.A simple complete sentence consists of a single clause. Other complete sentences consist of two or more clauses (seebelow).

99

100 CHAPTER 34. SENTENCE (LINGUISTICS)

34.2 Classification

34.2.1 By structure

One traditional scheme for classifying English sentences is by clause structure, the number and types of clauses inthe sentence with finite verbs.

• A simple sentence consists of a single independent clause with no dependent clauses.

• A compound sentence consists of multiple independent clauses with no dependent clauses. These clauses arejoined together using conjunctions, punctuation, or both.

• A complex sentence consists of one independent clause and at least one dependent clause.

• A compound–complex sentence (or complex–compound sentence) consists of multiple independent clauses, atleast one of which has at least one dependent clause.

34.2.2 By purpose

Sentences can also be classified based on their purpose:

• A declarative sentence or declaration, the most common type, commonly makes a statement: “I have to go towork.”

• An interrogative sentence or question is commonly used to request information—"Do I have to go to work?"—but sometimes not; see rhetorical question.

• An exclamatory sentence or exclamation is generally a more emphatic form of statement expressing emotion:“I have to go to work!"

• An imperative sentence or command tells someone to do something (and if done strongly may be consideredboth imperative and exclamatory): “Go to work.” or “Go to work!"

34.2.3 Major and minor sentences

A major sentence is a regular sentence; it has a subject and a predicate, e.g. “I have a ball.”. In this sentence, onecan change the persons, e.g. “We have a ball.”. However, a minor sentence is an irregular type of sentence that doesnot contain a main clause, e.g. “Mary!", “Precisely so.”, “Next Tuesday evening after it gets dark.”. Other examplesof minor sentences are headings (e.g. the heading of this entry), stereotyped expressions (“Hello!"), emotional ex-pressions (“Wow!"), proverbs, etc. These can also include nominal sentences like “The more, the merrier”. Thesemostly omit a main verb for the sake of conciseness, but may also do so in order to intensify the meaning around thenouns.[3]

Sentences that comprise a single word are called word sentences, and the words themselves sentence words.[4]

34.3 Length

After a slump in interest, sentence length came to be studied in the 1980s, mostly “with respect to other syntacticphenomena”.[5]

One definition of the average sentence length of a prose passage is the ratio of the number of words to the number ofsentences.[6] The textbookMathematical linguistics, by András Kornai, suggests that in “journalistic prose the mediansentence length is above 15 words”.[7] The average length of a sentence generally serves as a measure of sentencedifficulty or complexity.[8] In general, as the average sentence length increases, the complexity of the sentences alsoincreases.[9]

Another definition of “sentence length” is the number of clauses in the sentence, whereas the “clause length” is thenumber of phones in the clause.[10]

34.4. SEE ALSO 101

Research by Erik Schils and Pieter de Haan by sampling five texts showed that two adjacent sentences are more likelyto have similar lengths than two non-adjacent sentences, and almost certainly have similar length when in a work offiction. This countered the theory that “authors may aim at an alternation of long and short sentences”.[11] Sentencelength, as well as word difficulty, are both factors in the readability of a sentence.[12] However, other factors, such asthe presence of conjunctions, have been said to “facilitate comprehension considerably”.[13]

34.4 See also• Grammatical polarity

• Inflectional phrase

• Periodic sentence

• Sentence arrangement

• Sentence function

• T-unit

34.5 References[1] "'Sentence' – Definitions from Dictionary.com”. Dictionary.com. Retrieved 2008-05-23.

[2] Halliday, M.A.K. and Matthiessen, C.M.I.M. 2004. An Introduction to Functional Grammar. Arnold: p. 6.

[3] Exploring Language: Sentences

[4] Jan Noordegraaf (2001). “J. M. Hoogvliet as a teacher and theoretician”. In Marcel Bax, C. Jan-Wouter Zwart, and A. J.van Essen. Reflections on Language and Language Learning. John Benjamins B.V. p. 24. ISBN 90-272-2584-2.

[5] Těšitelová, Marie (1992). Quantitative Linguistics. p. 126. Retrieved December 15, 2011.

[6] “Calculate Average Sentence Length”. Linguistics Forum. Jun 23, 2011. Retrieved December 12, 2011.

[7] Kornai, András. Mathematical linguistics. p. 188. Retrieved December 15, 2011.

[8] Perera, Katherine. The assessment of sentence difficulty. p. 108. Retrieved December 15, 2011.

[9] Troia, Gary A. Instruction and assessment for struggling writers: evidence-based practices. p. 370. Retrieved December15, 2011.

[10] Reinhard Köhler, Gabriel Altmann, Raĭmond Genrikhovich Piotrovskiĭ (2005). Quantitative Linguistics. p. 352. RetrievedDecember 15, 2011. (Caption) Table 26.3: Sentence length (expressed by the number of clauses) and clause length(expressed by the number of phones) in a Turkish text

[11] Erik Schils, Pieter de Haan (1993). “Characteristics of Sentence Length in Running Text”. Oxford University Press.Retrieved December 12, 2011.

[12] Perera, Katherine. The assessment of sentence difficulty. p. 108. Retrieved December 15, 2011.

[13] Fries, Udo. Sentence Length, Sentence Complexity, and the Noun Phrase in 18th-Century News Publications. p. 21. RetrievedDecember 15, 2011.

Chapter 35

Simple non-inferential passage

A simple non-inferential passage is a type of nonargument characterized by the lack of a claim that anything is beingproved.[1] Simple non-inferential passages include warnings, pieces of advice, statements of belief or opinion, looselyassociated statements, and reports. Simple non-inferential passages are nonarguments because while the statementsinvolved may be premises, conclusions or both, the statements do not serve to infer a conclusion or support oneanother. This is distinct from a logical fallacy, which indicates an error in reasoning.

35.1 Types

35.1.1 Warnings

A warning is a type of simple non-inferential passage that serves to alert a person to any sort of potential danger.This can be as simple as a road sign indicating falling rock or a janitorial sign indicating a wet, slippery floor.

35.1.2 Piece of advice

Apiece of advice is a type of simple non-inferential passage that recommends some future action or course of conduct.A mechanic recommending regular oil changes or a doctor recommending that a patient refrain from smoking areexamples of pieces of advice.

35.1.3 Statements of belief or opinion

Main article: Opinion

A statement of belief or opinion is a type of simple non-inferential passage containing an expression of belief oropinion lacking an inferential claim. In A concise introduction to logic, Hurley uses the following example to illustrate:

We believe that our company must develop and produce outstanding products that will perform agreat service or fulfill a need for our customers. We believe that our business must be run at an adequateprofit and that the services and products we offer must be better than those offered by competitors— A concise introduction to logic, 10th edition

35.1.4 Loosely associated statements

Main article: Loosely associated statements

102

35.2. REFERENCES 103

A loosely associated statement is a type of simple non-inferential passage wherein statements about a general subjectare juxtaposed but make no inferential claim.[1] As a rhetorical device, loosely associated statements may be intendedby the speaker to infer a claim or conclusion, but because they lack a coherent logical structure any such interpretationis subjective as loosely associated statements prove nothing and attempt no obvious conclusion.[2] Loosely associatedstatements can be said to serve no obvious purpose, such as illustration or explanation.[3]

35.1.5 Reports

A report is a type of simple non-inferential passage wherein the statements serve to convey knowledge.

Even though more of the world is immunized than ever before, many old diseases have proven quoteresilient in the face of changing population and environmental conditions, especially in the developingworld. New diseases, such as AIDS, have taken their toll in both the North and the South— Steven L. Spiedel, World politics in a new era

The above is considered a report because it informs the reader without making any sort of claim, ethical or otherwise.However, the statements being made could be seen as a set of premises, and with the addition of a conclusion it wouldbe considered an argument.

35.2 References[1] Hurley, Patrick J. (2008). A Concise Introduction to Logic 10th ed. Thompson Wadsworth. p. 16. ISBN 0-495-50383-5.

[2] “The logic of arguments”. Retrieved April 28, 2012.

[3] “NONargument - Loosely associated statements”. Retrieved April 28, 2012.

Chapter 36

Special weather statement

A Special weather statement is a form of weather advisory. Special weather statements are issued by the NationalWeather Service of the United States (the NWS) and the Meteorological Service of Canada (the MSC). There are noset criteria for special weather statements in either country.

36.1 United States

A special weather statement may be issued by the NWS for hazards that have not yet reached warning or advisorystatus or that do not have a specific code of their own, such as widespread funnel clouds. They are also occasionallyused to clear counties from severe weather watches.[1] A common form of special weather statement is a significantweather alert. Occasionally special weather statements appear as heat advisories. An EAS activation can and may berequested on very rare occasions.

36.1.1 Examples of an American special weather statement

SPECIALWEATHER STATEMENTNATIONALWEATHER SERVICE BISMARCKND 1022 PMCDTWEDSEP 30 2015 NDZ012-021-010400- MCHENRY ND-MCLEAN ND- 1022 PM CDT WED SEP 30 2015 ...ASTRONGTHUNDERSTORMWILLAFFECTNORTHEASTERNMCLEANANDSOUTHCENTRALMCHENRYCOUNTIES... AT 1021 PM CDT...DOPPLER RADAR WAS TRACKING A STRONG THUNDERSTORM11 MILES SOUTHEAST OF BENEDICT...OR 25 MILES EAST OF GARRISON...MOVING NORTHEASTAT 15 MPH. PENNY SIZE HAIL WILL BE POSSIBLE WITH THIS STORM. LOCATIONS IMPACTED IN-CLUDE... BUTTE...RUSO AND STRAWBERRY LAKE. FREQUENT CLOUD TO GROUND LIGHTNINGIS OCCURRING WITH THIS STORM. LIGHTNING CAN STRIKE 15 MILES AWAY FROM A THUNDER-STORM. SEEK A SAFE SHELTER INSIDE A BUILDING OR VEHICLE. && LAT...LON 4785 10054 478510058 4770 10059 4764 10092 4778 10105 4793 10073 TIME...MOT...LOC 0321Z 244DEG 14KT 4774 10088 $$ABELING SPECIAL WEATHER STATEMENT NATIONAL WEATHER SERVICE LITTLE ROCK AR 426PM CDT WED APR 27 2011 ARZ003>007-012>016-021>025-030>034-037>047-052>057-062>069-280015-ARKANSAS-BAXTER-BOONE-BRADLEY-CALHOUN-CLARK-CLEBURNE-CLEVELAND-CONWAY-DALLAS-DESHA-DREW-FAULKNER-FULTON-GARLAND-GRANT-HOTSPRING- INDEPENDENCE-IZARD-JACKSON-JEFFERSON-JOHNSON-LINCOLN-LOGAN-LONOKE-MARION-MONROE-MONTGOMERY-NEWTON-OUACHITA-PERRY-PIKE-POLK-POPE- PRAIRIE-PULASKI-SALINE-SCOTT-SEARCY-SHARP-STONE-VANBUREN-WHITE-WOODRUFF-YELL- INCLUDINGTHECITIESOF...ALTHEIMER...AMITY...ARKADELPHIA... ARKANSASCITY...ASHFLAT...AUGUSTA...AVILLA...BATESVILLE... BAUXITE...BEARDEN...BEEBRANCH...BEEBE...BENTON...BISMARCK...BONNERDALE...BOONEVILLE...BRINKLEY...BRYANT...BULLSHOALS... CABOT...CALICOROCK...CAMDEN...CAVECITY...CENTER RIDGE... CHIDESTER...CLARENDON...CLARKSVILLE...CLINTON...CONWAY... COT-TONPLANT...DANVILLE...DARDANELLE...DEVALLSBLUFF...DEWITT... DEER...DESARC...DONALDSON...DRASCO...DUMAS...ELPASO...ENGLAND... FAIRFIELDBAY...FLIPPIN...FORDYCE...FOURCHE JUNCTION... GASSVILLE...GEORGETOWN...GILLETT...GLENWOOD...GOULD...GRADY...GRAVELLY...GREENBRIER...GREERSFERRY...GURDON...HAMPTON...HARDY... HARRISON...HASKELL...HAZEN...HEBERSPRINGS...HECTOR...HERMITAGE... HORSESHOEBEND...HOTSPRINGS...HOTSPRINGSVILLAGE...HOUSTON...HUMNOKE...HUMPHREY...JACKSONVILLE...JASPER...JESSIEVILLE... KINGSLAND...LACEY...LEADHILL...LEOLA...LESLIE...LITTLEROCK... LONOKE...MALVERN...MAMMOTHSPRING...MARSHALL...MAUMELLE... MAYFLOWER...MCCRORY...MCGEHEE...MELBOURNE...MENA...MONTICELLO...

104

36.2. CANADA 105

MOROBAY...MORRILTON...MOUNT IDA...MOUNTMAGAZINE... MOUNTAINHOME...MOUNTAINVIEW...MURFREESBORO...NEWPORT...NORFORK...NORMAN...NORTHLITTLEROCK...OILTROUGH...OKOLONA... OLA...OMAHA...OXFORD...OZONE...PARIS...PARON...PELSOR...PERRYVILLE...PINEBLUFF...PINERIDGE...PLEASANTPLAINS...POYEN... PRATTSVILLE...REDFIELD...RISON...ROHWER...ROSEBUD... RUSSELLVILLE...SALEM...SEARCY...SHERIDAN...SHERWOOD...STARCITY... STEPHENS...STUTTGART...SUMMIT...SWIFTON...THORNTON...TUCKERMAN...VILONIA...VIOLA...WALDRON...WARREN...WESTERNGROVE...WICKES... WRIGHTSVILLE...YCITY...YELLVILLE426 PM CDTWED APR 27 2011 ...COLD AIR FUNNELS POSSIBLE THIS AFTERNOON... THE ENVIRON-MENT THIS AFTERNOONHAS BECOME FAVORABLE FOR COLDAIR FUNNEL CLOUDS AS ANAREAOF UPPER LEVEL LOW PRESSURE MOVES THROUGH. COLD AIR FUNNELS RARELY REACH THEGROUND AND GENERALLY POSE LITTLE THREAT. $$

36.2 Canada

Special weather statements are issued by theMSC for weather events that are unusual or those that cause general incon-venience or public concern and cannot adequately be described in a weather forecast. These may include widespreadevents such as Arctic outflows, Alberta clippers, coastal fog banks, areas of possible thunderstorm development, andstrong winds such as chinooks and les suetes winds. They are written in a free style and may also reflect warnings ineffect near the United States border.

36.2.1 Example of a Canadian special weather statement

WOCN17 CWHX 141915 Special weather statement updated by Environment Canada at 4:45 PM NDT Tuesday14 May 2013. --------------------------------------------------------------------- Special weather statement for: ChurchillFalls and vicinity Churchill Valley Upper LakeMelville Eagle River Nain and vicinity Hopedale and vicinity Postville- Makkovik Rigolet and vicinity Cartwright to Black Tickle. Snow, ice pellets and freezing rain expected onWednes-day. --------------------------------------------------------------------- ==discussion== A low pressure system south ofNova Scotia will move northward to lie south of Goose Bay by Wednesday evening. Rain associated with this systemwill develop over central regions overnight tonight then change to snow or ice pellets with a risk of freezing rain beforemorning. In the Rigolet area, snow is expected to mix with or change to ice pellets, however an extended period offreezing rain is possible. Precipitation is expected to remain primarily as snow from Nain to Makkovik and in theChurchill Falls area. Though snowfall amounts are not expected to reach warning criteria at this time. The public isadvised to monitor future forecasts and warnings as warnings may be required or extended. Please monitor the latestforecasts and warnings from Environment Canada at www.weatheroffice.gc.ca End

36.3 See also• Severe weather terminology (United States)

• Severe weather terminology (Canada)

36.4 References[1] Special Weather Statement

36.5 External links• United States National Weather Service

• Canadian Weather Warning Terminology

Chapter 37

Statement (computer science)

In computer programming, a statement is the smallest standalone element of an imperative programming languagethat expresses some action to be carried out. It is an instruction written in a high-level language that commands thecomputer to perform a specified action.[1] A program written in such a language is formed by a sequence of one ormore statements. A statement may have internal components (e.g., expressions).Many languages (e.g. C) make a distinction between statements and definitions, with a statement only containingexecutable code and a definition declaring an identifier, while an expression evaluates to a value only.[2] A distinctioncan also be made between simple and compound statements; the latter may contain statements as components.

37.1 Kinds of statements

The following are the major generic kinds of statements with examples in typical imperative languages:

37.1.1 Simple statements

• assertion: assert(ptr != NULL);

• assignment: A:= A + 5

• goto: goto next;

• return: return 5;

• call: CLEARSCREEN()

37.1.2 Compound statements

• block: begin integer NUMBER; WRITE('Number? '); READLN(NUMBER); A:= A*NUMBER end

• do-loop: do { computation(&i); } while (i < 10);

• for-loop: for A:=1 to 10 do WRITELN(A) end

• if-statement: if A > 3 then WRITELN(A) else WRITELN(“NOT YET”); end

• switch-statement: switch (c) { case 'a': alert(); break; case 'q': quit(); break; }

• while-loop: while NOT EOF DO begin READLN end

• with-statement: with open(filename) as f: use(f)

106

37.2. SYNTAX 107

37.2 Syntax

The appearance of statements shapes the look of programs. Programming languages are characterized by the flavor ofstatements they use (e.g. the curly brace language family). Many statements are introduced by identifiers like if, whileor repeat. Often statement keywords are reserved such that they cannot be used as names of variables or functions.Imperative languages typically use special syntax for each statement, which looks quite different from function calls.Common methods to describe the syntax of statements are Backus–Naur Form and syntax diagrams.

37.3 Semantics

Semantically many statements differ from subroutine calls by their handling of parameters. Usually an actual sub-routine parameter is evaluated once before the subroutine is called. This contrasts to many statement parameters thatcan be evaluated several times (e.g. the condition of a while loop) or not at all (e.g. the loop body of a while loop).Technically such statement parameters are call-by-name parameters. Call-by-name parameters are evaluated whenneeded (see also lazy evaluation). When call-by-name parameters are available a statement like behaviour can beimplemented with subroutines (see Lisp). For languages without call-by-name parameters the semantic descriptionof a loop or conditional is usually beyond the capabilities of the language. Therefore standard documents often referto semantic descriptions in natural language.

37.4 Expressions

In most languages, statements contrast with expressions in that statements do not return results and are executedsolely for their side effects, while expressions always return a result and often do not have side effects at all. Amongimperative programming languages, Algol 68 is one of the few in which a statement can return a result. In languagesthat mix imperative and functional styles, such as the Lisp family, the distinction between expressions and statementsis not made: even expressions executed in sequential contexts solely for their side effects and whose return values arenot used are considered 'expressions’. In purely functional programming, there are no statements; everything is anexpression.This distinction is frequently observed in wording: a statement is executed, while an expression is evaluated. Thisis found in the exec and eval functions found in some languages: in Python both are found, with exec applied tostatements and eval applied to expressions.

37.5 Extensibility

Most languages have a fixed set of statements defined by the language, but there have been experiments with extensiblelanguages that allow the programmer to define new statements.

37.6 See also• Comparison of Programming Languages - Statements• Control flow• Expression (contrast)• Extensible languages

37.7 References[1] “statement”. webopedia. Retrieved 2015-03-03.

[2] Anders Kaseorg (2014-10-31). “What’s the difference between a statement and an expression in Python?". Quora.Archived from the original on 2014-10-31. Retrieved 2015-03-03.

108 CHAPTER 37. STATEMENT (COMPUTER SCIENCE)

37.8 External links• PC ENCYCLOPEDIA: Definition of: program statement

Chapter 38

Statement (logic)

In logic, a statement is either (a) a meaningful declarative sentence that is either true or false, or (b) that which a trueor false declarative sentence asserts. In the latter case, a statement is distinct from a sentence in that a sentence isonly one formulation of a statement, whereas there may be many other formulations expressing the same statement.Philosopher of language, Peter Strawson advocated the use of the term “statement” in sense (b) in preference toproposition. Strawson used the term “Statement” to make the point that two declarative sentences can make the samestatement if they say the same thing in different ways. Thus in the usage advocated by Strawson, “All men are mortal.”and “Every man is mortal.” are two different sentences that make the same statement.In either case a statement is viewed as a truth bearer.Examples of sentences that are (or make) statements:

• “Socrates is a man.”• “A triangle has three sides.”• “Madrid is the capital of Spain.”

Examples of sentences that are not (or do not make) statements:

• “Who are you?"• “Run!"• “Greenness perambulates.”• “I had one grunch but the eggplant over there.”• “The King of France is wise.”• “Broccoli tastes good.”• “Pegasus exists.”

The first two examples are not declarative sentences and therefore are not (or do not make) statements. The thirdand fourth are declarative sentences but, lacking meaning, are neither true nor false and therefore are not (or donot make) statements. The fifth and sixth examples are meaningful declarative sentences, but are not statements butrather matters of opinion or taste. Whether or not the sentence “Pegasus exists.” is a statement is a subject of debateamong philosophers. Bertrand Russell held that it is a (false) statement. Strawson held it is not a statement at all.

38.1 Statement as an abstract entity

In some treatments “statement” is introduced in order to distinguish a sentence from its informational content. Astatement is regarded as the information content of an information-bearing sentence. Thus, a sentence is related tothe statement it bears like a numeral to the number it refers to. Statements are abstract logical entities, while sentencesare grammatical entities.[1][2]

109

110 CHAPTER 38. STATEMENT (LOGIC)

38.2 See also• Claim (logic)

• Sentence (mathematical logic)

• Belief

• Concept

• Truthbearer - Statements

38.3 Notes[1] Rouse

[2] Ruzsa 2000, p. 16

38.4 References• A. G. Hamilton, Logic for Mathematicians, Cambridge University Press, 1980, ISBN 0-521-29291-3.

• Rouse, David L., “Sentences, Statements and Arguments”, A Practical Introduction to Formal Logic. (PDF)

• Ruzsa, Imre (2000), Bevezetés a modern logikába, Osiris tankönyvek, Budapest: Osiris, ISBN 963-379-978-3

• Xenakis, Jason (1956). “Sentence and Statement: Prof. Quine on Mr. Strawson”. Analysis 16 (4): 91–4.doi:10.2307/3326478. ISSN 1467-8284. JSTOR 3326478 – via JSTOR. (registration required (help)).

• PeterMillican, “Statements andModality: Strawson, Quine andWolfram”, http://philpapers.org/rec/MILSAM-2/

• P. F. Strawson, “On Referring” in Mind, Vol 59 No 235 (Jul 1950) P. F. Strawson (http://www.sol.lu.se/common/courses/LINC04/VT2010/Strawson1950.pdf/)

Chapter 39

Statement on the Co-operative Identity

The Statement on theCo-operative Identity, promulgated by the International Co-operative Alliance (ICA), definesand guides co-operatives worldwide. It contains the definition of a co-operative as a special form of organization, thevalues of co-operatives, and the currently accepted cooperative principles that direct their behavior and operation.The Statement with the latest revision of the cooperative principles was adopted by ICA in 1995.According to the Statement, a co-operative is defined as “an autonomous association of persons united voluntarily tomeet their common economic, social, and cultural needs and aspirations through a jointly owned and democraticallycontrolled enterprise.” Co-operatives “are based on the values of self-help, self-responsibility, democracy, equality,equity and solidarity. In the tradition of co-operative founders, co-operative members believe in the ethical values ofhonesty, openness, social responsibility and caring for others.”

• For a discussion of the seven cooperative principles see Rochdale Principles.

39.1 References• Statement on the Co-operative Identity, 1995 version.

111

Chapter 40

Theorem

For the Italian film, see Teorema (film).In mathematics, a theorem is a statement that has been proven on the basis of previously established statements,such as other theorems—and generally accepted statements, such as axioms. A theorem is a logical consequence ofthe axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accordwith the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of thetheorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentallydeductive, in contrast to the notion of a scientific law, which is experimental.[2]

Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from condi-tions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion isoften viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypothe-ses are true, without any further assumptions. However, the conditional could be interpreted differently in certaindeductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theoremsare often expressed in a natural language such as English. The same is true of proofs, which are often expressed aslogically organized and clearly worded informal arguments, intended to convince readers of the truth of the statementof the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Sucharguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express apreference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it isobviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the coreof mathematics, they are also central to its aesthetics. Theorems are often described as being “trivial”, or “difficult”,or “deep”, or even “beautiful”. These subjective judgments vary not only from person to person, but also with time:for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. Onthe other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connectionsbetween disparate areas of mathematics. Fermat’s Last Theorem is a particularly well-known example of such atheorem.

40.1 Informal account of theorems

Logically, many theorems are of the form of an indicative conditional: if A, then B. Such a theorem does not assertB, only that B is a necessary consequence of A. In this case A is called the hypothesis of the theorem (note that“hypothesis” here is something very different from a conjecture) and B the conclusion (formally, A and B are termedthe antecedent and consequent). The theorem “If n is an even natural number then n/2 is a natural number” is a typicalexample in which the hypothesis is "n is an even natural number” and the conclusion is "n/2 is also a natural number”.To be proven, a theorem must be expressible as a precise, formal statement. Nevertheless, theorems are usuallyexpressed in natural language rather than in a completely symbolic form, with the intention that the reader can producea formal statement from the informal one.It is common in mathematics to choose a number of hypotheses within a given language and declare that the theoryconsists of all statements provable from these hypotheses. These hypothesis form the foundational basis of the theoryand are called axioms or postulates. The field of mathematics known as proof theory studies formal languages, axioms

112

40.2. PROVABILITY AND THEOREMHOOD 113

and the structure of proofs.Some theorems are “trivial”, in the sense that they follow from definitions, axioms, and other theorems in obviousways and do not contain any surprising insights. Some, on the other hand, may be called “deep”, because their proofsmay be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself,or show surprising connections between disparate areas of mathematics.[3] A theorem might be simple to state andyet be deep. An excellent example is Fermat’s Last Theorem, and there are many other examples of simple yet deeptheorems in number theory and combinatorics, among other areas.Other theorems have a known proof that cannot easily be written down. The most prominent examples are the fourcolor theorem and the Kepler conjecture. Both of these theorems are only known to be true by reducing them to acomputational search that is then verified by a computer program. Initially, many mathematicians did not accept thisform of proof, but it has become more widely accepted. The mathematician Doron Zeilberger has even gone so far asto claim that these are possibly the only nontrivial results that mathematicians have ever proved.[4] Many mathemat-ical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometricidentities and hypergeometric identities.[5]

40.2 Provability and theoremhood

To establish a mathematical statement as a theorem, a proof is required, that is, a line of reasoning from axioms in thesystem (and other, already established theorems) to the given statement must be demonstrated. However, the proof isusually considered as separate from the theorem statement. Although more than one proof may be known for a singletheorem, only one proof is required to establish the status of a statement as a theorem. The Pythagorean theorem andthe law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs.

40.3 Relation with scientific theories

Theorems inmathematics and theories in science are fundamentally different in their epistemology. A scientific theorycannot be proven; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that aretestable by experiments. Any disagreement between prediction and experiment demonstrates the incorrectness of thescientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, arepurely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence inthe same way such evidence is used to support scientific theories.Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical the-orems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an ideaof what to prove, and in some cases even a plan for how to set about doing the proof. For example, the Collatzconjecture has been verified for start values up to about 2.88 × 1018. The Riemann hypothesis has been verified forthe first 10 trillion zeroes of the zeta function. Neither of these statements is considered proven.Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbersthat is now known to be false, but no explicit counterexample (i.e., a natural number n for which the Mertens functionM(n) equals or exceeds the square root of n) is known: all numbers less than 1014 have the Mertens property, and thesmallest number that does not have this property is only known to be less than the exponential of 1.59 × 1040, whichis approximately 10 to the power 4.3 × 1039. Since the number of particles in the universe is generally consideredless than 10 to the power 100 (a googol), there is no hope to find an explicit counterexample by exhaustive search.Note that the word “theory” also exists in mathematics, to denote a body of mathematical axioms, definitions andtheorems, as in, for example, group theory. There are also “theorems” in science, particularly physics, and in en-gineering, but they often have statements and proofs in which physical assumptions and intuition play an importantrole; the physical axioms on which such “theorems” are based are themselves falsifiable.

40.4 Terminology

A number of different terms for mathematical statements exist, these terms indicate the role statements play in aparticular subject. The distinction between different terms is sometimes rather arbitrary and the usage of some termshas evolved over time.

114 CHAPTER 40. THEOREM

• An axiom or postulate is a statement that is accepted without proof and regarded as fundamental to a subject.Historically these have been regarded as “self-evident”, but more recently they are considered assumptions thatcharacterize the subject of study. In classical geometry, axioms are general statements, while postulates arestatements about geometrical objects.[6] A definition is also accepted without proof since it simply gives themeaning of a word or phrase in terms of known concepts.

• An unproven statement that is believed true is called a conjecture (or sometimes a hypothesis, but with adifferent meaning from the one discussed above). To be considered a conjecture, a statement must usuallybe proposed publicly, at which point the name of the proponent may be attached to the conjecture, as withGoldbach’s conjecture. Other famous conjectures include the Collatz conjecture and the Riemann hypothesis.On the other hand, Fermat’s last theorem has always been known by that name, even before it was proven; itwas never known as “Fermat’s conjecture”.

• A proposition is a theorem of no particular importance. This term sometimes connotes a statement with asimple proof, while the term theorem is usually reserved for the most important results or those with long ordifficult proofs. In classical geometry, a proposition may be a construction that satisfies given requirements; forexample, Proposition 1 in Book I of Euclid’s elements is the construction of an equilateral triangle.[7]

• A lemma is a “helping theorem”, a proposition with little applicability except that it forms part of the proofof a larger theorem. In some cases, as the relative importance of different theorems becomes more clear, whatwas once considered a lemma is now considered a theorem, though the word “lemma” remains in the name.Examples include Gauss’s lemma, Zorn’s lemma, and the Fundamental lemma.

• A corollary is a proposition that follows with little proof from another theorem or definition.[8] Also a corollaryis used for a theorem restated for a more restricted special case. For example, the theorem that all angles ina rectangle are right angles has as corollary that all angles in a square (a special case of a rectangle) are rightangles.

• A converse of a theorem is a statement formed by interchanging what is given in a theorem and what is to beproved. For example, the isosceles triangle theorem states that if two sides of a triangle are equal then twoangles are equal. In the converse, the given (that two sides are equal) and what is to be proved (that two anglesare equal) are swapped, so the converse is the statement that if two angles of a triangle are equal then twosides are equal. In this example, the converse can be proven as another theorem, but this is often not the case.For example, the converse to the theorem that two right angles are equal angles is the statement that two equalangles must be right angles, and this is clearly not always the case.[9]

• A generalization is a theorem which includes a previously proven theorem as a special case and hence as acorollary.

There are other terms, less commonly used, that are conventionally attached to proven statements, so that certaintheorems are referred to by historical or customary names. For example:

• An identity is an equality, contained in a theorem, between two mathematical expressions that holds regardlessof what values are used for any variables or parameters appearing in the expressions. Examples include Euler’sformula and Vandermonde’s identity.

• A rule is a theorem, such as Bayes’ rule and Cramer’s rule, that establishes a useful formula.

• A law or a principle is a theorem that applies in a wide range of circumstances. Examples include the lawof large numbers, the law of cosines, Kolmogorov’s zero-one law, Harnack’s principle, the least upper boundprinciple, and the pigeonhole principle.[10]

A few well-known theorems have even more idiosyncratic names. The division algorithm (see Euclidean division) isa theorem expressing the outcome of division in the natural numbers and more general rings. The Bézout’s identityis a theorem asserting that the greatest common divisor of two numbers may be written as a linear combination ofthese numbers. The Banach–Tarski paradox is a theorem in measure theory that is paradoxical in the sense that itcontradicts common intuitions about volume in three-dimensional space.

40.5. LAYOUT 115

40.5 Layout

A theorem and its proof are typically laid out as follows:

Theorem (name of person who proved it and year of discovery, proof or publication).Statement of theorem (sometimes called the proposition).Proof.Description of proof.

End mark.

The end of the proof may be signalled by the letters Q.E.D. (quod erat demonstrandum) or by one of the tombstonemarks "□" or "∎" meaning “End of Proof”, introduced by Paul Halmos following their usage in magazine articles.The exact style depends on the author or publication. Many publications provide instructions or macros for typesettingin the house style.It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in thetheorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are thenused in the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs,or with their proofs presented after the proof of the theorem.Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Some-times, corollaries have proofs of their own that explain why they follow from the theorem.

40.6 Lore

It has been estimated that over a quarter of a million theorems are proved every year.[11]

The well-known aphorism, “A mathematician is a device for turning coffee into theorems”, is probably due to AlfrédRényi, although it is often attributed to Rényi’s colleague Paul Erdős (and Rényi may have been thinking of Erdős),who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking.[12]

The classification of finite simple groups is regarded by some to be the longest proof of a theorem. It comprisestens of thousands of pages in 500 journal articles by some 100 authors. These papers are together believed to givea complete proof, and several ongoing projects hope to shorten and simplify this proof.[13] Another theorem of thistype is the Four color theorem whose computer generated proof is too long for a human to read. It is certainly thelongest known proof of a theorem whose statement can be easily understood by a layman.

40.7 Theorems in logic

Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formedformulas) of a formal language. The statements of the language are strings of symbols and may be broadly dividedinto nonsense and well-formed formulas. A set of deduction rules, also called transformation rules or rules ofinference, must be provided. These deduction rules tell exactly when a formula can be derived from a set of premises.The set of well-formed formulas may be broadly divided into theorems and non-theorems. However, according toHofstadter, a formal system often simply defines all its well-formed formula as theorems.[14]

Different sets of derivation rules give rise to different interpretations of what it means for an expression to be atheorem. Some derivation rules and formal languages are intended to capture mathematical reasoning; the mostcommon examples use first-order logic. Other deductive systems describe term rewriting, such as the reduction rulesfor λ calculus.The definition of theorems as elements of a formal language allows for results in proof theory that study the structureof formal proofs and the structure of provable formulas. The most famous result is Gödel’s incompleteness theorem;by representing theorems about basic number theory as expressions in a formal language, and then representingthis language within number theory itself, Gödel constructed examples of statements that are neither provable nordisprovable from axiomatizations of number theory.

116 CHAPTER 40. THEOREM

A theorem may be expressed in a formal language (or “formalized”). A formal theorem is the purely formal analogueof a theorem. In general, a formal theorem is a type of well-formed formula that satisfies certain logical and syntacticconditions. The notation S is often used to indicate that S is a theorem.Formal theorems consist of formulas of a formal language and the transformation rules of a formal system. Specif-ically, a formal theorem is always the last formula of a derivation in some formal system each formula of which isa logical consequence of the formulas that came before it in the derivation. The initially accepted formulas in thederivation are called its axioms, and are the basis on which the theorem is derived. A set of theorems is called atheory.What makes formal theorems useful and of interest is that they can be interpreted as true propositions and theirderivations may be interpreted as a proof of the truth of the resulting expression. A set of formal theorems may bereferred to as a formal theory. A theorem whose interpretation is a true statement about a formal system is called ametatheorem.

40.7.1 Syntax and semantics

Main articles: Syntax (logic) and Formal semantics (logic)

The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a true proposition, whichintroduces semantics. Different deductive systems can yield other interpretations, depending on the presumptionsof the derivation rules (i.e. belief, justification or other modalities). The soundness of a formal system depends onwhether or not all of its theorems are also validities. A validity is a formula that is true under any possible inter-pretation, e.g. in classical propositional logic validities are tautologies. A formal system is considered semanticallycomplete when all of its tautologies are also theorems.

40.7.2 Derivation of a theorem

Main article: Formal proof

The notion of a theorem is very closely connected to its formal proof (also called a “derivation”). To illustrate howderivations are done, we will work in a very simplified formal system. Let us call ours FS Its alphabet consists onlyof two symbols { A, B } and its formation rule for formulas is:

FS

The single axiom of FS is:

ABBA.

The only rule of inference (transformation rule) for FS is:

Any occurrence of "A" in a theorem may be replaced by an occurrence of the string "AB" and the resultis a theorem.

Theorems in FS are defined as those formulae that have a derivation ending with that formula. For example

1. ABBA (Given as axiom)

2. ABBBA (by applying the transformation rule)

3. ABBBAB (by applying the transformation rule)

is a derivation. Therefore, "ABBBAB" is a theorem of FS . The notion of truth (or falsity) cannot be applied to theformula "ABBBAB" until an interpretation is given to its symbols. Thus in this example, the formula does not yetrepresent a proposition, but is merely an empty abstraction.Two metatheorems of FS are:

40.8. SEE ALSO 117

Every theorem begins with "A".Every theorem has exactly two "A“s.

40.7.3 Interpretation of a formal theorem

Main article: Interpretation (logic)

40.7.4 Theorems and theories

Main articles: Theory and Theory (mathematical logic)

40.8 See also

• Inference

• List of theorems

• Toy theorem

• Metamath – a language for developing strictly formalized mathematical definitions and proofs accompanied bya proof checker for this language and a growing database of thousands of proved theorems

40.9 Notes

[1] Elisha Scott Loomis. “The Pythagorean proposition: its demonstrations analyzed and classified, and bibliography of sourcesfor data of the four kinds of proofs” (PDF). Education Resources Information Center. Institute of Education Sciences (IES)of the U.S. Department of Education. Retrieved 2010-09-26. Originally published in 1940 and reprinted in 1968 byNational Council of Teachers of Mathematics.

[2] However, both theorems and scientific law are the result of investigations. See Heath 1897 Introduction, The terminologyof Archimedes, p. clxxxii:"theorem (θεὼρνμα) from θεωρεἳν to investigate”

[3] Weisstein, Eric W., “Deep Theorem”, MathWorld.

[4] Doron Zeilberger. “Opinion 51”.

[5] Petkovsek et al. 1996.

[6] Wentworth, G.; Smith, D.E. (1913). “Art. 46, 47”. Plane Geometry. Ginn & Co.

[7] Wentworth & Smith Art. 50

[8] Wentworth & Smith Art. 51

[9] Follows Wentworth & Smith Art. 79

[10] The word law can also refer to an axiom, a rule of inference, or, in probability theory, a probability distribution.

[11] Hoffman 1998, p. 204.

[12] Hoffman 1998, p. 7.

[13] An enormous theorem: the classification of finite simple groups, Richard Elwes, Plus Magazine, Issue 41 December 2006.

[14] Hofstadter 1980

118 CHAPTER 40. THEOREM

40.10 References• Heath, Sir Thomas Little (1897). The works of Archimedes. Dover. Retrieved 2009-11-15.

• Hoffman, P. (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathe-matical Truth. Hyperion, New York. ISBN 1-85702-829-5.

• Hofstadter, Douglas (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books.

• Hunter, Geofrfrey (1996) [1973]. Metalogic: An Introduction to the Metatheory of Standard First Order Logic.University of California Press. ISBN 0-520-02356-0.

• Mates, Benson (1972). Elementary Logic. Oxford University Press. ISBN 0-19-501491-X.

• Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996). A = B. A.K. Peters, Wellesley, Massachusetts.ISBN 1-56881-063-6.

40.11 External links• Weisstein, Eric W., “Theorem”, MathWorld.

• Theorem of the Day

40.11. EXTERNAL LINKS 119

The Pythagorean theorem has at least 370 known proofs[1]

120 CHAPTER 40. THEOREM

A planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with onlyfour colors. The four color theorem states that such colorings are possible for any planar map, but every known proof involves acomputational search that is too long to check by hand.

40.11. EXTERNAL LINKS 121

The Collatz conjecture: one way to illustrate its complexity is to extend the iteration from the natural numbers to the complex numbers.The result is a fractal, which (in accordance with universality) resembles the Mandelbrot set.

122 CHAPTER 40. THEOREM

Symbols andstrings of symbols

Well-formed formulas

Theorems

This diagram shows the syntactic entities that can be constructed from formal languages. The symbols and strings of symbols maybe broadly divided into nonsense and well-formed formulas. A formal language can be thought of as identical to the set of itswell-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.

Chapter 41

Truth claim

A truth claim is a proposition or statement that a particular person or belief system holds to be true. The term iscommonly used in philosophy in discussions of logic, metaphysics, and epistemology, particularly when discussingthe doctrinal statements of religions; however it is also used when discussing non-religious ideologies.

41.1 Types

41.1.1 Positive and negative

One major division of truth claims is that between positive and negative truth claims. Positive truth claims proclaimthe existence of an object or entity. Negative truth claims are the opposite and proclaim the non-existence of anobject or entity.

41.2 Religions

Religions which make strong absolutist truth claims stand in stark contrast to more relativist or universalist positions.

41.2.1 Hinduism

In the post-modern period Hinduism is generally known to be Universalist and accepts all other religions to be trueand valid. Mahatma Gandhi is credited to be the original proponent. However, this seems to be true from a socio-political perspective, since Hinduism itself is a congregation of large numbers of individual sects and peoples thathave long lived in harmony among themselves without persecuting each other. But if one goes into what is in thescriptures of more organized Hindu sects like Vedanta; we do find exclusive truth claims.Bhagavad Gita says -ye me matamidaM nityamanutishhThanti maanavaaH .shraddhaavanto.anasuuyanto muchyante te.api karmabhiH .. 3.31

Those who continuously practice what I preach they will be freed from Karma.ye tvetadabhyasuuyanto naanutishhThanti me matam.h .sarvaGYaanavimuuDhaa.nstaanviddhi nashhTaanachetasaH .. 3.32

But those who, out of envy, disregard these teachings and do not practice them regularly, are to be considered bereftof all knowledge, befooled, and doomed to ignorance and bondage.A Hindu is expected to examine a truth claim based on his intellect. He can add or improve upon the vast ocean ofHindu philosophy. This is unlike some other religions, where, because of the truth claim that the entire book is arevelation from their God, a single verse proved wrong discredits the entire book. This leads to extreme fanaticismon the part of their followers.

123

124 CHAPTER 41. TRUTH CLAIM

Traditionally Hinduism (more specifically Vedanta) considers itself to be eternal religion (Sanatana Dharma). Fun-damental belief is that all beings are divine, that the human condition is one of ignorance to not recognize this divinityinside, and that direct experience of God is achievable for all human beings.

41.3 Science

Main article: Philosophy of science

It has been debated as to whether science makes any truth claims of its own, or if it is a set of methods for evaluatingor falsifying other truth claims.

41.3.1 Agnosticism

Agnosticism makes the claim that the existence or non-existence of any deity is unknown and possibly unknowable.This is a claim about what we can or do know, not a metaphysical claim as to the nature of the world.

41.3.2 Atheism

Main article: Negative and positive atheism

The truth claims of atheism are divided between negative atheism, which does not accept the positive truth claims ofreligions, and positive atheism, which specifically claims the non-existence of deities (and other spiritual phenomena).

41.4 See also• Truth

• World view

Chapter 42

Truth-apt

In philosophy, truth-apt denotes statements that could be uttered in some context (without their meaning beingaltered) and would then express a true or false proposition.[1]

Truth-apt sentences are capable of being true or false, unlike questions or commands. Whether paradoxical sentences,prescriptions (especially moral claims), or attitudes are truth-apt is sometimes controversial.

42.1 See also• Non-cognitivism

• Cognitivism (ethics)

• Fictionalism

42.2 References[1] SimonBlackburn. TheOxfordDictionary of Philosophy (2 rev. ed.) OxfordUniversity Press. http://www.oxfordreference.

com/view/10.1093/acref/9780199541430.001.0001/acref-9780199541430-e-3154

125

Chapter 43

Vision statement

A vision statement is a declaration of an organization's objectives, ideally based on economic foresight, intended toguide its internal decision-making.[1]

43.1 Definition and structure

A vision statement is a company’s road map, indicating both what the company wants to become and guiding trans-formational initiatives by setting a defined direction for the company’s growth. Vision statements undergo minimalrevisions during the life of a business, unlike operational goals which may be updated from year-to-year. Visionstatements can range in length from short sentences to multiple pages. Vision statements are also formally writtenand referenced in company documents rather than, for example, general principles informally articulated by seniormanagement. [2] A vision statement is not limited to business organizations and may also be used by non-profit orgovernmental entities.[3]

A consensus does not exist on the characteristics of a “good” or “bad” vision statement. Commonly cited traitsinclude:[4]

• concise: able to be easily remembered and repeated

• clear: defines a prime goal

• future-oriented: describes where the company is going rather than the current state

• stable: offers a long-term perspective and is unlikely to be impacted by market or technology changes

• challenging: not something that can be easily met and discarded

• abstract: general enough to encompass all of the organization’s interests and strategic direction

• inspiring: motivates employees and is something that employees view as desireable

43.1.1 Mission vs. Vision Statement

Mission statements and vision statements fill different purposes. A mission statement describes an organization’spurpose and answers the questions “What business are we in?" and “What is our business for?" A vision statementprovides strategic direction and describes what the owner or founder wants the company to achieve in the future.[5]

43.2 Purpose

Vision statements may fill the following functions for a company:[2]

• Serve as foundations for a broader strategic plan

126

43.3. CHALLENGES 127

• Motivate existing employees and attract potential employees by clearly categorizing the company’s goals andattracting like-minded individuals

• Focus company efforts and facilitate the creation of core competencies by directing the company to only focuson strategic opportunities that advance the company’s vision

• Help companies differentiate from competitors. For example, profit is a common business goal, and visionstatements typically describe how a company will become profitable rather than list profit directly as the long-term vision

43.2.1 Relevance

While a consensus does not exist on the value of mission and vision statements, literature supporting the relevanceof these documents to companies outweighs those opposed to them. This may be due to, among other reasons,the positive value of the tools in communicating to internal and external stakeholders or retrospective attempts tolegitimize the use of these tools.[5]

43.3 Challenges

Creating and implementing vision statements presents challenges to organizations. They can be challenging to writebecause they must balance being forward-looking and describing an ideal state without becoming so idealistic thatthe vision is unattainable. Vision statements can be an employee dissatisfier when staff feel the company’s visionis filled with business buzzwords unrelated to the company’s services or when the vision does not match day-to-daycompany policy; for example, a vision statement that includes root cause problem solving whilemanagers are rewardedfor fixing problems quickly rather than resolving systemic issues.[2] A vision statement may need to be paired withcompany initiatives to communicate and reinforce the vision, ensure processes align with the vision, and empowerand incentivize employees to take actions that support the company vision.[4]

43.4 References[1] http://www.businessdictionary.com/definition/vision-statement.html#ixzz3OGy3r4Gn

[2] Lipton, Mark (Summer 1996). “Demystifying the Development of an Organizational Vision” (PDF). Sloan ManagementReview 37 (4): 83. Retrieved 2015-08-15.

[3] Ozdem, Guven (2011). “An Analysis of the Mission and Vision Statements on the Strategic Plans of Higher EducationInstitutions” (PDF). Educational Sciences: Theory and Practice: 1887–1894. Retrieved 2015-08-15.

[4] Kantabutra, Sooksan; Avery, Gayle (2010). “The power of vision: statements that resonate” (PDF). Journal of BusinessStrategy 31 (1): 37–45.

[5] Kofi Darbi, William Phanuel (July 2012). “Of Mission and Vision Statements and Their Potential Impact on EmployeeBehaviour and Attitudes: The Case of A Public But Profit-Oriented Tertiary Institution” (PDF). International Journal ofBusiness and Social Science 3 (14). Retrieved 2015-08-15.

128 CHAPTER 43. VISION STATEMENT

43.5 Text and image sources, contributors, and licenses

43.5.1 Text• Apophantic Source: https://en.wikipedia.org/wiki/Apophantic?oldid=642734946 Contributors: Gregbard, Magioladitis, SchreiberBike,

Yobot, Omnipaedista, Xindhus, Faizan, Eminence2012 and Anonymous: 4• Artist’s statement Source: https://en.wikipedia.org/wiki/Artist’{}s_statement?oldid=655829913Contributors: Darkwind, Andrewman327,

PaulHanson, Sparkit, Meika, SmackBot, Can't sleep, clown will eat me, Saxbryn, Cyrusc, Besieged, Missvain, MarshBot, Bus stop, Al-lenbham, Laval, ClueBot, NuclearWarfare, M.O.X, Iamericmercer, NellieBly, Yobot, Petropoxy (Lithoderm Proxy), FrescoBot, Many-texts, ClueBot NG, Wfgp, MusikAnimal, Lugia2453 and Anonymous: 29

• Attending physician statement Source: https://en.wikipedia.org/wiki/Attending_physician_statement?oldid=665631467 Contributors:Jfdwolff, Rich Farmbrough, Gary, PaulHanson, RJFJR, Carabinieri, Exit2DOS2000, SmackBot, Anastrophe, Alaibot, Y2kcrazyjoker4,Mfriedel, Wilhelmina Will, Addbot, PigFlu Oink, Delusion23, TylerDurden8823, Ashleyleia and Anonymous: 3

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• Conjecture Source: https://en.wikipedia.org/wiki/Conjecture?oldid=698245792 Contributors: AxelBoldt, The Anome, Fubar Obfusco,Stevertigo, Chas zzz brown, Michael Hardy, Ixfd64, Eric119, Ahoerstemeier, Glenn, Revolver, Charles Matthews, Timwi, Jm34harvey,Dysprosia, Gandalf61, Henrygb, Rasmus Faber, Mattflaschen, Tobias Bergemann, Ancheta Wis, Centrx, Giftlite, Andries, BenFrantz-Dale, Wikipedia benefits, Icairns, Hkpawn~enwiki, Yuriz, Chris Howard, Discospinster, Rich Farmbrough, Guanabot, 1pezguy, Dcoetzee-Bot~enwiki, Elwikipedista~enwiki, Porton, Bobo192, Obradovic Goran, Haham hanuka, Anthony Appleyard, Sligocki, Axeman89, OlegAlexandrov, GrouchyDan, Justinlebar, Oliphaunt, Davidfstr, Graham87, BD2412, Qwertyus, Yurik, Eyu100, Salix alba, Tp, FlaBot,Mathbot, Kri, Gwernol, Algebraist, Wavelength, Sceptre, Ytcracker, Zwobot, Cadillac, GrinBot~enwiki, SmackBot, Jab843, Ohnoit-sjamie, Ppntori, Bluebot, Hgrosser, BesselDekker, Byelf2007, Lambiam, Mouse Nightshirt, Valfontis, Scetoaux, Noah Salzman, MrStephen, George100, JForget, CRGreathouse, CBM, Gregbard, Doctormatt, Wang ty87916, Danny lost, JAnDbot, Andonic, Magio-laditis, VoABot II, Careless hx, Juansidious, J.delanoy, McSly, Ohms law, 42n82rst, Cometstyles, VolkovBot, Am Fiosaigear~enwiki,Hqb, DieBuche, Lova Falk, SieBot, Pallab1234, This, that and the other, ~enwiki, Antonio Lopez, Sjn28, Sunrise, Catrope, EscapeOrbit, DEMcAdams, Mr. Granger, ClueBot, Mild Bill Hiccup, Blanchardb, Psychonomics, Cuckowski, NuclearWarfare, Aitias, Ver-sus22, Qwfp, Editor2020, Cogitus1, Favonian, Qwrk, Legobot, Luckas-bot, Yobot, Fraggle81, THEN WHO WAS PHONE?, Reindra,AnomieBOT, AdjustShift, Phlembowper99, Larseven, Лев Дубовой, David Schwein, Lotje, EmausBot, Set theorist, Dcirovic, ZéroBot,Stbunco, Trititaty, Rmashhadi, Garfieldperfect, ClueBot NG, Wcherowi, SusikMkr, LJosil, Startire, MerlIwBot, BlueMoonset, BG19bot,Johny Five, ChrisGualtieri, YFdyh-bot, Brirush, Choor monster, Purnendu Karmakar, Comp.arch, A Certain Lack of Grandeur, Jackm-cbarn, AddWittyNameHere, Plesantdreams, Tcc astronaut, Loraof, Macosojr, Gavnerfsk and Anonymous: 146

• Corollary Source: https://en.wikipedia.org/wiki/Corollary?oldid=695160887 Contributors: Tarquin, Michael Hardy, Glinos, TakuyaMu-rata, Rl, Robbot, Giftlite, Bepp, DanielCD, Kwamikagami, Toussaint, Me and, Katieh5584, Finell, Rutja76, BiT, SashatoBot, Bjanku-loski06en~enwiki, Hvn0413, Werdan7, SmokeyJoe, BranStark, Iridescent, Heqs, CBM, Gregbard, J. W. Love, Deflective, Ipthief, Ma-gioladitis, Americanhero, Hdt83, WinterSpw, Jordo ex, VolkovBot, The Tetrast, Geometry guy, Kmhkmh, Ebasconp, ClueBot, Zackwadghiri, Jotterbot, BodhisattvaBot, Dnael, Zc Abc, Addbot, Rainbow-five, AnomieBOT, Rubinbot, Fvendi, Erik9bot, Kwiki, WP ad-dict 0, PiRSquared17, TheStrayCat, Dinamik-bot, U002764, DexDor, Dmwpowers, EmausBot, Racerx11, Markmassie, ClueBot NG,Lugia2453, Nigellwh and Anonymous: 32

• Corresponding conditional Source: https://en.wikipedia.org/wiki/Corresponding_conditional?oldid=678540031Contributors: SatyrTN,Jason Quinn, Rich Farmbrough, Zenohockey, Bobo192, Gary, SmackBot, Chris the speller, Beefyt, JRSpriggs, CBM,Wykebjs, Gregbard,Fabrictramp, Philogo, Graymornings, Otr500, Gemtpm, FrescoBot, Helpful Pixie Bot, Lugia2453, Sheep0x and Anonymous: 4

• Elevator pitch Source: https://en.wikipedia.org/wiki/Elevator_pitch?oldid=682756793 Contributors: Atlan, Edward, Ronz, Saltine, Lu-mos3, Pigsonthewing, Tobias Bergemann, David Gerard, Brian Kendig, Bohemiotx, Drichardson, Elembis, Josephgrossberg, Abelson,Bender235, Jarsyl, Blahma, Woohookitty, Graham87, ConradKilroy, Jweiss11, Engineer1, Tawker, FlaBot, SchuminWeb, AndriuZ,Whoisjohngalt, RussBot, Shaddack, StuRat, Mike Selinker, Wainstead, NiTenIchiRyu, Veinor, SmackBot, Closetoeuphoria, StefanoC,Ohnoitsjamie, Kuru, Dfrench, Fan-1967, Iridescent, Badly~enwiki, Courcelles, Danlev, Mikebrand, Thijs!bot, Seaphoto, Thepainguy,Tedickey, Steven Walling, Rbrewer42, Glynth, Jös, Richswier, TXiKiBoT, Perohanych, Davin, Yintan, Infi01, Themoleskin, HighInBC,DBrnstn, Nnemo, XLinkBot, AgnosticPreachersKid, Addbot, MrOllie, Alpinwolf, Yobot, Themfromspace, Fourmiz59, CaptainComma,AnomieBOT, ThaddeusB, Jim1138, JackieBot, Citation bot, AbigailAbernathy, Shadowjams, TawsifSalam, Saxman58, Biker Biker,Joe12387, Dinamik-bot, Manyberg, TjBot, EmausBot, John of Reading, Meisele, Westley Turner, Mjenksny, ClueBot NG, JordoCo,Helpful Pixie Bot, 220 of Borg, Mohdfirdausaziz, SteenthIWbot, YanikB, Maverick RoR, DCyphert, Rebootuni, D.S. Cordoba-Bahle,Sweepy and Anonymous: 85

• Eternal statement Source: https://en.wikipedia.org/wiki/Eternal_statement?oldid=594851659 Contributors: Melaen, Gregbard, Cyde-bot, SchreiberBike, DannyJIguess? and SpencerK123456

• Fact Source: https://en.wikipedia.org/wiki/Fact?oldid=700143617 Contributors: The Epopt, Malcolm Farmer, Fredbauder, Waveguy,AaronAgassi, Patrick, Michael Hardy, Jahsonic, Ixfd64, Dcljr, IZAK, Ihcoyc, DavidWBrooks, Julesd, Glenn, Reddi, Tpbradbury, Fur-rykef, Hyacinth, David Shay, David.Monniaux, Banno, Psychonaut, Mirv, Rfc1394, Ebeisher, Unyounyo, Alan Liefting, Dave6, Giftlite,Polsmeth, Wolfkeeper, Fudoreaper, Kenny sh, Fastfission, Tsca, Everyking, TomViza, Bensaccount, Yekrats, Antandrus, JimWae, JHCC,Neutrality, Joyous!, Clemwang, Adashiel, Discospinster, Rich Farmbrough, KillerChihuahua, Vsmith, Dave souza, Arianna the First,Rgdboer, Shanes, Bobo192, Longhair, Smalljim, Neg, Townmouse, Cherlin, Mattduval, Nsaa, Jumbuck, Alansohn, Ungtss, Atlant,Lightdarkness, Ynhockey, DreamGuy, Wtmitchell, Velella, Velho, Peter Hitchmough, Jeff3000, Waldir, Wayward, Gimboid13, Ste-fanomione, Dysepsion, Paxsimius, RuM, BD2412, Dpv, SteveW, Rjwilmsi, Mayumashu, Coemgenus, Vegaswikian, Bhadani, FlaBot,PlatypeanArchcow, Twipley, Alfred Centauri, Flowerparty, RexNL, TeaDrinker, Gareth E. Kegg, DVdm, Wavelength, RussBot, Tjss,Honshuzen, Clemondo, Gaius Cornelius, CambridgeBayWeather, NawlinWiki, Rick Norwood, Wiki alf, Astral, Trovatore, Emersoni,Alex43223, Lockesdonkey, Wangi, Brat32, DeadEyeArrow, Brisvegas, Tomisti, Enormousdude, StuRat, Cjwright79, GrinBot~enwiki,DVD RW, Wayne Goode, SmackBot, McGeddon, Bomac, Davewild, Apartmento, Xaosflux, Gilliam, Ohnoitsjamie, Kmarinas86, Christhe speller, Hibbleton, Robth, Firetrap9254, Tsca.bot, Can't sleep, clown will eat me, Yidisheryid, Rrburke, The tooth, Veni Markovski,Nakon, RJN, Richard001, Dcamp314, DMacks, Just plain Bill, Acdx, Where, Drunken Pirate, Eliyak, Geoffrey Wickham, Scientiz-zle, JoshuaZ, Nonsuch, Aleenf1, Cyberstrike2000x, 16@r, Sandb, Waggers, Dodo bird, K, Dreftymac, Theone00, JoeBot, Manwithear,

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UncleDouggie, Twas Now, Ewulp, The Artist Formerly Known as BenFranklin, Tawkerbot2, Dlohcierekim, Chris55, Eastlaw, Ale jrb, In-sanephantom, Misteraznkid, CWY2190, Skoch3, Alaymehta, Osa23, Gregbard, Rudjek, Cydebot, Peterdjones, UncleBubba, Gogo Dodo,Adolphus79, 879(CoDe), DumbBOT, Omicronpersei8, Epbr123, Rusl, Marek69, Srose, Lenk~enwiki, AntiVandalBot, Luna Santin, GuyMacon, Seaphoto, Tozoku, Carolmooredc, TimVickers, D. Webb, Danny lost, Zedla, Pixelface, JAnDbot, Narssarssuaq, Leuko, Husond,Barek, MER-C, Smerdis, GurchBot, Andy Ross, Bencherlite, Bongwarrior, VoABot II, Nhorning, Rgfolsom, DerHexer, Lelkesa, Án-gel Luis Alfaro, Ronbtni, Arjun01, AlexiusHoratius, Nono64, N4nojohn, J.delanoy, Trusilver, Uncle Dick, VAcharon, TomS TDotO,Katalaveno, Ncmvocalist, Bargains001, Anonywiki, Jurassic4, NewEnglandYankee, Shshshsh, STBotD, WJBscribe, Jamesontai, Squidsand Chips, LeFront, Idioma-bot, ACSE, VolkovBot, Butwhatdoiknow, Philip Trueman, Fran Rogers, TXiKiBoT, Oshwah, Fenrir656,Asarlaí, Fddcarey, Qxz, Voorlandt, Anna Lincoln, Jane Fairfax, Dizzynoise, Sandry~enwiki, Broadbot, Colbot, Wiae, Seaneseor, Miran-dalayne, Richwil, Monty845, HiDrNick, Captaintoadman, AlleborgoBot, Randian, Newbyguesses, SieBot, Jauerback, Toddst1, Tiptoety,MaynardClark, Oxymoron83, Lightmouse, RyanParis, Sunrise, Benny the wayfarer, Firefly322, Missing Ace, ClueBot, Binksternet, Go-rillaWarfare, Wikikosti, Snigbrook, Ovangle, Drmies, Neoballmon III, Jerainseltran, Excirial, HUMPBRO, Jusdafax, Ykhwong, Brewsohare, Shadowgoth69, ChampionshipManager, Wasthere, 7, Burner0718, Mr. Gerbear, XLinkBot, Aaron north, Pichpich, Yanksfan21,Sergay, SilvonenBot, Airplaneman, RyanCross, Addbot, Fyrael, Bananicaa, Mr. Wheely Guy, D0762, Randoman90, PranksterTurtle,Favonian, Doniago, Kaboomgoesthekat, Tide rolls, Plani, Krano, Teles, Gail, Luckas-bot, Yobot, Alewo27, 6u56u, Cflm001, Mind-builder, Azcolvin429, Rubinbot, Bsimmons666, Jim1138, IRP, Materialscientist, Citation bot, 2 port usb hub, THEKEETH, Xqbot,Belasted, Ched, Mathonius, Josh51568, Omgiitzmegz, FrescoBot, Wikipe-tan, Mark Renier, Whatje, Recognizance, Xefer, Dsjhtf-shtgh, Holly92, DivineAlpha, Cannolis, SixPurpleFish, Viswa.gogineni, Pinethicket, 10metreh, Hoo man, Serols, 9E2, Kuthar, Lotje,GregKaye, Vrenator, Capt. James T. Kirk, Factloverlol, Uzigrunt12, Artms~enwiki, Tbhotch, Dj6ual, Ron.sangal, Mean as custard,Jpcull, Bento00, Legiteditor2, GoodInkInc, Meesher, J36miles, EmausBot, Tippy2115, RA0808, Tommy2010, Hasan123-786, Info-manic12, AvicBot, A 1AAJ, Pingu.dbl96, Mythical Editor, Bertmanforever, L Kensington, Sonicspeedx, ClueBot NG, Gibbytheoptical,MelbourneStar, Nschang57, Sp89fairy, Widr, Rthoms27, Helpful Pixie Bot, Userman113fred, Tommyjoewood, Doug4321, Darekaon,Skifunkster2011, RSmiggy, Wiki13, Andrew.baggott, Alex.Ramek, Joydeep, CitationCleanerBot, Callmepgr, BLACK COMMANDO,Againstmywill, BattyBot, Eduardofeld, Dooplissmum, Quadratic42, Davidlwinkler, Stinga1, Rogerbacon666, Pi^(e*i)=−1, Swainybb2k,Beverfar, JKempf2013, Sadiqsalih3, Haminoon, NottNott, Quenhitran, AddWittyNameHere, Arfæst Ealdwrítere, JaconaFrere, KH-1,Loraof, Liance, Heide2705, Mrdoctorandprofessor, Quonxy, Wisdom Scholar, Kavya8, Xtom8, KWSmith80, TedRose 2, CLCStudent,Sunshine2night, Mic331 and Anonymous: 480

• False statement Source: https://en.wikipedia.org/wiki/False_statement?oldid=687259483Contributors: SimonP,Alan Liefting, Hooperbloob,PaulHanson, BD2412, Computor, Emiao, ColdRedRain, Ageekgal, PurplePlatypus, Gilliam, BrainMagMo, DanielRigal, Alaibot, Pbroks13,Bobo2000, Oshwah, Liko81, Int21h, Selenemeow, Twinsday, Gnome de plume, Addbot, Fgnievinski, TechBot, Pinklitigation, Wik-iTrenches, Machine Elf 1735, John of Reading, AvicAWB, GreaseballNYC, Why should I have a User Name? and Anonymous: 15

• I Am aMan! Source: https://en.wikipedia.org/wiki/I_Am_a_Man!?oldid=691225416 Contributors: Neo-Jay, USchick, EmausBot, Johnof Reading, ClueBot NG, Alvin Lee, TBrandley, Khazar2, 7Sidz, RyanTQuinn, Pussyslayer696966 and Anonymous: 4

• I-message Source: https://en.wikipedia.org/wiki/I-message?oldid=700701890 Contributors: Sunray, Andycjp, Andrewpmk, Rjwilmsi,Bgwhite, Siddhant, Malcolma, Gadget850, Cazort, Alaibot, Pharaoh of the Wizards, Coppertwig, Skullers, Funandtrvl, VanishedUsersdu9aya9fs787sads, ClueBot, The Thing That Should Not Be, Addbot, AnomieBOT, Tavatar, Citation bot, Omnipaedista, Eugene-elgato,Citation bot 1, Trappist the monk, RenamedUser01302013, AvicAWB, ClueBot NG, Theopolisme, Helpful Pixie Bot, Calabe1992,Wiki13, MusikAnimal, Christian Stroppel, Pratyya Ghosh, Monkbot, Yev Yev and Anonymous: 15

• Illocutionary act Source: https://en.wikipedia.org/wiki/Illocutionary_act?oldid=687700693 Contributors: Dieter Simon, Michael Hardy,Sethmahoney, Peter Damian (original account), Javier Carro, Lucidish, Espoo, Jumbuck, Fbd, RJCraig, Ish ishwar, Velho, BMF81,Bhny, Sasuke Sarutobi, Modify, Rob GWeemhoff, SmackBot, Dster, DroEsperanto, Fuzzform, Relpaed, Lambiam, JorisvS, Filippowiki,Calmargulis, BFD1, Gregbard, Jasperdoomen, Lindsay658, Hunt.topher, D. Webb, Albi the Newen, .anacondabot, Tiepies, Magioladitis,Anarchia, Jackfork, SieBot, PipepBot, Wordwright, M.boli, Addbot, Twinkie eater91, Lightbot, Ettrig, THEN WHO WAS PHONE?,Glenfarclas, Omnipaedista, Born2bgratis, Hpvpp, Rmashhadi, KLBot2, Hans-Jürgen Streicher~enwiki, Luizpuodzius, Tiocaima7n, SFK2,Monsegu 2, Dr Lindsay B Yeates, Kaveh.almasi and Anonymous: 41

• Leave and Earnings Statement Source: https://en.wikipedia.org/wiki/Leave_and_Earnings_Statement?oldid=684785626 Contributors:Dale Arnett, PaulHanson, Woohookitty, Kajmal, Old Moonraker, Kafziel, SmackBot, Quidam65, Hmains, Vanished user 9i39j3, IsaacCrumm, Skapur, Katydidit, Kumioko (renamed),Mild Bill Hiccup, TravelingDude 2000, Erik9bot, TheGrimReaperNS, I.Am.The.Napster,ClueBot NG, Riley Huntley and Anonymous: 14

• Locutionary act Source: https://en.wikipedia.org/wiki/Locutionary_act?oldid=695959904 Contributors: The Anome, Michael Hardy,Kzzl, Xezbeth, RHaworth, Rob GWeemhoff, Trickstar, SmackBot, Bluebot, Nbarth, Tamfang, Skapur, Gregbard, Goldenrowley, Cinna-mon42, Japadrum, Nightmiler, Fadesga, Mild Bill Hiccup, Vegas Bleeds Neon, Addbot, Amirobot, Erik9bot, Overclax, BG19bot, SFK2,Monsegu 2, Kaveh.almasi and Anonymous: 3

• Loosely associated statements Source: https://en.wikipedia.org/wiki/Loosely_associated_statements?oldid=592910308 Contributors:Gregbard, Miracle Pen, Saedon and Anonymous: 1

• Maxim (philosophy) Source: https://en.wikipedia.org/wiki/Maxim_(philosophy)?oldid=687521149 Contributors: Andres, Furrykef,Fudoreaper, Karol Langner, Imjustmatthew, D6, Omphaloscope, Quiddity, Chobot, YurikBot, Tomisti, Caco de vidro, Sardanaphalus,SmackBot, Shaggorama, Meitme, Cybercobra, Jon Awbrey, Mineralè, Gregbard, Aiko, AntiVandalBot, Enix150, Magarmach, Flyer22Reborn, Fadesga, Wikijens, DragonBot, Londonclanger, Addbot, Mann jess, Erik9bot, I dream of horses, Jacobisq, Faust~enwiki, Za-spino, Daveco333, Helpful Pixie Bot, CsDix and Anonymous: 20

• Meaningless statement Source: https://en.wikipedia.org/wiki/Meaningless_statement?oldid=543699327 Contributors: Bryan Derksen,Chato, Fubar Obfusco, Julesd, Rossami, Kimiko, CimonAvaro, PaulHanson, Alai, ZendarPC, Aristotle Pagaltzis, SmackBot, Xyzzyplugh,Olsdude, CRGreathouse, Cumulus Clouds, Gregbard, Zalgo, Addbot, Erik9bot, FrescoBot, Machine Elf 1735 and Anonymous: 12

• Mission statement Source: https://en.wikipedia.org/wiki/Mission_statement?oldid=699856515Contributors: Lousyd, Kku, GTBacchus,Pcastellina, Rainer Wasserfuhr~enwiki, Jni, Robbot, HaeB, Art Carlson, Orangemike, Dav4is, Gyrofrog, MisfitToys, Neutrality, Gazpa-cho, Rich Farmbrough, John FitzGerald, Rgdboer, Bobo192, Orbst, PaulHanson, Arthena, Curious1i, Ceyockey, Scjessey, Pol098, Vary,RobertG, Divinus, Gurch, Bgwhite, FrankTobia, RadioFan, Voidxor, SmackBot, Delldot, Jab843, Canthusus, Gilliam, MalafayaBot,Octahedron80, DHN-bot~enwiki, Egsan Bacon, Fmiddleton, BesselDekker, Decltype, Andrew c, Kukini, Birdman1, SashatoBot, Kuru,

130 CHAPTER 43. VISION STATEMENT

Sir Nicholas de Mimsy-Porpington, Brian Gunderson, Minna Sora no Shita, $cammer, Skalman, Blehfu, Slippyd, Whereizben, Aris-teo, Thijs!bot, Mdotley, MikeLynch, Tesammon, MER-C, I80and, Magioladitis, Tedickey, WhatamIdoing, Drm310, Jackson Pee-bles, MartinBot, Oregongirl0407, Keith D, WotherspoonSmith, Jargon777, J.delanoy, Octopus-Hands, Icseaturtles, SneakyPiet, IonNegru, C.m.jones, Rpeh, TreasuryTag, Egurr, Lear’s Fool, Philip Trueman, TXiKiBoT, Grimpen, Vipinhari, Technopat, Mosmof, As-pire3623WXCi, LeaveSleaves, Mmurdoch, Propter Hoc, HiDrNick, Ohiostandard, Technion, SieBot, MurrayG002, Dawn Bard, SE7, Jonjoy 1999, How2, BlueAzure, Happysailor, Avnjay, KoshVorlon, Emesee, Brylie, Mygerardromance, Busy Stubber, Ochendzki, Twinsday,ClueBot, All Hallow’s Wraith, Mpseattle7, Arjunaraoc, Excirial, Hello Control, PixelBot, Rkskool, Mahue, Vdhegde, Brianboulton, Hunt-thetroll, Saebjorn, PCHS-NJROTC, Jenhilde, GordonUS, Helloboy 92, Libcub, WikHead, WikiDao, MystBot, Addbot, Blethering Scot,MrOllie, Nfs994, CarsracBot, Commasalot, Amulet Heart, Glane23, Z. Patterson, ChenzwBot, Luckas Blade, David0811, Luckas-bot,Yobot, Fraggle81, Guy1890, Brougham96, SwisterTwister, AnomieBOT, IRP, Crecy99, Materialscientist, Carlsotr, LilHelpa, Xqbot,JulianDelphiki, Shadowjams, SD5, Thehelpfulbot, Griffinofwales, Sushiflinger, Prari, VS6507, VI, Web10koh, Anindex, Pinethicket, Idream of horses, RedBot, Salvidrim!, Trappist the monk, ItsZippy, DARTH SIDIOUS 2, Sisterdarby, Jonbon903, Maq10solja, Noom-mos, DASHBot, Abyss of enchantment, Faceless Enemy, Greg.giersch, Wikipelli, Dcirovic, ZéroBot, Midas02, A930913, Wayne Slam,Tot12, ClueBot NG, O.Koslowski, Widr, Helpful Pixie Bot, Electriccatfish2, BG19bot, Chaosknightzhao, JeremyG92, Snow Blizzard,Glacialfox, Anbu121, BattyBot, IPWAI, EuroCarGT, Webclient101, Ocinternational, ToddBallowe, Erik a hanson, Faizan, Cosmicmom,Glaisher, TMGJannic, Dwarf99, Vozul, CamelCase, Andrew the hacker, Zceirbi, Muggmuggs and Anonymous: 312

• Normative statement Source: https://en.wikipedia.org/wiki/Normative_statement?oldid=641599440 Contributors: Pearle, John Quig-gin, Gwernol, SmackBot, Bluebot, LoonyLuke, Gregbard, Jarry1250, Tescoid, 7&6=thirteen, Dthomsen8, PatrickFlaherty, Yobot, Frag-gle81, RjwilmsiBot, ClueBot NG, Widr, Helpful Pixie Bot, Marcus Grant, BattyBot, Faizan, Seppi333, Verascope and Anonymous:17

• Objection (argument) Source: https://en.wikipedia.org/wiki/Objection_(argument)?oldid=666654835Contributors: Piotrus, Rich Farm-brough, SmackBot, Byelf2007, Grumpyyoungman01, Neelix, Gregbard, Al Lemos, CommonsDelinker, Haikon, Newbyguesses, Kivaan,Denisarona, Fadesga, Addbot, PranksterTurtle, Ptbotgourou, Erik9bot, Pollinosisss, Wikielwikingo, Mjbmrbot and Anonymous: 12

• Opening statement Source: https://en.wikipedia.org/wiki/Opening_statement?oldid=683321417 Contributors: Michael Hardy, Friedo,JamesMLane, Paul August, PaulHanson, BD2412, Jake Wartenberg, Lockley, Hibana, SmackBot, Psiphiorg, Eastlaw, Prosecutrix, Greg-bard, BlueAg09, Kevinmon, Uncle Dick, ClueBot, The Thing That Should Not Be, Kirby Victory Dance, Rror, Chasnor15, Erik9bot,Aetius41, Fæ, ClueBot NG, Jack Greenmaven, MusikAnimal, Suuzen, Contributor613 and Anonymous: 37

• Positive statement Source: https://en.wikipedia.org/wiki/Positive_statement?oldid=693521749 Contributors: Altenmann, RJHall, The-Project, John Quiggin, Tlroche, Catamorphism, Amalthea, SmackBot, Quaque, Can't sleep, clown will eat me, Xyzzyplugh, Soumyasch,Gregbard, Gltimmons, Jarry1250, Briholt, 7&6=thirteen, PatrickFlaherty, Jarble, Helpful Pixie Bot, Electriccatfish2, BG19bot, Mengyuan.Zhaoand Anonymous: 12

• Precept Source: https://en.wikipedia.org/wiki/Precept?oldid=691706585 Contributors: Steinsky, Nat Krause, JHCC, Thorwald, Dbach-mann, Stbalbach, Jnestorius, Ricky81682, Sandylouise, BD2412, Vary, Doc glasgow, DVdm, Gdrbot, Portress, RussBot, Jugander,SmackBot, Gilliam, Deeb, Meco, Nekohakase, CmdrObot, Neelix, Gregbard, Vaquero100, Cydebot, Sacca, Binarybits, Calaka, Indon,Peterhi, V.V zzzzz, Rucha58, Aesopos, Warren Forsythe, Ronhjones, Fraggle81, AnomieBOT, J04n, Haeinous, DrilBot, Skyerise, Jim-steele9999, Ὁ οἶστρος, ClueBot NG, BoltonSM3, Camyoung54, YiFeiBot, JimRenge, Eurodyne, PeterTVC, KasparBot and Anonymous:28

• Presidential Statement Source: https://en.wikipedia.org/wiki/Presidential_Statement?oldid=544573415Contributors: Rich Farmbrough,JJJJust, Mrzaius, Joriki, RussBot, Midway, Colonel Warden, Gregbard, TreasuryTag, Freeman501, Addbot, Erik9bot, EmausBot, HelpfulPixie Bot and Anonymous: 1

• Prior consistent statements and prior inconsistent statements Source: https://en.wikipedia.org/wiki/Prior_consistent_statements_and_prior_inconsistent_statements?oldid=613740223 Contributors: Ellsworth, PullUpYourSocks, BD2412, Edivorce, Gregbard, Cyde-bot, Lainestl, Fourthwaved, Snotbot and Anonymous: 5

• Proposition Source: https://en.wikipedia.org/wiki/Proposition?oldid=689631127 Contributors: AxelBoldt, Mav, Toby Bartels, Zoe,Stevertigo, K.lee, Michael Hardy, Zeno Gantner, TakuyaMurata, Minesweeper, Evercat, Sethmahoney, Conti, Reddi, Greenrd, Markhurd,Hyacinth, Banno, RedWolf, Ojigiri~enwiki, Timrollpickering, Tobias Bergemann, Giftlite, Jason Quinn, Stevietheman, Antandrus, Su-perborsuk, Sebbe, Amicuspublilius, Martpol, Hapsiainen, Vanished user lp09qa86ft, Chalst, Phiwum, Duesentrieb, Bobo192, Larryv,MPerel, Helix84, V2Blast, Ish ishwar, Emvee~enwiki, RJFJR, Bobrayner, Philthecow, Velho, Woohookitty, Kzollman, Isnow, Patl,Brolin Empey, Lakitu~enwiki, Fresheneesz, Bornhj, YurikBot, Hairy Dude, Rick Norwood, Wknight94, Finell, SmackBot, Evanreyes,Ignacioerrico, Bluebot, Jaymay, DHN-bot~enwiki, Cybercobra, Richard001, Lacatosias, Jon Awbrey, Vina-iwbot~enwiki, Byelf2007,Harryboyles, SilkTork, Ckatz, 16@r, Grumpyyoungman01, Stwalkerster, Caiaffa, Levineps, Iridescent, JoeBot, Gveret Tered, Eastlaw,CRGreathouse, CBM, Sdorrance, Andkore, Gregbard, Juansempere, Yesterdog, Thijs!bot, Barticus88, Kredal, AllenFerguson, Voyag-ing, NSH001, JAnDbot, MER-C, Leolaursen, Bookinvestor, Connormah, VoABot II, WhatamIdoing, Pomte, Tgeairn, J.delanoy, Ali,Ginsengbomb, Katalaveno, Coppertwig, Nieske, Funandtrvl, King Lopez, ABF, TXiKiBoT, Philogo, Tracerbullet11, Cnilep, Barkeep,SieBot, Legion fi, Oxymoron83, OKBot, ClueBot, The Thing That Should Not Be, Watchduck, Estirabot, Hans Adler, Hugo Herbelin,DumZiBoT, Makotoy, Crazy Boris with a red beard, Dthomsen8, Dwnelson, SilvonenBot, Good Olfactory, Addbot, Andrewghutchison,LAAFan, Luckas-bot, TheSuave, Denyss, THENWHOWAS PHONE?, Ehuss, KamikazeBot, AnomieBOT, E235, Yalckram, Wortafad,ArthurBot, Luis Felipe Schenone, Omnipaedista, FrescoBot, BrideOfKripkenstein, Motomuku, Pinethicket, A8UDI, Monkeymanman,Gamewizard71, FoxBot, Lotje, TheMesquito, Daliot, EmausBot, John of Reading, Eekerz, Look2See1, Honestrosewater, Britannic124,Bollyjeff, Coasterlover1994, Chewings72, ClueBot NG, Wcherowi, MelbourneStar, Satellizer, Masssly, Helpful Pixie Bot, Hans-JürgenStreicher~enwiki, ,زكريا ChrisGualtieri, Jochen Burghardt, Eyesnore, Purnendu Karmakar, DetectiveKraken, SanketDash, Ashika Bieber,Eavestn and Anonymous: 165

• Propositional formula Source: https://en.wikipedia.org/wiki/Propositional_formula?oldid=697111267 Contributors: Michael Hardy,Hyacinth, Timrollpickering, Tobias Bergemann, Filemon, Giftlite, Golbez, PWilkinson, Klparrot, Bookandcoffee, Woohookitty, Linas,Mindmatrix, Tabletop, BD2412, Kbdank71, Rjwilmsi, Bgwhite, YurikBot, Hairy Dude, RussBot, Open2universe, SmackBot, Hmains,Chris the speller, Bluebot, Colonies Chris, Tsca.bot, Jon Awbrey, Muhammad Hamza, Lambiam, Wvbailey, Wizard191, Iridescent,Happy-melon, ChrisCork, CBM, Gregbard, Cydebot, Julian Mendez, Nick Number, Arch dude, Djihed, R'n'B, Raise exception, Wiae,Billinghurst, Spinningspark, WRK, Maelgwnbot, Jaded-view, Mild Bill Hiccup, Neuralwarp, Addbot, Yobot, Adelpine, AnomieBOT,Neurolysis, LilHelpa, The Evil IP address, Kwiki, Klbrain, Kevin Gorman, Helpful Pixie Bot, BG19bot, PhnomPencil, Wolfmanx122,Jochen Burghardt, Mark viking, Knife-in-the-drawer, JJMC89 and Anonymous: 18

43.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 131

• Proxy statement Source: https://en.wikipedia.org/wiki/Proxy_statement?oldid=697631187 Contributors: Edward, Nurg, Pascal666,PaulHanson, Tim!, DocendoDiscimus, SmackBot, JPH-FM, Ohnoitsjamie, Kuru, Rldoromor, Stefan2, Codemorse, Billinghurst, Im-perfectlyInformed, Yobot, AnomieBOT, Eks287, Tribunal56 and Anonymous: 14

• Risk andSafety Statements Source: https://en.wikipedia.org/wiki/Risk_and_Safety_Statements?oldid=606473599Contributors: Sonett72,Cacycle, ArnoldReinhold, PaulHanson, Physchim62, RobotE, LeonardoRob0t, KaiserbBot, Beetstra, Lunisneko, JAnDbot, Howa0082,Acalamari, Arkwatem, Auntof6, Kbdankbot, Leszek Jańczuk, Fraggle81, Choij, Brane.Blokar, Dzsi, MerlLinkBot, Wayne Riddock,Raublexxion, L Kensington, Joebrick46, UAwiki, Francesca Cattaneo, RandomLittleHelper, *thing goes and Anonymous: 15

• Safety statement Source: https://en.wikipedia.org/wiki/Safety_statement?oldid=651727590 Contributors: Rich Farmbrough, PaulHan-son, Stephen, RHaworth, Tabletop, Jmorgan, Sceptre, CambridgeBayWeather, Rwxrwxrwx, SmackBot, Canthusus, Betacommand, Hu12,Karlhahn, Blackrock36, R'n'B, Katharineamy, Jeepday, C. J. Harrington, CWii, Flyingidiot, Scarian, Kbdankbot, Addbot, AnomieBOT,ClueBot NG, P0PP4B34R732, Khazar2, Epicgenius and Anonymous: 13

• Scope statement Source: https://en.wikipedia.org/wiki/Scope_statement?oldid=662417207Contributors: AndreasKaufmann, D6,Mould-ing, PaulHanson, Bernburgerin, Tony1, Bweaver, SmackBot, Fake User, Fabrictramp, DGG, Gjd001, SieBot, Pm master, Sanya3, Dead-Bot, Splach, DARaynor, Addbot, Rvintila, Macbookair3140, EmausBot, Jazzy Diva, Stephie Raymond, Angelo Mascaro, ClueBot NG,MerlIwBot, BattyBot, Robvdhorst and Anonymous: 18

• Sentence (linguistics) Source: https://en.wikipedia.org/wiki/Sentence_(linguistics)?oldid=696968041 Contributors: Stevertigo, MichaelHardy, Liftarn, Ahoerstemeier, Rob Hooft, Furrykef, Shizhao, Nufy8, RedWolf, Benc, Ruakh, Dbenbenn, Eran, Meursault2004, Andy-cjp, Beland, PDH, Burschik, Rick Burns, Mike Rosoft, Discospinster, Kdammers, Florian Blaschke, Mani1, Art LaPella, Stesmo, Jojitfb, Ncik~enwiki, Howrealisreal, Sowelilitokiemu, Melaen, Dejvid, Hojimachong, Woohookitty, Linas, Vikramkr, Jonathan de BoynePollard, GregorB, Umofomia, Stefanomione, Jendeyoung, Mandarax, Graham87, Sjö, Rjwilmsi, Mayumashu, Amire80, Stardust8212,Ianthegecko, Margosbot~enwiki, Chobot, Banaticus, Raelx, YurikBot, Wavelength, Phantomsteve, Bhny, Trondtr, Eleassar, AdiJapan,Everyguy, E Wing, Vicarious, Demon Slayer, SmackBot, Lestrade, InverseHypercube, Zerida, Bigbluefish, Chairman S., Arcan~enwiki,Commander Keane bot, Gilliam, Mazeface, Miguel Andrade, Lazar Taxon, Ardemus, -xfi-, Battamer, Kukini, Byelf2007, Ergative rlt,Rigadoun, JorisvS, IronGargoyle, Anand Karia, Rockpickle85, Keith-264, Levineps, Igoldste, Wolfdog, Randalllin, Schweiwikist, Knight-Lago, WeggeBot, Esalen~enwiki, Gregbard, Qrc2006, FilipeS, A D 13, Gimmetrow, PamD, JamesAM, Epbr123, N5iln, TXiKi, NER-IUM, Nick Number, Escarbot, AntiVandalBot, Antique Rose, Gökhan, JAnDbot, Casmith 789, Hostilebanana, VoABot II, Nimic86,Felicia222, Pere prlpz, MartinBot, Arjun01, R'n'B, Jesant13, Coin945, NewEnglandYankee, Alitha, Mlle thenardier, Treisijs, Winter-Spw, Suaven, Idioma-bot, VolkovBot, Andrea moro, Thomas.W, Philip Trueman, TXiKiBoT, BotKung, Ilyaroz, Gen. von Klinkerhoffen,Enviroboy, Cnilep, AlleborgoBot, Neparis, S.Örvarr.S, Dan Polansky, SieBot, Legion fi, Radon210, Oysterguitarist, Belinrahs, Hidden-fromview, Oxymoron83, Daniale93, ClueBot, Mild Bill Hiccup, Robby.is.on, Cfsenel, Excirial, MacedonianBoy, Terra Xin, Tnxman307,Chris.let, Rui Gabriel Correia, Bald Zebra, Vegetator, Horselover Frost, The Zig, Fastily, Snowmonster, ZooFari, Addbot, Brumski,NjardarBot, Jim10701, Quercus solaris, Rtz-bot, Ehrenkater, Zien3, Luckas-bot, Yobot, Anypodetos, Nallimbot, AnomieBOT, 1exec1,Rjanag, Piano non troppo, Flewis, Robinvar, ArthurBot, Xqbot, GrouchoBot, Backpackadam, In fact, Green Cardamom, JeffreyCTK,Telofy, Endofskull, BenzolBot, Kwiki, Briardew, DrilBot, Rushbugled13, Skyerise, White Shadows, FoxBot, Lostonsahil, Tjo3ya, Bron-teStormCherrypoppins4, RjwilmsiBot, EmausBot, WikitanvirBot, Themindsurgeon, Donner60, Carmichael, Petrb, Gwen-chan, ClueBotNG, Chadgadya, Gareth Griffith-Jones, SithShumail, Cntras, Widr, MerlIwBot, Helpful Pixie Bot, SchroCat, BG19bot, Annabelle Lukin,Thefactmasterofchucknorris, Geilamir, Victor Yus, Makecat-bot, Eyesnore, George8211, Crystallizedcarbon and Anonymous: 267

• Simple non-inferential passage Source: https://en.wikipedia.org/wiki/Simple_non-inferential_passage?oldid=578957871 Contributors:Salimfadhley, BD2412, Gregbard, LilHelpa and Saedon

• Special weather statement Source: https://en.wikipedia.org/wiki/Special_weather_statement?oldid=683576767 Contributors: PaulHan-son, Mrschimpf, ArielGold, SmackBot, Robomaeyhem, WxGopher, TimVickers, Southern Illinois SKYWARN, Xenon54, DumZiBoT,NellieBly, Yobot, Ren97, Salvio giuliano, NGUFreezee and Anonymous: 12

• Statement (computer science) Source: https://en.wikipedia.org/wiki/Statement_(computer_science)?oldid=699722300 Contributors:Mxn, Murray Langton, Lowellian, Wlievens, Haeleth, Macrakis, Rich Farmbrough, Danakil, Spoon!, Hooperbloob, SemperBlotto, TonySidaway, Ruud Koot, Sstrader, Roboto de Ajvol, Bhny, SmackBot, Jibjibjib, Skizzik, Nbarth, Fonebone, Derek farn, Dreftymac, GeorgPeter, Gregbard, Valodzka, Calaka, WinBot, PaperConfessional, Magioladitis, Hebrews412, Haacked, Hans Dunkelberg, Thomas Larsen,Raise exception, Remi0o, VolkovBot, SieBot, Callmejosh, Addbot, Prim Ethics, Luckas-bot, Synchronism, AnomieBOT, ВолодимирГруша, Doulos Christos, Erik9bot, Sae1962, HRoestBot, Shinjikun, Flamerecca, ZéroBot, Elaz85, Amr.rs, ChrisGualtieri, Pbarb, Me,Myself, and I are Here, Monkbot, KasparBot and Anonymous: 24

• Statement (logic) Source: https://en.wikipedia.org/wiki/Statement_(logic)?oldid=693543602 Contributors: Giftlite, Jason Quinn, Dis-cospinster, Woohookitty, Xover, BD2412, Rjwilmsi, OneWeirdDude, Fresheneesz, Rick Norwood, SmackBot, Delfeye, Neo-Jay, Physis,Gregbard, Al Lemos, R'n'B, Tgeairn, Philogo, OKBot, Francvs, C xong, Hans Adler, 51kwad, Addbot, ב ,.דניאל GB fan, TensaiKashou,Hriber, Akerans, ClueBot NG, Helpful Pixie Bot, Shaun, Zach Lipsitz, Khazar2, Saehry, Yamaha5 and Anonymous: 35

• Statement on theCo-operative Identity Source: https://en.wikipedia.org/wiki/Statement_on_the_Co-operative_Identity?oldid=602420358Contributors: Kaihsu, Tim Ivorson, AmishThrasher, PaulHanson, Baxrob, Wavelength, RussBot, Grafen, Bluebot, Dustingc, GreatBig-Circles, Noah Salzman, Barticus88, Terrypin, Brett epic, Bsilkey, Zlerman, DumZiBoT, MystBot, Addbot, Ptbotgourou, Teilolondon,CsDix, Amandawerhane and Anonymous: 1

• Theorem Source: https://en.wikipedia.org/wiki/Theorem?oldid=697737703 Contributors: AxelBoldt, Mav, Zundark, The Anome, Tar-quin, Tbackstr, XJaM, Aldie, Michael Hardy, Zeno Gantner, TakuyaMurata, Bagpuss, Glenn, Tim Retout, Rotem Dan, Andres, CharlesMatthews, Dcoetzee, Bemoeial, Hyacinth, Traroth, SirPeebles, Moriel~enwiki, Josh Cherry, Fredrik, MathMartin, Ojigiri~enwiki, Tim-rollpickering, Hadal, Alan Liefting, Snobot, Ancheta Wis, Tosha, Giftlite, Monedula, Fropuff, Fishal, Chowbok, Alaz, MarkSweep,Karol Langner, Jacob grace, Pmanderson, Tyler McHenry, Hkpawn~enwiki, Tzanko Matev, Joyous!, EugeneZelenko, Discospinster,Rich Farmbrough, Paul August, Bender235, Gauge, Tompw, El C, Edwinstearns, Billymac00, Sasquatch, Alansohn, Gary, Sciurinæ,Joriki, Igny, Ruud Koot, Eras-mus, Mekong Bluesman, Tslocum, Graham87, BD2412, Gmelli, Sdornan, Salix alba, Jrtayloriv, AndriuZ,M7bot, Chobot, MithrandirMage, Sbrools, DVdm, Algebraist, Roboto de Ajvol, Borgx, Chaos, Trovatore, Dbfirs, Bota47, Tomisti,Arthur Rubin, Kier07, Anclation~enwiki, Curpsbot-unicodify, Erudy, Finell, Sardanaphalus, SmackBot, RDBury, Bomac, Skizzik, Christhe speller, Fuzzform, MalafayaBot, DHN-bot~enwiki, Sholto Maud, Cybercobra, Acdx, SashatoBot, Lambiam, IronGargoyle, Craig-block, Lalaith, Autonova, Mike Fikes, Zero sharp, JRSpriggs, CRGreathouse, Hi.ro, CBM, Gregbard, Cydebot, R Harris, Moxmalin,Thijs!bot, Epbr123, Mrcs, James086, Nick Number, AntiVandalBot, Widefox, Thenub314, .anacondabot, Magioladitis, Dvptl, Animum,

132 CHAPTER 43. VISION STATEMENT

Gwern, Stephenchou0722, Pomte, Hippasus the Younger, Fcsuper, Jeepday, Nznancy, Coppertwig, Haseldon, Tparameter, Fjbfour,Dessources, DavidCBryant, Caiodnh, Austinmohr, VolkovBot, Psmythirl, Am Fiosaigear~enwiki, TXiKiBoT, Rei-bot, IKiddo, Voor-landt, Philogo, Geometry guy, Wiae, Graymornings, Dmcq, HiDrNick, DestroyerofDreams, Hthoreau2, Newbyguesses, SieBot, Respir,Huzzah018, Oxymoron83, SimonTrew, OKBot, Kumioko (renamed), DesolateReality, Wjemather, Loren.wilton, ClueBot, Blanchardb,Excirial, Alexbot, TheSnacks, Hans Adler, Muzz 2008, MonoBot, XLinkBot, Burkaja, Marc van Leeuwen, SilvonenBot, Badgernet,Addbot, DrThunder88, Some jerk on the Internet, Rainbow-five, CanadianLinuxUser, CarsracBot, Dr. Universe, Favonian, Numbo3-bot, Flatfish89, Stepfordswife, Zorrobot, Legobot, Cote d'Azur, Luckas-bot, Yobot, II MusLiM HyBRiD II, Kristen Eriksen, LilHelpa,Xqbot, Doezxcty, Capricorn42, Almabot, Miym, GrouchoBot, Jubb-green, RibotBOT, FrescoBot, Sirtywell, Haeinous, Heptadecagon,BigDwiki, RedBot, Eric wisniewski, EmausBot, ZéroBot, The Nut, Xzenu, GZ-Bot, H3llBot, D.Lazard, ChuispastonBot, RockMagnetist,ClueBot NG, Helpful Pixie Bot, Howald, Jibun, bukiyou desu kara, Khazar2, Oracions, Wywin, Yamaha5, WillemienH, Loraof, EoRdE6,BabyChastie, Nbro, KasparBot, Blakktaktiks, Youknowwhatimsayin, AmitMakwana008 and Anonymous: 112

• Truth claim Source: https://en.wikipedia.org/wiki/Truth_claim?oldid=644784453 Contributors: ADM, Rjwilmsi, Malcolma, SmackBot,Onorem, Sejtam,Wolfdog, Gregbard, Cydebot, Genedoug, Andrewaskew, M krishna 71, Flyer22 Reborn, Canis Lupus, Addbot, Erik9bot,EmausBot, BattyBot, ChrisGualtieri, DoctorKubla, Goldspotter, Mogism, Purnendu Karmakar, MsGingerHoneycutt and Anonymous: 7

• Truth-apt Source: https://en.wikipedia.org/wiki/Truth-apt?oldid=609854856 Contributors: Gregbard and Anders Sandberg• Vision statement Source: https://en.wikipedia.org/wiki/Vision_statement?oldid=697789207 Contributors: Kku, Mydogategodshat, Lu-

mos3, Bearcat, AnomieBOT, Amaury, Anindex, ClueBot NG, BG19bot, Iaritmioawp, HFiratUnlucayakli, Alaynestone and Anonymous:2

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134 CHAPTER 43. VISION STATEMENT

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