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  • foral l xCalgary Remix/AccessibleAn Introduction toFormal Logic

    P. D. MagnusTim Buttonwith addit ions byJ. Robert Loftisremixed and revised byAaron Thomas-BolducRichard Zach

    Winter 2018

  • forall x : Calgary Remix

    An Introduction toFormal Logic

    By P. D. MagnusTim Button

    with addit ions byJ. Robert Loftis

    remixed and revised byAaron Thomas-Bolduc

    Richard Zach

  • Winter 2018

  • This book is based on forallx: Cambridge, byTim ButtonUniversity of Cambridge

    used under a CC BY-SA 3.0 license, which is basedin turn on forallx, by

    P.D. MagnusUniversity at Albany, State University of New


    used under a CC BY-SA 3.0 license,and was remixed, revised, & expanded byAaron Thomas-Bolduc & Richard ZachUniversity of Calgary

    It includes additional material from forallx by P.D.Magnus and Metatheory by Tim Button, both used undera CC BY-SA 3.0 license, and from forallx: LorainCounty Remix, by Cathal Woods and J. Robert Loftis,used under a CC BY-SA 4.0 license.

    This work is licensed under a Creative CommonsAttribution-ShareAlike 4.0 license. You are free to copy andredistribute the material in any medium or format, and remix,

  • transform, and build upon the material for any purpose, evencommercially, under the following terms:

    . You must give appropriate credit, provide a link to thelicense, and indicate if changes were made. You may doso in any reasonable manner, but not in any way thatsuggests the licensor endorses you or your use.

    . If you remix, transform, or build upon the material, youmust distribute your contributions under the samelicense as the original.

    The LATEX source for this book is available onGitHub. This version is revision 28b0d8d(2017-11-25).

    The preparation of this textbook was made possibleby a grant from the Taylor Institute for Teaching andLearning.

  • ContentsPreface ix

    I Key notions of logic 1

    1 Arguments 22 Valid arguments 103 Other logical notions 19

    II Truth-functional logic 32

    4 First steps to symbolization 335 Connectives 416 Sentences of TFL 717 Use and mention 82


  • Contents vi

    III Truth tables 91

    8 Characteristic truth tables 929 Truth-functional connectives 9610 Complete truth tables 10611 Semantic concepts 11912 Truth table shortcuts 13713 Partial truth tables 147

    IV Natural deduction for TFL 159

    14 The very idea of natural deduction 16015 Basic rules for TFL 16516 Additional rules for TFL 20817 Proof-theoretic concepts 22018 Proof strategies 22719 Derived rules 23020 Soundness and completeness 243

    V First-order logic 258

    21 Building blocks of FOL 25922 Sentences with one quantifier 275

  • Contents vii

    23 Multiple generality 30024 Identity 32425 Definite descriptions 33426 Sentences of FOL 350

    VI Interpretations 360

    27 Extensionality 36128 Truth in FOL 37429 Semantic concepts 38930 Using interpretations 39231 Reasoning about all interpretations 405

    VII Natural deduction for FOL 413

    32 Basic rules for FOL 41433 Conversion of quantifiers 43934 Rules for identity 44335 Derived rules 45036 Proof-theoretic and semantic concepts 453

  • Contents viii

    VIII Advanced Topics 459

    37 Normal forms and expressive completeness 460

    Appendices 481

    A Symbolic notation 481B Alternative proof systems 487C Quick reference 498

    Glossary 513

  • PrefaceAs the title indicates, this is a textbook on formallogic. Formal logic concerns the study of a certainkind of language which, like any language, canserve to express states of affairs. It is a formallanguage, i.e., its expressions (such as sentences)are defined formally. This makes it a very usefullanguage for being very precise about the states ofaffairs its sentences describe. In particular, informal logic is is impossible to be ambiguous. Thestudy of these languages centres on therelationship of entailment between sentences, i.e.,which sentences follow from which other sentences.Entailment is central because by understanding itbetter we can tell when some states of affairs mustobtain provided some other states of affairs obtain.


  • Preface x

    But entailment is not the only important notion. Wewill also consider the relationship of beingconsistent, i.e., of not being mutually contradictory.These notions can be defined semantically, usingprecise definitions of entailment based oninterpretations of the languageorproof-theoretically, using formal systems ofdeduction.

    Formal logic is of course a centralsub-discipline of philosophy, where the logicalrelationship of assumptions to conclusions reachedfrom them is important. Philosophers investigatethe consequences of definitions and assumptionsand evaluate these definitions and assumptions onthe basis of their consequences. It is also importantin mathematics and computer science. Inmathematics, formal languages are used todescribe not everyday states of affairs, butmathematical states of affairs. Mathematicians arealso interested in the consequences of definitionsand assumptions, and for them it is equallyimportant to establish these consequences (whichthey call theorems) using completely precise and

  • Preface xi

    rigorous methods. Formal logic provides suchmethods. In computer science, formal logic isapplied to describe the state and behaviours ofcomputational systems, e.g., circuits, programs,databases, etc. Methods of formal logic canlikewise be used to establish consequences of suchdescriptions, such as whether a circuit is error-free,whether a program does what its intended to do,whether a database is consistent or if something istrue of the data in it.

    The book is divided into eight parts. Part Iintroduces the topic and notions of logic in aninformal way, without introducing a formal languageyet. Parts IIIV concern truth-functional languages.In it, sentences are formed from basic sentencesusing a number of connectives (or, and, not, if. . . then) which just combine sentences into morecomplicated ones. We discuss logical notions suchas entailment in two ways: semantically, using themethod of truth tables (in Part III) andproof-theoretically, using a system of formalderivations (in Part IV). PartsVVII deal with amore complicated language, that of first-order

  • Preface xii

    logic. It includes, in addition to the connectives oftruth-functional logic, also names, predicates,identity, and the so-called quantifiers. Theseadditional elements of the language make it muchmore expressive than the truth-functional language,and well spend a fair amount of time investigatingjust how much one can express in it. Again, logicalnotions for the language of first-order logic aredefined semantically, using interpretations, andproof-theoretically, using a more complex versionof the formal derivation system introduced inPart IV. Part VIII covers an advanced topic: that ofexpressive adequacy of the truth-functionalconnectives.

    In the appendices youll find a discussion ofalternative notations for the languages we discussin this text, of alternative derivation systems, and aquick reference listing most of the important rulesand definitions. The central terms are listed in aglossary at the very end.

    This book is based on a text originally writtenby P. D. Magnus and revised and expanded by Tim

  • Preface xiii

    Button and independently by J. Robert Loftis. AaronThomas-Bolduc and Richard Zach have combinedelements of these texts into the present version,changed some of the terminology and examples,and added material of their own. The resulting textis licensed under a Creative CommonsAttribution-ShareAlike 4.0 license.

  • Part I

    Key notions oflogic


  • Chapter 1


    Logic is the business of evaluating arguments;sorting the good from the bad.

    In everyday language, we sometimes use theword argument to talk about belligerent shoutingmatches. If you and a friend have an argument inthis sense, things are not going well between thetwo of you. Logic is not concerned with suchteeth-gnashing and hair-pulling. They are notarguments, in our sense; they are disagreements.

    An argument, as we will understand it, issomething more like this:

    It is raining heavily.2

  • Chapter 1. Arguments 3

    If you do not take an umbrella, you will getsoaked.

    .. You should take an umbrella.

    We here have a series of sentences. The three dotson the third line of the argument are readtherefore. They indicate that the final sentenceexpresses the conclusion of the argument. Thetwo sentences before that are the premises of theargument. If you believe the premises, then theargument (perhaps) provides you with a reason tobelieve the conclusion.

    This is the sort of thing that logicians areinterested in. We will say that an argument is anycollection of premises, together with a conclusion.

    This Part discusses some basic logical notionsthat apply to arguments in a natural language likeEnglish. It is important to begin with a clearunderstanding of what arguments are and of what itmeans for an argument to be valid. Later we willtranslate arguments from English into a formallanguage. We want formal validity, as defined in the

  • Chapter 1. Arguments 4

    formal language, to have at least some of theimportant features of natural-language validity.

    In the example just given, we used individualsentences to express both of the argumentspremises, and we used a third sentence to expressthe arguments conclusion. Many arguments areexpressed in this way, but a single sentence cancontain a complete argument. Consider:

    I was wearing my sunglasses; so it musthave been sunny.

    This argument has one premise followed by aconclusion.

    Many arguments start with premises, and endwith a conclusion, but not all of them. The argumentwith which this section began might equally havebeen presented with the conclusion at thebeginning, like so:

    You should take an umbrella. After all, itis raining heavily. And if you do not take

  • Chapter 1. Arguments 5

    an umbrella, you will get soaked.

    Equally, it might have been presented with theconclusion in the middle:

    It is raining heavily. Accordingly, youshould take an umbrella, given that if youdo not take an umbrella, you will getsoaked.

    When approaching an argument, we want to knowwhether or not the conclusion follows from thepremises. So the first thing to do is to separate outthe conclusion from the premises. As a guideline,the following words are often used to indicate anarguments conclusion:

    so, therefore, hence, thus, accordingly,consequently

    Similarly, these expressions often indicate that weare dealing with a premise, rather than aconclusion:

  • Chapter 1. Arguments 6

    since, because, given that

    But in analysing an argument, there is no substitutefor a good nose.

    1.1 Sentences

    To be perfectly general, we can define an argumentas a series of sentences. The sentences at thebeginning of the series are premises. The finalsentence in the series is the conclusion. If thepremises are true and the argument is a good one,then you have a reason to accept the conclusion.

    In logic, we are only interested in sentencesthat can figure as a premise or conclusion of anargument. So we will say that a sentence issomething that can be true or false.

    You should not confuse the idea of a sentencethat can be true or false with the difference betweenfact and opinion. Often, sentences in logic willexpress things that would count as facts such asKierkegaard was a hunchback or Kierkegaard

  • Chapter 1. Arguments 7

    liked almonds. They can also express things thatyou might think of as matters of opinion such as,Almonds are tasty.

    Also, there are things that would count assentences in a linguistics or grammar course thatwe will not count as sentences in logic.Questions In a grammar class, Are you sleepyyet? would count as an interrogative sentence.Although you might be sleepy or you might be alert,the question itself is neither true nor false. For thisreason, questions will not count as sentences inlogic. Suppose you answer the question: I am notsleepy. This is either true or false, and so it is asentence in the logical sense. Generally,questions will not count as sentences, butanswers will.

    What is this course about? is not a sentence(in our sense). No one knows what this course isabout is a sentence.Imperatives Commands are often phrased asimperatives like Wake up!, Sit up straight, and so

  • Chapter 1. Arguments 8

    on. In a grammar class, these would count asimperative sentences. Although it might be good foryou to sit up straight or it might not, the command isneither true nor false. Note, however, thatcommands are not always phrased as imperatives.You will respect my authority is either true orfalse either you will or you will not and so itcounts as a sentence in the logical sense.Exclamations Ouch! is sometimes called anexclamatory sentence, but it is neither true norfalse. We will treat Ouch, I hurt my toe! asmeaning the same thing as I hurt my toe. Theouch does not add anything that could be true orfalse.

    Practice exercises

    At the end of some chapters, there are problemsthat review and explore the material covered in thechapter. There is no substitute for actually workingthrough some problems, because logic is moreabout a way of thinking than it is about memorizingfacts.

  • Chapter 1. Arguments 9

    Highlight the phrase which expresses theconclusion of each of these arguments:

    1. It is sunny. So I should take my sunglasses.2. It must have been sunny. I did wear mysunglasses, after all.

    3. No one but you has had their hands in thecookie-jar. And the scene of the crime islittered with cookie-crumbs. Youre the culprit!

    4. Miss Scarlett and Professor Plum were in thestudy at the time of the murder. ReverendGreen had the candlestick in the ballroom, andwe know that there is no blood on his hands.Hence Colonel Mustard did it in the kitchenwith the lead-piping. Recall, after all, that thegun had not been fired.

  • Chapter 2


    In 1, we gave a very permissive account of what anargument is. To see just how permissive it is,consider the following:

    There is a bassoon-playing dragon in theCathedra Romana.

    .. Salvador Dali was a poker player.

    We have been given a premise and a conclusion. Sowe have an argument. Admittedly, it is a terribleargument, but it is still an argument.


  • Chapter 2. Valid arguments 11

    2.1 Two ways that argumentscan go wrong

    It is worth pausing to ask what makes the argumentso weak. In fact, there are two sources of weakness.First: the arguments (only) premise is obviouslyfalse. The Popes throne is only ever occupied by ahat-wearing man. Second: the conclusion does notfollow from the premise of the argument. Even ifthere were a bassoon-playing dragon in the Popesthrone, we would not be able to draw anyconclusion about Dalis predilection for poker.

    What about the main argument discussed in 1?The premises of this argument might well be false.It might be sunny outside; or it might be that youcan avoid getting soaked without taking anumbrella. But even if both premises were true, itdoes not necessarily show you that you should takean umbrella. Perhaps you enjoy walking in the rain,and you would like to get soaked. So, even if bothpremises were true, the conclusion mightnonetheless be false.

  • Chapter 2. Valid arguments 12

    The general point is as follows. For anyargument, there are two ways that it might go wrong:

    One or more of the premises might be false. The conclusion might not follow from thepremises.

    To determine whether or not the premises of anargument are true is often a very important matter.However, that is normally a task best left to expertsin the field: as it might be, historians, scientists, orwhomever. In our role as logicians, we are moreconcerned with arguments in general. So we are(usually) more concerned with the second way inwhich arguments can go wrong.

    So: we are interested in whether or not aconclusion follows from some premises. Dont,though, say that the premises infer the conclusion.Entailment is a relation between premises andconclusions; inference is something we do. (So ifyou want to mention inference when the conclusionfollows from the premises, you could say that onemay infer the conclusion from the premises.)

  • Chapter 2. Valid arguments 13

    2.2 Validity

    As logicians, we want to be able to determine whenthe conclusion of an argument follows from thepremises. One way to put this is as follows. Wewant to know whether, if all the premises were true,the conclusion would also have to be true. Thismotivates a definition:An argument is valid if and only if it is impos-sible for all of the premises to be true and theconclusion false.The crucial thing about a valid argument is that

    it is impossible for the premises to be true whilst theconclusion is false. Consider this example:

    Oranges are either fruits or musicalinstruments.Oranges are not fruits.

    .. Oranges are musical instruments.

    The conclusion of this argument is ridiculous.Nevertheless, it follows from the premises. If bothpremises were true, then the conclusion just has to

  • Chapter 2. Valid arguments 14

    be true. So the argument is valid.This highlights that valid arguments do not

    need to have true premises or true conclusions.Conversely, having true premises and a trueconclusion is not enough to make an argument valid.Consider this example:

    London is in England.Beijing is in China.

    .. Paris is in France.

    The premises and conclusion of this argument are,as a matter of fact, all true, but the argument isinvalid. If Paris were to declare independence fromthe rest of France, then the conclusion would befalse, even though both of the premises wouldremain true. Thus, it is possible for the premises ofthis argument to be true and the conclusion false.The argument is therefore invalid.

    The important thing to remember is that validityis not about the actual truth or falsity of thesentences in the argument. It is about whether it is

  • Chapter 2. Valid arguments 15

    possible for all the premises to be true and theconclusion false. Nonetheless, we will say that anargument is sound if and only if it is both valid andall of its premises are true.

    2.3 Inductive arguments

    Many good arguments are invalid. Consider thisone:

    In January 1997, it rained in London.In January 1998, it rained in London.In January 1999, it rained in London.In January 2000, it rained in London.

    .. It rains every January in London.This argument generalises from observations aboutseveral cases to a conclusion about all cases. Sucharguments are called inductive arguments. Theargument could be made stronger by addingadditional premises before drawing the conclusion:In January 2001, it rained in London; In January2002. . . . But, however many premises of this formwe add, the argument will remain invalid. Even if it

  • Chapter 2. Valid arguments 16

    has rained in London in every January thus far, itremains possible that London will stay dry nextJanuary.

    The point of all this is that inductiveargumentseven good inductive argumentsarenot (deductively) valid. They are not watertight.Unlikely though it might be, it is possible for theirconclusion to be false, even when all of theirpremises are true. In this book, we will set aside(entirely) the question of what makes for a goodinductive argument. Our interest is simply in sortingthe (deductively) valid arguments from the invalidones.

    Practice exercises

    A. Which of the following arguments are valid?Which are invalid?

    1. Socrates is a man.2. All men are carrots... Socrates is a carrot.

  • Chapter 2. Valid arguments 17

    1. Abe Lincoln was either born in Illinois or hewas once president.

    2. Abe Lincoln was never president... Abe Lincoln was born in Illinois.

    1. If I pull the trigger, Abe Lincoln will die.2. I do not pull the trigger... Abe Lincoln will not die.

    1. Abe Lincoln was either from France or fromLuxemborg.

    2. Abe Lincoln was not from Luxemborg... Abe Lincoln was from France.

    1. If the world were to end today, then I would notneed to get up tomorrow morning.

    2. I will need to get up tomorrow morning... The world will not end today.

    1. Joe is now 19 years old.2. Joe is now 87 years old... Bob is now 20 years old.

  • Chapter 2. Valid arguments 18

    B. Could there be:

    1. A valid argument that has one false premiseand one true premise?

    2. A valid argument that has only false premises?3. A valid argument with only false premises anda false conclusion?

    4. An invalid argument that can be made valid bythe addition of a new premise?

    5. A valid argument that can be made invalid bythe addition of a new premise?

    In each case: if so, give an example; if not, explainwhy not.

  • Chapter 3

    Other logicalnotions

    In 2, we introduced the idea of a valid argument.We will want to introduce some more ideas that areimportant in logic.

    3.1 Joint possibility

    Consider these two sentences:B1. Janes only brother is shorter than her.B2. Janes only brother is taller than her.Logic alone cannot tell us which, if either, of thesesentences is true. Yet we can say that if the first


  • Chapter 3. Other logical notions 20

    sentence (B1) is true, then the second sentence(B2) must be false. Similarly, if B2 is true, then B1must be false. It is impossible that both sentencesare true together. These sentences are inconsistentwith each other, they cannot all be true at the sametime. This motivates the following definition:Sentences are jointly possible if and only if itis possible for them all to be true together.Conversely, B1 and B2 are jointly impossible.We can ask about the possibility of any number

    of sentences. For example, consider the followingfour sentences:

    G1. There are at least four giraffes at the wildanimal park.

    G2. There are exactly seven gorillas at the wildanimal park.

    G3. There are not more than two martians at thewild animal park.

    G4. Every giraffe at the wild animal park is amartian.

  • Chapter 3. Other logical notions 21

    G1 and G4 together entail that there are at leastfour martian giraffes at the park. This conflicts withG3, which implies that there are no more than twomartian giraffes there. So the sentences G1G4 arejointly impossible. They cannot all be true together.(Note that the sentences G1, G3 and G4 are jointlyimpossible. But if sentences are already jointlyimpossible, adding an extra sentence to the mix willnot make them jointly possible!)

    3.2 Necessary truths, necessaryfalsehoods, andcontingency

    In assessing arguments for validity, we care aboutwhat would be true if the premises were true, butsome sentences just must be true. Consider thesesentences:

    1. It is raining.2. Either it is raining here, or it is not.3. It is both raining here and not raining here.

  • Chapter 3. Other logical notions 22

    In order to know if sentence 1 is true, you wouldneed to look outside or check the weather channel.It might be true; it might be false. A sentence whichis capable of being true and capable of being false(in different circumstances, of course) is calledcontingent.

    Sentence 2 is different. You do not need to lookoutside to know that it is true. Regardless of whatthe weather is like, it is either raining or it is not.That is a necessary truth.

    Equally, you do not need to check the weatherto determine whether or not sentence 3 is true. Itmust be false, simply as a matter of logic. It mightbe raining here and not raining across town; it mightbe raining now but stop raining even as you finishthis sentence; but it is impossible for it to be bothraining and not raining in the same place and at thesame time. So, whatever the world is like, it is notboth raining here and not raining here. It is anecessary falsehood.

  • Chapter 3. Other logical notions 23

    Necessary equivalence

    We can also ask about the logical relationsbetween two sentences. For example:

    John went to the store after he washed thedishes.John washed the dishes before he went to thestore.

    These two sentences are both contingent, sinceJohn might not have gone to the store or washeddishes at all. Yet they must have the sametruth-value. If either of the sentences is true, thenthey both are; if either of the sentences is false,then they both are. When two sentences necessarilyhave the same truth value, we say that they arenecessarily equivalent.

    Summary of logical notions

    . An argument is (deductively) valid if it isimpossible for the premises to be true and theconclusion false; it is invalid otherwise.

  • Chapter 3. Other logical notions 24

    . A necessary truth is a sentence that must betrue, that could not possibly be false.

    . A necessary falsehood is a sentence thatmust be false, that could not possibly be true.

    . A contingent sentence is neither a necessarytruth nor a necessary falsehood. It may be truebut could have been false, or vice versa.

    . Two sentences are necessarily equivalent ifthey must have the same truth value.

    . A collection of sentences is jointly possible ifit is possible for all these sentences to be truetogether; it is jointly impossible otherwise.

    Practice exercises

    A. For each of the following: Is it a necessary truth,a necessary falsehood, or contingent?

    1. Caesar crossed the Rubicon.2. Someone once crossed the Rubicon.3. No one has ever crossed the Rubicon.

  • Chapter 3. Other logical notions 25

    4. If Caesar crossed the Rubicon, then someonehas.

    5. Even though Caesar crossed the Rubicon, noone has ever crossed the Rubicon.

    6. If anyone has ever crossed the Rubicon, it wasCaesar.

    B. For each of the following: Is it a necessary truth,a necessary falsehood, or contingent?1. Elephants dissolve in water.2. Wood is a light, durable substance useful forbuilding things.

    3. If wood were a good building material, it wouldbe useful for building things.

    4. I live in a three story building that is twostories tall.

    5. If gerbils were mammals they would nursetheir young.

    C. Which of the following pairs of sentences arenecessarily equivalent?

    1. Elephants dissolve in water.

  • Chapter 3. Other logical notions 26

    If you put an elephant in water, it willdisintegrate.

    2. All mammals dissolve in water.If you put an elephant in water, it willdisintegrate.

    3. George Bush was the 43rd president.Barack Obama is the 44th president.

    4. Barack Obama is the 44th president.Barack Obama was president immediately afterthe 43rd president.

    5. Elephants dissolve in water.All mammals dissolve in water.

    D. Which of the following pairs of sentences arenecessarily equivalent?

    1. Thelonious Monk played piano.John Coltrane played tenor sax.

    2. Thelonious Monk played gigs with JohnColtrane.John Coltrane played gigs with TheloniousMonk.

    3. All professional piano players have big hands.

  • Chapter 3. Other logical notions 27

    Piano player Bud Powell had big hands.4. Bud Powell suffered from severe mentalillness.All piano players suffer from severe mentalillness.

    5. John Coltrane was deeply religious.John Coltrane viewed music as an expressionof spirituality.

    E. Consider the following sentences:

    G1 There are at least four giraffes at the wildanimal park.

    G2 There are exactly seven gorillas at the wildanimal park.

    G3 There are not more than two Martians at thewild animal park.

    G4 Every giraffe at the wild animal park is aMartian.

  • Chapter 3. Other logical notions 28

    Now consider each of the following collectionsof sentences. Which are jointly possible? Which arejointly impossible?

    1. Sentences G2, G3, and G42. Sentences G1, G3, and G43. Sentences G1, G2, and G44. Sentences G1, G2, and G3

    F. Consider the following sentences.

    M1 All people are mortal.M2 Socrates is a person.M3 Socrates will never die.M4 Socrates is mortal.

    Which combinations of sentences are jointlypossible? Mark each possible or impossible.

    1. Sentences M1, M2, and M32. Sentences M2, M3, and M43. Sentences M2 and M3

  • Chapter 3. Other logical notions 29

    4. Sentences M1 and M45. Sentences M1, M2, M3, and M4

    G. Which of the following is possible? If it ispossible, give an example. If it is not possible,explain why.1. A valid argument that has one false premiseand one true premise

    2. A valid argument that has a false conclusion3. A valid argument, the conclusion of which is anecessary falsehood

    4. An invalid argument, the conclusion of which isa necessary truth

    5. A necessary truth that is contingent6. Two necessarily equivalent sentences, both ofwhich are necessary truths

    7. Two necessarily equivalent sentences, one ofwhich is a necessary truth and one of which iscontingent

    8. Two necessarily equivalent sentences thattogether are jointly impossible

    9. A jointly possible collection of sentences thatcontains a necessary falsehood

  • Chapter 3. Other logical notions 30

    10. A jointly impossible set of sentences thatcontains a necessary truth

    H. Which of the following is possible? If it ispossible, give an example. If it is not possible,explain why.

    1. A valid argument, whose premises are allnecessary truths, and whose conclusion iscontingent

    2. A valid argument with true premises and afalse conclusion

    3. A jointly possible collection of sentences thatcontains two sentences that are notnecessarily equivalent

    4. A jointly possible collection of sentences, allof which are contingent

    5. A false necessary truth6. A valid argument with false premises7. A necessarily equivalent pair of sentences thatare not jointly possible

    8. A necessary truth that is also a necessaryfalsehood

  • Chapter 3. Other logical notions 31

    9. A jointly possible collection of sentences thatare all necessary falsehoods

  • Part II



  • Chapter 4

    First steps tosymbolization

    4.1 Validity in virtue of form

    Consider this argument:

    It is raining outside.If it is raining outside, then Jenny is miserable.

    .. Jenny is miserable.

    and another argument:

    Jenny is an anarcho-syndicalist.


  • Chapter 4. First steps to symbolization 34

    If Jenny is an anarcho-syndicalist, then Dipanis an avid reader of Tolstoy.

    .. Dipan is an avid reader of Tolstoy.

    Both arguments are valid, and there is astraightforward sense in which we can say that theyshare a common structure. We might express thestructure thus:

    AIf A, then C

    .. C

    This looks like an excellent argument structure.Indeed, surely any argument with this structurewill be valid, and this is not the only good argumentstructure. Consider an argument like:

    Jenny is either happy or sad.Jenny is not happy.

    .. Jenny is sad.

    Again, this is a valid argument. The structure hereis something like:

  • Chapter 4. First steps to symbolization 35

    A or Bnot-A

    .. B

    A superb structure! Here is another example:

    Its not the case that Jim both studied hard andacted in lots of plays.Jim studied hard

    .. Jim did not act in lots of plays.

    This valid argument has a structure which we mightrepresent thus:

    not-(A and B)A

    .. not-B

    These examples illustrate an important idea, whichwe might describe as validity in virtue of form.The validity of the arguments just considered hasnothing very much to do with the meanings ofEnglish expressions like Jenny is miserable,

  • Chapter 4. First steps to symbolization 36

    Dipan is an avid reader of Tolstoy, or Jim acted inlots of plays. If it has to do with meanings at all, itis with the meanings of phrases like and, or, not,and if. . . , then. . . .

    In Parts IIIV, we are going to develop a formallanguage which allows us to symbolize manyarguments in such a way as to show that they arevalid in virtue of their form. That language will betruth-functional logic, or TFL.

    4.2 Validity for special reasons

    There are plenty of arguments that are valid, but notfor reasons relating to their form. Take an example:

    Juanita is a vixen.. Juanita is a fox

    It is impossible for the premise to be true and theconclusion false. So the argument is valid.However, the validity is not related to the form ofthe argument. Here is an invalid argument with thesame form:

  • Chapter 4. First steps to symbolization 37

    Juanita is a vixen.. Juanita is a cathedral

    This might suggest that the validity of the firstargument is keyed to the meaning of the wordsvixen and fox. But, whether or not that is right, itis not simply the shape of the argument that makesit valid. Equally, consider the argument:

    The sculpture is green all over... The sculpture is not red all over.

    Again, it seems impossible for the premise to betrue and the conclusion false, for nothing can beboth green all over and red all over. So theargument is valid, but here is an invalid argumentwith the same form:

    The sculpture is green all over... The sculpture is not shiny all over.

    The argument is invalid, since it is possible to begreen all over and shiny all over. (One might paint

  • Chapter 4. First steps to symbolization 38

    their nails with an elegant shiny green varnish.)Plausibly, the validity of the first argument is keyedto the way that colours (or colour-words) interact,but, whether or not that is right, it is not simply theshape of the argument that makes it valid.

    The important moral can be stated as follows.At best, TFL will help us to understandarguments that are valid due to their form.

    4.3 Atomic sentences

    We started isolating the form of an argument, in4.1, by replacing subsentences of sentences withindividual letters. Thus in the first example of thissection, it is raining outside is a subsentence of Ifit is raining outside, then Jenny is miserable, andwe replaced this subsentence with A.

    Our artificial language, TFL, pursues this ideaabsolutely ruthlessly. We start with some atomicsentences. These will be the basic building blocksout of which more complex sentences are built. Wewill use uppercase Roman letters for atomic

  • Chapter 4. First steps to symbolization 39

    sentences of TFL. There are only twenty-six lettersof the alphabet, but there is no limit to the numberof atomic sentences that we might want to consider.By adding subscripts to letters, we obtain newatomic sentences. So, here are five different atomicsentences of TFL:

    A , P , P1 , P2 , A234We will use atomic sentences to represent, orsymbolize, certain English sentences. To do this,we provide a symbolization key, such as thefollowing:

    A: It is raining outsideC : Jenny is miserable

    In doing this, we are not fixing this symbolizationonce and for all. We are just saying that, for thetime being, we will think of the atomic sentence ofTFL, A , as symbolizing the English sentence It israining outside, and the atomic sentence of TFL,C , as symbolizing the English sentence Jenny ismiserable. Later, when we are dealing with

  • Chapter 4. First steps to symbolization 40

    different sentences or different arguments, we canprovide a new symbolization key; as it might be:

    A: Jenny is an anarcho-syndicalistC : Dipan is an avid reader of Tolstoy

    It is important to understand that whatever structurean English sentence might have is lost when it issymbolized by an atomic sentence of TFL. From thepoint of view of TFL, an atomic sentence is just aletter. It can be used to build more complexsentences, but it cannot be taken apart.

  • Chapter 5


    In the previous chapter, we considered symbolizingfairly basic English sentences with atomicsentences of TFL. This leaves us wanting to dealwith the English expressions and, or, not, andso forth. These are connectivesthey can be usedto form new sentences out of old ones. In TFL, wewill make use of logical connectives to buildcomplex sentences from atomic components. Thereare five logical connectives in TFL. This tablesummarises them, and they are explainedthroughout this section.

    These are not the only connectives of English ofinterest. Others are, e.g., unless, neither . . . nor. . . , and because. We will see that the first two


  • Chapter 5. Connectives 42

    symbol what it is called rough meaning negation It is not the case that. . . conjunction Both. . . and . . . disjunction Either. . . or . . . conditional If . . . then . . . biconditional . . . if and only if . . . can be expressed by the connectives we willdiscuss, while the last cannot. Because, incontrast to the others, is not truth functional.

    5.1 Negation

    Consider how we might symbolize these sentences:

    1. Mary is in Barcelona.2. It is not the case that Mary is in Barcelona.3. Mary is not in Barcelona.

    In order to symbolize sentence 1, we will need anatomic sentence. We might offer this symbolizationkey:

    B : Mary is in Barcelona.

  • Chapter 5. Connectives 43

    Since sentence 2 is obviously related to sentence 1,we will not want to symbolize it with a completelydifferent sentence. Roughly, sentence 2 meanssomething like It is not the case that B. In order tosymbolize this, we need a symbol for negation. Wewill use . Now we can symbolize sentence 2 withB .

    Sentence 3 also contains the word not, and itis obviously equivalent to sentence 2. As such, wecan also symbolize it with B .A sentence can be symbolized as A if it canbe paraphrased in English as It is not the casethat. . . .It will help to offer a few more examples:

    4. The widget can be replaced.5. The widget is irreplaceable.6. The widget is not irreplaceable.

    Let us use the following representation key:

    R : The widget is replaceable

  • Chapter 5. Connectives 44

    Sentence 4 can now be symbolized by R . Movingon to sentence 5: saying the widget is irreplaceablemeans that it is not the case that the widget isreplaceable. So even though sentence 5 does notcontain the word not, we will symbolize it asfollows: R .

    Sentence 6 can be paraphrased as It is not thecase that the widget is irreplaceable. Which canagain be paraphrased as It is not the case that it isnot the case that the widget is replaceable. So wemight symbolize this English sentence with the TFLsentence R .

    But some care is needed when handlingnegations. Consider:

    7. Jane is happy.8. Jane is unhappy.

    If we let the TFL-sentence H symbolize Jane ishappy, then we can symbolize sentence 7 as H .However, it would be a mistake to symbolizesentence 8 with H . If Jane is unhappy, then she

  • Chapter 5. Connectives 45

    is not happy; but sentence 8 does not mean thesame thing as It is not the case that Jane is happy.Jane might be neither happy nor unhappy; shemight be in a state of blank indifference. In order tosymbolize sentence 8, then, we would need a newatomic sentence of TFL.

    5.2 Conjunction

    Consider these sentences:

    9. Adam is athletic.10. Barbara is athletic.11. Adam is athletic, and Barbara is also athletic.

    We will need separate atomic sentences of TFL tosymbolize sentences 9 and 10; perhaps

    A: Adam is athletic.B : Barbara is athletic.

    Sentence 9 can now be symbolized as A , andsentence 10 can be symbolized as B . Sentence 11

  • Chapter 5. Connectives 46

    roughly says A and B. We need another symbol, todeal with and. We will use . Thus we willsymbolize it as (A B). This connective is calledconjunction. We also say that A and B are thetwo conjuncts of the conjunction (A B).

    Notice that we make no attempt to symbolizethe word also in sentence 11. Words like both andalso function to draw our attention to the fact thattwo things are being conjoined. Maybe they affectthe emphasis of a sentence, but we will not (andcannot) symbolize such things in TFL.

    Some more examples will bring out this point:

    12. Barbara is athletic and energetic.13. Barbara and Adam are both athletic.14. Although Barbara is energetic, she is not

    athletic.15. Adam is athletic, but Barbara is more athletic

    than him.

    Sentence 12 is obviously a conjunction. Thesentence says two things (about Barbara). In

  • Chapter 5. Connectives 47

    English, it is permissible to refer to Barbara onlyonce. It might be tempting to think that we need tosymbolize sentence 12 with something along thelines of B and energetic. This would be a mistake.Once we symbolize part of a sentence as B , anyfurther structure is lost, as B is an atomic sentenceof TFL. Conversely, energetic is not an Englishsentence at all. What we are aiming for is somethinglike B and Barbara is energetic. So we need to addanother sentence letter to the symbolization key.Let E symbolize Barbara is energetic. Now theentire sentence can be symbolized as (B E ).

    Sentence 13 says one thing about two differentsubjects. It says of both Barbara and Adam thatthey are athletic, even though in English we use theword athletic only once. The sentence can beparaphrased as Barbara is athletic, and Adam isathletic. We can symbolize this in TFL as (B A),using the same symbolization key that we havebeen using.

    Sentence 14 is slightly more complicated. Theword although sets up a contrast between the first

  • Chapter 5. Connectives 48

    part of the sentence and the second part.Nevertheless, the sentence tells us both thatBarbara is energetic and that she is not athletic. Inorder to make each of the conjuncts an atomicsentence, we need to replace she with Barbara.So we can paraphrase sentence 14 as, BothBarbara is energetic, and Barbara is not athletic.The second conjunct contains a negation, so weparaphrase further: Both Barbara is energetic andit is not the case that Barbara is athletic. Now wecan symbolize this with the TFL sentence (E B).Note that we have lost all sorts of nuance in thissymbolization. There is a distinct difference in tonebetween sentence 14 and Both Barbara is energeticand it is not the case that Barbara is athletic. TFLdoes not (and cannot) preserve these nuances.

    Sentence 15 raises similar issues. There is acontrastive structure, but this is not something thatTFL can deal with. So we can paraphrase thesentence as Both Adam is athletic, and Barbara ismore athletic than Adam. (Notice that we onceagain replace the pronoun him with Adam.) Howshould we deal with the second conjunct? We

  • Chapter 5. Connectives 49

    already have the sentence letter A , which is beingused to symbolize Adam is athletic, and thesentence B which is being used to symbolizeBarbara is athletic; but neither of these concernstheir relative athleticity. So, to symbolize the entiresentence, we need a new sentence letter. Let theTFL sentence R symbolize the English sentenceBarbara is more athletic than Adam. Now we cansymbolize sentence 15 by (A R).A sentence can be symbolized as (AB) if it canbe paraphrased in English as Both. . . , and. . . ,or as . . . , but . . . , or as although . . . , . . . .You might be wondering why we put brackets

    around the conjunctions. The reason for this isbrought out by considering how negation mightinteract with conjunction. Consider:16. Its not the case that you will get both soup and

    salad.17. You will not get soup but you will get salad.Sentence 16 can be paraphrased as It is not thecase that: both you will get soup and you will get

  • Chapter 5. Connectives 50

    salad. Using this symbolization key:

    S1 : You will get soup.S2 : You will get salad.

    We would symbolize both you will get soup and youwill get salad as (S1 S2). To symbolize sentence16, then, we simply negate the whole sentence,thus: (S1 S2).

    Sentence 17 is a conjunction: you will not getsoup, and you will get salad. You will not get soupis symbolized by S1 . So to symbolize sentence17 itself, we offer (S1 S2).

    These English sentences are very different, andtheir symbolizations differ accordingly. In one ofthem, the entire conjunction is negated. In theother, just one conjunct is negated. Brackets helpus to keep track of things like the scope of thenegation.

  • Chapter 5. Connectives 51

    5.3 Disjunction

    Consider these sentences:

    18. Either Fatima will play videogames, or she willwatch movies.

    19. Either Fatima or Omar will play videogames.

    For these sentences we can use this symbolizationkey:

    F : Fatima will play videogames.O : Omar will play videogames.M : Fatima will watch movies.

    However, we will again need to introduce a newsymbol. Sentence 18 is symbolized by (F M ).The connective is called disjunction. We also saythat F and M are the disjuncts of the disjunction(F M ).

    Sentence 19 is only slightly more complicated.There are two subjects, but the English sentenceonly gives the verb once. However, we can

  • Chapter 5. Connectives 52

    paraphrase sentence 19 as Either Fatima will playvideogames, or Omar will play videogames. Nowwe can obviously symbolize it by (F O ) again.A sentence can be symbolized as (AB) if it canbe paraphrased in English as Either. . . , or. . . .Each of the disjuncts must be a sentence.Sometimes in English, the word or is used in a

    way that excludes the possibility that both disjunctsare true. This is called an exclusive or. Anexclusive or is clearly intended when it says, on arestaurant menu, Entrees come with either soup orsalad: you may have soup; you may have salad;but, if you want both soup and salad, then you haveto pay extra.

    At other times, the word or allows for thepossibility that both disjuncts might be true. This isprobably the case with sentence 19, above. Fatimamight play videogames alone, Omar might playvideogames alone, or they might both play.Sentence 19 merely says that at least one of themplays videogames. This is called an inclusive or.The TFL symbol always symbolizes an inclusive

  • Chapter 5. Connectives 53

    or.It might help to see negation interact with

    disjunction. Consider:

    20. Either you will not have soup, or you will nothave salad.

    21. You will have neither soup nor salad.22. You get either soup or salad, but not both.

    Using the same symbolization key as before,sentence 20 can be paraphrased in this way: Eitherit is not the case that you get soup, or it is not thecase that you get salad. To symbolize this in TFL,we need both disjunction and negation. It is not thecase that you get soup is symbolized by S1 . It isnot the case that you get salad is symbolized byS2 . So sentence 20 itself is symbolized by(S1 S2).

    Sentence 21 also requires negation. It can beparaphrased as, It is not the case that either youget soup or you get salad. Since this negates theentire disjunction, we symbolize sentence 21 with

  • Chapter 5. Connectives 54

    (S1 S2).Sentence 22 is an exclusive or. We can break

    the sentence into two parts. The first part says thatyou get one or the other. We symbolize this as(S1 S2). The second part says that you do not getboth. We can paraphrase this as: It is not the caseboth that you get soup and that you get salad.Using both negation and conjunction, we symbolizethis with (S1 S2). Now we just need to put thetwo parts together. As we saw above, but canusually be symbolized with . Sentence 22 canthus be symbolized as ((S1 S2) (S1 S2)).

    This last example shows something important.Although the TFL symbol always symbolizesinclusive or, we can symbolize an exclusive or inTFL. We just have to use a few of our other symbolsas well.

    5.4 Conditional

    Consider these sentences:

  • Chapter 5. Connectives 55

    23. If Jean is in Paris, then Jean is in France.24. Jean is in France only if Jean is in Paris.

    Lets use the following symbolization key:

    P : Jean is in Paris.F : Jean is in France

    Sentence 23 is roughly of this form: if P, then F.We will use the symbol to symbolize this if. . . ,then. . . structure. So we symbolize sentence 23 by(P F ). The connective is called the conditional.Here, P is called the antecedent of the conditional(P F ), and F is called the consequent.

    Sentence 24 is also a conditional. Since theword if appears in the second half of the sentence,it might be tempting to symbolize this in the sameway as sentence 23. That would be a mistake. Yourknowledge of geography tells you that sentence 23is unproblematically true: there is no way for Jeanto be in Paris that doesnt involve Jean being inFrance. But sentence 24 is not so straightforward:were Jean in Dieppe, Lyons, or Toulouse, Jean

  • Chapter 5. Connectives 56

    would be in France without being in Paris, therebyrendering sentence 24 false. Since geographyalone dictates the truth of sentence 23, whereastravel plans (say) are needed to know the truth ofsentence 24, they must mean different things.

    In fact, sentence 24 can be paraphrased as IfJean is in France, then Jean is in Paris. So we cansymbolize it by (F P ).A sentence can be symbolized as A B if itcan be paraphrased in English as If A, then Bor A only if B.

    In fact, many English expressions can berepresented using the conditional. Consider:

    25. For Jean to be in Paris, it is necessary thatJean be in France.

    26. It is a necessary condition on Jeans being inParis that she be in France.

    27. For Jean to be in France, it is sufficient thatJean be in Paris.

    28. It is a sufficient condition on Jeans being inFrance that she be in Paris.

  • Chapter 5. Connectives 57

    If we think deeply about it, all four of thesesentences mean the same as If Jean is in Paris,then Jean is in France. So they can all besymbolized by P F .

    It is important to bear in mind that theconnective tells us only that, if the antecedentis true, then the consequent is true. It says nothingabout a causal connection between two events (forexample). In fact, we lose a huge amount when weuse to symbolize English conditionals. We willreturn to this in 9.3 and 11.5.

    5.5 Biconditional

    Consider these sentences:

    29. Laika is a dog only if she is a mammal30. Laika is a dog if she is a mammal31. Laika is a dog if and only if she is a mammal

    We will use the following symbolization key:

    D : Laika is a dog

  • Chapter 5. Connectives 58

    M : Laika is a mammal

    Sentence 29, for reasons discussed above, can besymbolized by D M .

    Sentence 30 is importantly different. It can beparaphrased as, If Laika is a mammal then Laika isa dog. So it can be symbolized by M D .

    Sentence 31 says something stronger thaneither 29 or 30. It can be paraphrased as Laika is adog if Laika is a mammal, and Laika is a dog only ifLaika is a mammal. This is just the conjunction ofsentences 29 and 30. So we can symbolize it as(D M ) (M D ). We call this a biconditional,because it entails the conditional in both directions.

    We could treat every biconditional this way. So,just as we do not need a new TFL symbol to dealwith exclusive or, we do not really need a new TFLsymbol to deal with biconditionals. Because thebiconditional occurs so often, however, we will usethe symbol for it. We can then symbolizesentence 31 with the TFL sentence D M .

  • Chapter 5. Connectives 59

    The expression if and only if occurs a lotespecially in philosophy, mathematics, and logic.For brevity, we can abbreviate it with the snappierword iff. We will follow this practice. So if withonly one f is the English conditional. But iff withtwo fs is the English biconditional. Armed with thiswe can say:A sentence can be symbolized as A B if itcan be paraphrased in English as A iff B; thatis, as A if and only if B.A word of caution. Ordinary speakers of English

    often use if . . . , then. . . when they really mean touse something more like . . . if and only if . . . .Perhaps your parents told you, when you were achild: if you dont eat your greens, you wont getany dessert. Suppose you ate your greens, but thatyour parents refused to give you any dessert, on thegrounds that they were only committed to theconditional (roughly if you get dessert, then youwill have eaten your greens), rather than thebiconditional (roughly, you get dessert iff you eatyour greens). Well, a tantrum would rightly ensue.

  • Chapter 5. Connectives 60

    So, be aware of this when interpreting people; butin your own writing, make sure you use thebiconditional iff you mean to.

    5.6 Unless

    We have now introduced all of the connectives ofTFL. We can use them together to symbolize manykinds of sentences. An especially difficult case iswhen we use the English-language connectiveunless:

    32. Unless you wear a jacket, you will catch a cold.33. You will catch a cold unless you wear a jacket.

    These two sentences are clearly equivalent. Tosymbolize them, we will use the symbolization key:

    J : You will wear a jacket.D : You will catch a cold.

    Both sentences mean that if you do not wear ajacket, then you will catch a cold. With this in mind,we might symbolize them as J D .

  • Chapter 5. Connectives 61

    Equally, both sentences mean that if you do notcatch a cold, then you must have worn a jacket.With this in mind, we might symbolize them asD J .

    Equally, both sentences mean that either youwill wear a jacket or you will catch a cold. With thisin mind, we might symbolize them as J D .

    All three are correct symbolizations. Indeed, inchapter 11 we will see that all three symbolizationsare equivalent in TFL.If a sentence can be paraphrased as Unless A,B, then it can be symbolized as AB.Again, though, there is a little complication.

    Unless can be symbolized as a conditional; but aswe said above, people often use the conditional (onits own) when they mean to use the biconditional.Equally, unless can be symbolized as adisjunction; but there are two kinds of disjunction(exclusive and inclusive). So it will not surprise youto discover that ordinary speakers of English oftenuse unless to mean something more like the

  • Chapter 5. Connectives 62

    biconditional, or like exclusive disjunction.Suppose someone says: I will go running unless itrains. They probably mean something like I will gorunning iff it does not rain (i.e. the biconditional),or either I will go running or it will rain, but notboth (i.e. exclusive disjunction). Again: be awareof this when interpreting what other people havesaid, but be precise in your writing.

    Practice exercises

    A. Using the symbolization key given, symbolizeeach English sentence in TFL.

    M : Those creatures are men in suits.C : Those creatures are chimpanzees.G : Those creatures are gorillas.

    1. Those creatures are not men in suits.2. Those creatures are men in suits, or they arenot.

    3. Those creatures are either gorillas orchimpanzees.

  • Chapter 5. Connectives 63

    4. Those creatures are neither gorillas norchimpanzees.

    5. If those creatures are chimpanzees, then theyare neither gorillas nor men in suits.

    6. Unless those creatures are men in suits, theyare either chimpanzees or they are gorillas.

    B. Using the symbolization key given, symbolizeeach English sentence in TFL.

    A: Mister Ace was murdered.B : The butler did it.C : The cook did it.D : The Duchess is lying.E : Mister Edge was murdered.F : The murder weapon was a frying pan.

    1. Either Mister Ace or Mister Edge wasmurdered.

    2. If Mister Ace was murdered, then the cook didit.

    3. If Mister Edge was murdered, then the cookdid not do it.

  • Chapter 5. Connectives 64

    4. Either the butler did it, or the Duchess is lying.5. The cook did it only if the Duchess is lying.6. If the murder weapon was a frying pan, thenthe culprit must have been the cook.

    7. If the murder weapon was not a frying pan,then the culprit was either the cook or thebutler.

    8. Mister Ace was murdered if and only if MisterEdge was not murdered.

    9. The Duchess is lying, unless it was MisterEdge who was murdered.

    10. If Mister Ace was murdered, he was done inwith a frying pan.

    11. Since the cook did it, the butler did not.12. Of course the Duchess is lying!

    C. Using the symbolization key given, symbolizeeach English sentence in TFL.

    E1 : Ava is an electrician.E2 : Harrison is an electrician.F1 : Ava is a firefighter.F2 : Harrison is a firefighter.

  • Chapter 5. Connectives 65

    S1 : Ava is satisfied with her career.S2 : Harrison is satisfied with his career.

    1. Ava and Harrison are both electricians.2. If Ava is a firefighter, then she is satisfied withher career.

    3. Ava is a firefighter, unless she is anelectrician.

    4. Harrison is an unsatisfied electrician.5. Neither Ava nor Harrison is an electrician.6. Both Ava and Harrison are electricians, butneither of them find it satisfying.

    7. Harrison is satisfied only if he is a firefighter.8. If Ava is not an electrician, then neither isHarrison, but if she is, then he is too.

    9. Ava is satisfied with her career if and only ifHarrison is not satisfied with his.

    10. If Harrison is both an electrician and afirefighter, then he must be satisfied with hiswork.

    11. It cannot be that Harrison is both an electricianand a firefighter.

    12. Harrison and Ava are both firefighters if and

  • Chapter 5. Connectives 66

    only if neither of them is an electrician.

    D. Using the symbolization key given, symbolizeeach English-language sentence in TFL.

    J1 : John Coltrane played tenor sax.J2 : John Coltrane played soprano sax.J3 : John Coltrane played tubaM1 : Miles Davis played trumpetM2 : Miles Davis played tuba

    1. John Coltrane played tenor and soprano sax.2. Neither Miles Davis nor John Coltrane playedtuba.

    3. John Coltrane did not play both tenor sax andtuba.

    4. John Coltrane did not play tenor sax unless healso played soprano sax.

    5. John Coltrane did not play tuba, but MilesDavis did.

    6. Miles Davis played trumpet only if he alsoplayed tuba.

  • Chapter 5. Connectives 67

    7. If Miles Davis played trumpet, then JohnColtrane played at least one of these threeinstruments: tenor sax, soprano sax, or tuba.

    8. If John Coltrane played tuba then Miles Davisplayed neither trumpet nor tuba.

    9. Miles Davis and John Coltrane both playedtuba if and only if Coltrane did not play tenorsax and Miles Davis did not play trumpet.

    E. Give a symbolization key and symbolize thefollowing English sentences in TFL.

    1. Alice and Bob are both spies.2. If either Alice or Bob is a spy, then the codehas been broken.

    3. If neither Alice nor Bob is a spy, then the coderemains unbroken.

    4. The German embassy will be in an uproar,unless someone has broken the code.

    5. Either the code has been broken or it has not,but the German embassy will be in an uproarregardless.

    6. Either Alice or Bob is a spy, but not both.

  • Chapter 5. Connectives 68

    F. Give a symbolization key and symbolize thefollowing English sentences in TFL.1. If there is food to be found in the pridelands,then Rafiki will talk about squashed bananas.

    2. Rafiki will talk about squashed bananas unlessSimba is alive.

    3. Rafiki will either talk about squashed bananasor he wont, but there is food to be found in thepridelands regardless.

    4. Scar will remain as king if and only if there isfood to be found in the pridelands.

    5. If Simba is alive, then Scar will not remain asking.

    G. For each argument, write a symbolization keyand symbolize all of the sentences of the argumentin TFL.1. If Dorothy plays the piano in the morning, thenRoger wakes up cranky. Dorothy plays piano inthe morning unless she is distracted. So ifRoger does not wake up cranky, then Dorothymust be distracted.

  • Chapter 5. Connectives 69

    2. It will either rain or snow on Tuesday. If itrains, Neville will be sad. If it snows, Nevillewill be cold. Therefore, Neville will either besad or cold on Tuesday.

    3. If Zoog remembered to do his chores, thenthings are clean but not neat. If he forgot, thenthings are neat but not clean. Therefore, thingsare either neat or clean; but not both.

    H. For each argument, write a symbolization keyand symbolize the argument as well as possible inTFL. The part of the passage in italics is there toprovide context for the argument, and doesnt needto be symbolized.1. It is going to rain soon. I know because my legis hurting, and my leg hurts if its going to rain.

    2. Spider-man tries to figure out the badguys plan. If Doctor Octopus gets theuranium, he will blackmail the city. I amcertain of this because if Doctor Octopus getsthe uranium, he can make a dirty bomb, and ifhe can make a dirty bomb, he will blackmailthe city.

  • Chapter 5. Connectives 70

    3. A westerner tries to predict the policies ofthe Chinese government. If the Chinesegovernment cannot solve the water shortagesin Beijing, they will have to move the capital.They dont want to move the capital. Thereforethey must solve the water shortage. But theonly way to solve the water shortage is todivert almost all the water from the Yangziriver northward. Therefore the Chinesegovernment will go with the project to divertwater from the south to the north.

    I. We symbolized an exclusive or using , ,and . How could you symbolize an exclusive orusing only two connectives? Is there any way tosymbolize an exclusive or using only oneconnective?

  • Chapter 6

    Sentences ofTFLThe sentence either apples are red, or berries areblue is a sentence of English, and the sentence(A B) is a sentence of TFL. Although we canidentify sentences of English when we encounterthem, we do not have a formal definition ofsentence of English. But in this chapter, we willoffer a complete definition of what counts as asentence of TFL. This is one respect in which aformal language like TFL is more precise than anatural language like English.


  • Chapter 6. Sentences of TFL 72

    6.1 Expressions

    We have seen that there are three kinds of symbolsin TFL:

    Atomic sentences A , B , C , . . . , Zwith subscripts, as needed A1 , B1 , Z1 , A2 , A25 , J375 , . . .

    Connectives , , ,,

    Brackets ( , )We define an expression of TFL as any string ofsymbols of TFL. Take any of the symbols of TFL andwrite them down, in any order, and you have anexpression of TFL.

    6.2 Sentences

    Of course, many expressions of TFL will be totalgibberish. We want to know when an expression ofTFL amounts to a sentence.

  • Chapter 6. Sentences of TFL 73

    Obviously, individual atomic sentences like A and G13 should count as sentences. We can formfurther sentences out of these by using the variousconnectives. Using negation, we can get A andG13 . Using conjunction, we can get (A G13),(G13 A), (A A), and (G13 G13). We could alsoapply negation repeatedly to get sentences likeA or apply negation along with conjunction to getsentences like (A G13) and (G13 G13). Thepossible combinations are endless, even startingwith just these two sentence letters, and there areinfinitely many sentence letters. So there is nopoint in trying to list all the sentences one by one.

    Instead, we will describe the process by whichsentences can be constructed. Consider negation:Given any sentence A of TFL, A is a sentence ofTFL. (Why the funny fonts? We return to this in7.3.)

    We can say similar things for each of the otherconnectives. For instance, if A and B are sentencesof TFL, then (AB) is a sentence of TFL. Providingclauses like this for all of the connectives, we arrive

  • Chapter 6. Sentences of TFL 74

    at the following formal definition for a sentence ofTFL:

    1. Every atomic sentence is a sentence.2. If A is a sentence, then A is a sentence.3. If A and B are sentences, then (AB) isa sentence.

    4. If A and B are sentences, then (AB) isa sentence.

    5. If A and B are sentences, then (A B) isa sentence.

    6. If A and B are sentences, then (A B) isa sentence.

    7. Nothing else is a sentence.Definitions like this are called recursive.

    Recursive definitions begin with some specifiablebase elements, and then present ways to generateindefinitely many more elements by compoundingtogether previously established ones. To give you abetter idea of what a recursive definition is, we can

  • Chapter 6. Sentences of TFL 75

    give a recursive definition of the idea of anancestor of mine. We specify a base clause.

    My parents are ancestors of mine.

    and then offer further clauses like:

    If x is an ancestor of mine, then xs parents areancestors of mine.

    Nothing else is an ancestor of mine.

    Using this definition, we can easily check to seewhether someone is my ancestor: just checkwhether she is the parent of the parent of. . . one ofmy parents. And the same is true for our recursivedefinition of sentences of TFL. Just as the recursivedefinition allows complex sentences to be built upfrom simpler parts, the definition allows us todecompose sentences into their simpler parts.Once we get down to atomic sentences, then weknow we are ok.

    Lets consider some examples.

  • Chapter 6. Sentences of TFL 76

    Suppose we want to know whether or not D is a sentence of TFL. Looking at the second clauseof the definition, we know that D is a sentenceif D is a sentence. So now we need to askwhether or not D is a sentence. Again looking atthe second clause of the definition, D is asentence if D is. So, D is a sentence if D is asentence. Now D is an atomic sentence of TFL, sowe know that D is a sentence by the first clause ofthe definition. So for a compound sentence likeD , we must apply the definition repeatedly.Eventually we arrive at the atomic sentences fromwhich the sentence is built up.

    Next, consider the example (P (Q R)).Looking at the second clause of the definition, thisis a sentence if (P (Q R)) is, and this is asentence if both P and (Q R) are sentences.The former is an atomic sentence, and the latter is asentence if (Q R) is a sentence. It is. Looking atthe fourth clause of the definition, this is a sentenceif both Q and R are sentences, and both are!

    Ultimately, every sentence is constructed nicely

  • Chapter 6. Sentences of TFL 77

    out of atomic sentences. When we are dealing witha sentence other than an atomic sentence, we cansee that there must be some sentential connectivethat was introduced last, when constructing thesentence. We call that connective the main logicaloperator of the sentence. In the case of D ,the main logical operator is the very first sign. Inthe case of (P (Q R)), the main logicaloperator is . In the case of ((E F ) G), themain logical operator is .

    The recursive structure of sentences in TFL willbe important when we consider the circumstancesunder which a particular sentence would be true orfalse. The sentence D is true if and only if thesentence D is false, and so on through thestructure of the sentence, until we arrive at theatomic components. We will return to this point inPart III.

    The recursive structure of sentences in TFLalso allows us to give a formal definition of thescope of a negation (mentioned in 5.2). The scopeof a is the subsentence for which is the main

  • Chapter 6. Sentences of TFL 78

    logical operator. Consider a sentence like:(P ((R B) Q ))

    which was constructed by conjoining P with((R B) Q ). This last sentence was constructedby placing a biconditional between (R B) andQ . The former of these sentencesa subsentenceof our original sentenceis a sentence for which is the main logical operator. So the scope of thenegation is just (R B). More generally:The scope of a connective (in a sentence) isthe subsentence for which that connective isthe main logical operator.

    6.3 Bracketing conventions

    Strictly speaking, the brackets in (Q R) are anindispensable part of the sentence. Part of this isbecause we might use (Q R) as a subsentence ina more complicated sentence. For example, wemight want to negate (Q R), obtaining (Q R).If we just had Q R without the brackets and put anegation in front of it, we would have Q R . It is

  • Chapter 6. Sentences of TFL 79

    most natural to read this as meaning the same thingas (Q R), but as we saw in 5.2, this is verydifferent from (Q R).

    Strictly speaking, then, Q R is not asentence. It is a mere expression.

    When working with TFL, however, it will makeour lives easier if we are sometimes a little lessthan strict. So, here are some convenientconventions.

    First, we allow ourselves to omit the outermostbrackets of a sentence. Thus we allow ourselves towrite Q R instead of the sentence (Q R).However, we must remember to put the bracketsback in, when we want to embed the sentence into amore complicated sentence!

    Second, it can be a bit painful to stare at longsentences with many nested pairs of brackets. Tomake things a bit easier on the eyes, we will allowourselves to use square brackets, [ and ], insteadof rounded ones. So there is no logical differencebetween (P Q ) and [P Q ], for example.

  • Chapter 6. Sentences of TFL 80

    Combining these two conventions, we canrewrite the unwieldy sentence

    (((H I ) (I H )) (J K ))

    rather more clearly as follows:[(H I ) (I H )

    ] (J K )

    The scope of each connective is now much easier topick out.

    Practice exercises

    A. For each of the following: (a) Is it a sentence ofTFL, strictly speaking? (b) Is it a sentence of TFL,allowing for our relaxed bracketing conventions?

    1. (A)2. J374 J3743. F4. S5. (G G)6. (A (A F )) (D E )7. [(Z S) W ] [J X ]8. (F D J) (C D )

  • Chapter 6. Sentences of TFL 81

    B. Are there any sentences of TFL that contain noatomic sentences? Explain your answer.C. What is the scope of each connective in thesentence [

    (H I ) (I H )] (J K )

  • Chapter 7

    Use andmention

    In this Part, we have talked a lot about sentences.So we should pause to explain an important, andvery general, point.

    7.1 Quotation conventions

    Consider these two sentences:

    Justin Trudeau is the Prime Minister. The expression Justin Trudeau is composedof two uppercase letters and eleven lowercaseletters


  • Chapter 7. Use and mention 83

    When we want to talk about the Prime Minister, weuse his name. When we want to talk about thePrime Ministers name, we mention that name,which we do by putting it in quotation marks.

    There is a general point here. When we want totalk about things in the world, we just use words.When we want to talk about words, we typicallyhave to mention those words. We need to indicatethat we are mentioning them, rather than usingthem. To do this, some convention is needed. Wecan put them in quotation marks, or display themcentrally in the page (say). So this sentence:

    Justin Trudeau is the Prime Minister.

    says that some expression is the Prime Minister.Thats false. The man is the Prime Minister; hisname isnt. Conversely, this sentence:

    Justin Trudeau is composed of two uppercaseletters and eleven lowercase letters.

    also says something false: Justin Trudeau is a man,

  • Chapter 7. Use and mention 84

    made of flesh rather than letters. One finalexample:

    Justin Trudeau is the name of JustinTrudeau.

    On the left-hand-side, here, we have the name of aname. On the right hand side, we have a name.Perhaps this kind of sentence only occurs in logictextbooks, but it is true nonetheless.

    Those are just general rules for quotation, andyou should observe them carefully in all your work!To be clear, the quotation-marks here do notindicate indirect speech. They indicate that you aremoving from talking about an object, to talkingabout the name of that object.

    7.2 Object language andmetalanguage

    These general quotation conventions are ofparticular importance for us. After all, we are

  • Chapter 7. Use and mention 85

    describing a formal language here, TFL, and so weare often mentioning expressions from TFL.

    When we talk about a language, the languagethat we are talking about is called the objectlanguage. The language that we use to talk aboutthe object language is called the metalanguage.

    For the most part, the object language in thischapter has been the formal language that we havebeen developing: TFL. The metalanguage isEnglish. Not conversational English exactly, butEnglish supplemented with some additionalvocabulary which helps us to get along.

    Now, we have used italic uppercase letters foratomic sentences of TFL:

    A , B , C , Z , A1 , B4 , A25 , J375 , . . .These are sentences of the object language (TFL).They are not sentences of English. So we must notsay, for example:

    D is an atomic sentence of TFL.

  • Chapter 7. Use and mention 86

    Obviously, we are trying to come out with anEnglish sentence that says something about theobject language (TFL), but D is a sentence of TFL,and not part of English. So the preceding isgibberish, just like:

    Schnee ist wei is a German sentence.

    What we surely meant to say, in this case, is:

    Schnee ist wei is a German sentence.

    Equally, what we meant to say above is just:

    D is an atomic sentence of TFL.

    The general point is that, whenever we want to talkin English about some specific expression of TFL,we need to indicate that we are mentioning theexpression, rather than using it. We can eitherdeploy quotation marks, or we can adopt somesimilar convention, such as placing it centrally inthe page.

  • Chapter 7. Use and mention 87

    7.3 Metavariables

    However, we do not just want to talk about specificexpressions of TFL. We also want to be able to talkabout any arbitrary sentence of TFL. Indeed, wehad to do this in 6.2, when we presented therecursive definition of a sentence of TFL. We useduppercase script letters to do this, namely:

    A,B, C, D, . . .These symbols do not belong to TFL. Rather, theyare part of our (augmented) metalanguage that weuse to talk about any expression of TFL. To repeatthe second clause of the recursive definition of asentence of TFL, we said:2. If A is a sentence, then A is a sentence.

    This talks about arbitrary sentences. If we hadinstead offered:

    If A is a sentence, then A is a sentence.this would not have allowed us to determinewhether B is a sentence. To emphasize, then:

  • Chapter 7. Use and mention 88

    A is a symbol (called a metavariable) in aug-mented English, which we use to talk about anyTFL expression. A is a particular atomic sen-tence of TFL.But this last example raises a further

    complication for our quotation conventions. Wehave not included any quotation marks in thesecond clause of our recursive definition. Shouldwe have done so?

    The problem is that the expression on theright-hand-side of this rule is not a sentence ofEnglish, since it contains . So we might try towrite:

    2. If A is a sentence, then A is a sentence.

    But this is no good: A is not a TFL sentence,since A is a symbol of (augmented) English ratherthan a symbol of TFL.

    What we really want to say is something likethis:

  • Chapter 7. Use and mention 89

    2. If A is a sentence, then the result ofconcatenating the symbol with the sentenceA is a sentence.

    This is impeccable, but rather long-winded. But wecan avoid long-windedness by creating our ownconventions. We can perfectly well stipulate that anexpression like A should simply be read directlyin terms of rules for concatenation. So, officially,the metalanguage expression A simplyabbreviates:

    the result of concatenating the symbol with the sentence A

    and similarly, for expressions like (AB), (AB),etc.

    7.4 Quotation conventions forarguments

    One of our main purposes for using TFL is to studyarguments, and that will be our concern in Parts III

  • Chapter 7. Use and mention 90

    and IV. In English, the premises of an argument areoften expressed by individual sentences, and theconclusion by a further sentence. Since we cansymbolize English sentences, we can symbolizeEnglish arguments using TFL. Thus we might askwhether the argument whose premises are the TFLsentences A and A C , and whose conclusion isthe TFL sentence C , is valid. However, it is quite amouthful to write that every time. So instead we willintroduce another bit of abbreviation. This:

    A1 ,A2 , . . . ,An .. Cabbreviates:

    the argument with premises A1 ,A2 , . . . ,Anand conclusion C

    To avoid unnecessary clutter, we will not regard thisas requiring quotation marks around it. (Note, then,that .. is a symbol of our augmentedmetalanguage, and not a new symbol of TFL.)

  • Part III

    Truth tables


  • Chapter 8

    Characteristictruth tables

    Any non-atomic sentence of TFL is composed ofatomic sentences with sentential connectives. Thetruth value of the compound sentence depends onlyon the truth value of the atomic sentences thatcomprise it. In order to know the truth value of(D E ), for instance, you only need to know thetruth value of D and the truth value of E .

    We introduced five connectives in chapter 5, sowe simply need to explain how they map betweentruth values. For convenience, we will abbreviateTrue with T and False with F. (But just to beclear, the two truth values are True and False; the


  • Chapter 8. Characteristic truth tables 93

    truth values are not letters!)Negation. For any sentence A: If A is true, thenA is false. If A is true, then A is false. We cansummarize this in the characteristic truth table fornegation:

    A AT FF T

    Conjunction. For any sentences A and B, AB istrue if and only if both A and B are true. We cansummarize this in the characteristic truth table forconjunction:


    Note that conjunction is symmetrical. The truthvalue for AB is always the same as the truth valuefor BA.

  • Chapter 8. Characteristic truth tables 94

    Disjunction. Recall that always representsinclusive or. So, for any sentences A and B, ABis true if and only if either A or B is true. We cansummarize this in the characteristic truth table fordisjunction:


    Like conjunction, disjunction is symmetrical.Conditional. Were just going to come clean andadmit it: Conditionals are a right old mess in TFL.Exactly how much of a mess they are isphilosophically contentious. Well discuss a fewof the subtleties in 9.3 and 11.5. For now, we aregoing to stipulate the following: A B is false ifand only if A is true and B is false. We cansummarize this with a characteristic truth table forthe conditional.

  • Chapter 8. Characteristic truth tables 95


    The conditional is asymmetrical. You cannot swapthe antecedent and consequent without changingthe meaning of the sentence, because A B has avery different truth table from B A.Biconditional. Since a biconditional is to be thesame as the conjunction of a conditional running ineach direction, we will want the truth table for thebiconditional to be:


    Unsurprisingly, the biconditional is symmetrical.

  • Chapter 9


    9.1 The idea oftruth-functionality

    Lets introduce an important idea.


  • Chapter 9. Truth-functional connectives 97

    A connective is truth-functional iff the truthvalue of a sentence with that connective as itsmain logical operator is uniquely determinedby the truth value(s) of the constituent sen-tence(s).Every connective in TFL is truth-functional.

    The truth value of a negation is uniquely determinedby the truth value of the unnegated sentence. Thetruth value of a conjunction is uniquely determinedby the truth value of both conjuncts. The truth valueof a disjunction is uniquely determined by the truthvalue of both disjuncts, and so on. To determine thetruth value of some TFL sentence, we only need toknow the truth value of its components.

    This is what gives TFL its name: it istruth-functional logic.

    In plenty of languages there are connectivesthat are not truth-functional. In English, forexample, we can form a new sentence from anysimpler sentence by prefixing it with It isnecessarily the case that. . . . The truth value of this

  • Chapter 9. Truth-functional connectives 98

    new sentence is not fixed solely by the truth valueof the original sentence. For consider two truesentences:

    1. 2 + 2 = 42. Shostakovich wrote fifteen string quartets

    Whereas it is necessarily the case that 2 + 2 = 4, itis not necessarily the case that Shostakovichwrote fifteen string quartets. If Shostakovich haddied earlier, he would have failed to finish Quartetno. 15; if he had lived longer, he might have writtena few more. So It is necessarily the case that. . . isa connective of English, but it is nottruth-functional.

    9.2 Symbolizing versustranslating

    All of the connectives of TFL are truth-functional,but more than that: they really do nothing but mapus between truth values.

  • Chapter 9. Truth-functional connectives 99

    When we symbolize a sentence or an argumentin TFL, we ignore everything besides thecontribution that the truth values of a componentmight make to the truth value of the whole. Thereare subtleties to our ordinary claims that far outstriptheir mere truth values. Sarcasm; poetry; snideimplicature; emphasis; these are important parts ofeveryday discourse, but none of this is retained inTFL. As remarked in 5, TFL cannot capture thesubtle differences between the following Englishsentences:

    1. Dana is a logician and Dana is a nice person2. Although Dana is a logician, Dana is a niceperson

    3. Dana is a logician despite being a nice person4. Dana is a nice person, but also a logician5. Danas being a logician notwithstanding, he isa nice person

    All of the above sentences will be symbolized withthe same TFL sentence, perhaps L N .

    We keep saying that we use TFL sentences to

  • Chapter 9. Truth-functional connectives 100

    symbolize English sentences. Many othertextbooks talk about translating English sentencesinto TFL. However, a good translation shouldpreserve certain facets of meaning, andas wehave just pointed outTFL just cannot do that. Thisis why we will speak of symbolizing Englishsentences, rather than of translating them.

    This affects how we should understand oursymbolization keys. Consider a key like:

    L: Dana is a logician.N : Dana is a nice person.

    Other textbooks will understand this as a stipulationthat the TFL sentence L should mean that Dana isa logician, and that the TFL sentence N shouldmean that Dana is a nice person, but TFL just istotally unequipped to deal with meaning. Thepreceding symbolization key is doing no more andno less than stipulating that the TFL sentence L should take the same truth value as the Englishsentence Dana is a logician (whatever that mightbe), and that the TFL sentence N should take the

  • Chapter 9. Truth-functional connectives 101

    same truth value as the English sentence Dana is anice person (whatever that might be).When we treat a TFL sentence as symbolizingan English sentence, we are stipulating that theTFL sentence is to take the same truth value asthat English sentence.

    9.3 Indicative versussubjunctive conditionals

    We want to bring home the point that TFL can onlydeal with truth functions by considering the case ofthe conditional. When we introduced thecharacteristic truth table for the material conditionalin 8, we did not say anything to justify it. Lets nowoffer a justification, which follows DorothyEdgington.1

    Suppose that Lara has drawn some shapes on apiece of paper, and coloured some of them in. We

    1Dorothy Edgington, Conditionals, 2006, in the Stan-ford Encyclopedia of Philosophy (

  • Chapter 9. Truth-functional connectives 102

    have not seen them, but nevertheless claim:

    If any shape is grey, then that shape isalso circular.

    As it happens, Lara has drawn the following:

    A C D

    In this case, our claim is surely true. Shapes C andD are not grey, and so can hardly presentcounterexamples to our claim. Shape A is grey,but fortunately it is also circular. So my claim hasno counterexamples. It must be true. That meansthat each of the following instances of our claimmust be true too:

    If A is grey, then it is circular (true antecedent,true consequent)

    If C is grey, then it is circular(false antecedent,true consequent)

    If D is grey, then it is circular (falseantecedent, false consequent)

  • Chapter 9. Truth-functional connectives 103

    However, if Lara had drawn a fourth shape, thus:

    A B C D

    then our claim would be false. So it must be thatthis claim is false:

    If B is grey, then it is circular (true antecedent,false consequent)

    Now, recall that every connective of TFL has to betruth-functional. This means that merely the truthvalues of the antecedent and consequent mustuniquely determine the truth value of theconditional as a whole. Thus, from the truth valuesof our four claimswhich provide us with allpossible combinations of truth and falsity inantecedent and consequentwe can read off thetruth table for the material conditional.

    What this argument shows is that is thebest candidate for a truth-functional conditional.Otherwise put, it is the best conditional that TFL

  • Chapter 9. Truth-functional connectives 104

    can provide. But is it any good, as a surrogate forthe conditionals we use in everyday language?Consider two sentences:

    1. If Mitt Romney had won the 2012 election,then he would have been the 45th President ofthe USA.

    2. If Mitt Romney had won the 2012 election,then he would have turned into a helium-filledballoon and floated away into the night sky.

    Sentence 1 is true; sentence 2 is false, but bothhave false antecedents and false consequents. Sothe truth value of the whole sentence is not uniquelydetermined by the truth value of the parts. Do notjust blithely assume that you can adequatelysymbolize an English if . . . , then . . . with TFLs.

    The crucial point is that sentences 1 and 2employ subjunctive conditionals, rather thanindicative conditionals. They ask us to imaginesomething contrary to factMitt Romney lost the2012 electionand then ask us to evaluate what

  • Chapter 9. Truth-functional connectives 105

    would have happened in that case. Suchconsiderations just cannot be tackled using .

    We will say more about the difficulties withconditionals in 11.5. For now, we will contentourselves with the observation that is the onl