Assignment II

January 20, 2015

Notation: f∗ denotes the truth function corresponding to a logical connective ∗;fα denotes the truth function determined by the wff α.

Q1. Suppose we assign the following truth function to the conditional →:

A B A→ BT T TT F FF T TF F F

Show that with this truth function for →, the argument form p → q ∴ ¬q → ¬p isinvalid.

Q2. Consider the relation ≡ of logical equivalence on the set of all PC-wffs. Prove thatthere is one equivalence class containing exactly the tautologies, and another with exactlythe contradictions.

Q3. Prove that there is a bijection between the set of all truth functions from {T, F}nto {T, F} and the quotient set Fn/ ≡n.

Q4. Prove the following for any set Γ ∪∆ ∪ {α} of wffs in PC.

1. Γ |= α, if and only if Γ ∪ {¬α} is contradictory.

2. If Γ is satisfiable and ∆ ⊆ Γ then ∆ is satisfiable.

3. If Γ is contradictory and Γ ⊆ ∆ then ∆ is contradictory.

4. (Overlap) If α ∈ Γ then Γ |= α.

5. (Dilution) If Γ |= α, Γ ⊆ ∆ then ∆ |= α.

1

6. (Cut)If Γ |= α and ∆ |= β, for each β ∈ Γ then ∆ |= α.

7. Γ ∪ {α} |= β, if and only if Γ |= α→ β.

8. If Γ |= α ∧ ¬α for some wff α, then Γ |= β, for any wff β.

Q5. Prove that the following are equivalent.

1. Γ is satisfiable, if and only if Γ is finitely satisfiable.

2. Γ is contradictory, if and only if there is a finite subset of Γ that is contradictory.

3. For any wff α, Γ |= α, if and only if there is a finite subset ∆ of Γ such that ∆ |= α.

Definition 1. A set C of connectives is adequate if for every n ≥ 1 and every n-arytruth function f , there is a wff α with propositional variables p1, p2, . . . , pn such that (1)fα = f ; (2) the connectives that occur in α are from the set C.

Q6. Let ∗ be a binary connective such that {∗} is adequate. Prove that f∗(T, T ) = Fand f∗(F, F ) = T . Note: v : PV → {T, F} extends by v(α ∗ β) = f∗(v(α), v(β)).

Q7. Show that the only adequate binary connectives are NOR and NAND.Hint: By the preceding exercise, the only possibilities for a binary connective that byitself is adequate are the following:

A B A ∗1 B A ∗2 B A ∗3 B A ∗4 BT T F F F FT F F T T FF T T F T FF F T T T T

Now ∗3 and ∗4 are NAND and NOR, so you must show that neither {∗1} nor {∗2} isadequate.

Q8. Show that {∧,∨,→} is not an adequate set of connectives.

Q9.(Negation normal form) The set of wffs in negation normal form (NNF) is definedas follows:[F1] p and ¬p are in NNF for every p ∈ PV ;[F2] If α and β are in NNF, so are α ∧ β and α ∨ β.[F3] Every wff in NNF is obtained by a finite number of applications of F1 and F2.The only connectives that occur in wffs in NNF are ¬,∧ and ∨; moreover the onlyoccurrences of ¬ are before propositional variables.

2

Note that p ∨ ¬q is in NNF but ¬(p ∨ ¬q) is not. But ¬(p ∨ ¬q) is logically equivalentto ¬p ∧ q, which is in NNF.Show that every wff is logically equivalent to a wff in NNF.

3