3.4 Exercises

undecidable. For neither theoremhood nor tautologousness in classical predicate logic is it possible to devise a mechanical test that is guaranteed to yield, for any formula, a correct yes-or-no classification in a finite number of steps. On the other hand, first-order tautologousness (and hence theoremhood) is semi-decidable: there are for example mechanical tests based on resolution or semantic tableaux that will, after a finite number of steps, always yield a correct yes classification for any first-order formula that is a tautology. But such tests do not yield full decidability because they may fail to yield any answer within a finite number of steps for formulas that are not tautologies.7

3.4 Exercises

SECTION 3.2

1Determine the truth-value of each of the following formulas on an interpretation that makes the following assignments:

D:set of positive integers

I(O) = {<u>: u D and u is odd}

I(S) = {<u1, u2>: u1 D, u2 D, and u1 squared is u2}

I(E) = {<u1, u2>: u1 D, u2 D, and u1 evenly divides u2}

I(a) = 1

I(b) = 2

I(c) = 3

a.Oa Oc

b.Oa → Ob

c.( x)Ox

d.( x)Exx

e.( x)¬Sxx

f.( x)Sxb

g.( x) ( y)((Ox Oy) → Exy)

h.( x)( y)(Exy Eyx)

i.( x)( y)(¬Ox Exy)

j.( x)( y)(Sxy → Exy)

k.( x)( y)(Sxy Syx)

l.( x)(Ox → ( y)( z)(Sxy Syz))

m.( x)( y)Sxy

n.( y)( x)Sxy

7Church (1936) proved that theoremhood in first-order logic is undecidable. See Part 4 of Hunter (1971) for a general proof of undecidability in various first-order systems. A semi–decision procedure for first-order classical logic based on semantic tableaux is presented in Smullyan (1968, pp. 59–60).

SECTION 3.3

2a. Prove that every instance of the axiom schemata CL 4 is a tautology in classical first-order logic.

b.Prove that every instance of the axiom schemata CL 5 is a tautology in classical first-order logic.

3Construct a derivation that justifies derived axiom schema CL D7.

4Construct a derivation that shows that ¬( x) (¬Tx ¬¬Tx) is a theorem of CL A. You will first need to use the definitions of the existential quantifier and conjunction to write the formula without those operators.

5Show that the following rule can be derived in CL A:

EG (Existential Generalization). From P(a/x) infer ( x)P where x is any individual variable

6Construct derivations to show that the following formulas are theorems of CL A (rewriting each formula to obtain a formula that contains only negation, the conditional, and the universal quantifier as operators):

a.( x)( y)Lxy → ( x)( y)Lxy

b.( x)(Fa → Gx) → (Fa → ( x)Gx)

c.( x)( y)Lxy → ( y)( x)Lxy

d.( x)( y)Lxy → ( x)Lxx

Clauses 1–5 produce truth-tables that look exactly like our previous classical truth-tables, except that we now have 1 in place of T and 0 in place of F:

Note that conjunction could equivalently be defined in terms of multiplication as V(P Q) = V(P) · V(Q). This will become important when we turn to fuzzy logic, where truth-values are always defined numerically. And disjunction is almost like addition, except in the case where both disjuncts have the value 1, so an alternative clause for disjunction is V(P Q) = min (1, V(P) + V(Q)).

Having substituted 1 and 0 for true and false in clauses defining the values of complex formulas, we need to make the same substitutions in other semantic definitions: a tautology of classical propositional logic is a formula that always has the value 1; a contradiction is a formula that always has the value 0; an argument is valid if its conclusion has the value 1 whenever its premises have the value 1; and so on. Thus we will arrive at the same set of tautologies as we had based on the values true and false, and other semantic results also remain the same. The difference is that now we can compute values for formulas numerically rather than simply referring to truth-tables. For example, we can show that A ¬A is a tautology of classical propositional logic as follows: V(A ¬A) = max (V(A), 1–V(A)), and when V(A) = 1 the maximum is V(A), that is, 1, while when V(A) = 0 the maximum is 1 – V(A), that is, 1. Thus the formula always has the value 1 and is therefore a tautology.

We can similarly define numerical values for formulas of classical first-order logic. Instead of talking about a variable assignment satisfying or not satisfying a formula, we may talk instead about formulas having the value 1 or 0 on variable assignments. We’ll designate the value that a formula P has on a variable assignment v on an interpretation I with the notation Iv(P):

1.Iv(Pt1 . . . tn) = 1 if <I*(t1), . . . ,I*(tn)> I(P), where I*(ti) is I(ti) if ti is a constant and is V(ti) if ti is a variable, and IV(Pt1 . . . tn) = 0 otherwise.

2.Iv(¬P) = 1 – Iv(P).

3.Iv(P Q) = min(Iv(P), Iv(Q))

4.Iv(P Q) = max(Iv(P), Iv(Q))

5.Iv(P → Q) = max(1 – I(P), Iv(Q))

6.Iv(P ↔ Q) = min(max(1 – Iv(P), Iv(Q)), max(1 – Iv(Q), Iv(P)))

7.Iv(( x)P) = min{Iv (P): v is an x-variant of v}

8.Iv(( x)P) = max{Iv (P): v is an x-variant of v}