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3.4 Exercises

55

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).

56

Review of Classical First-Order Logic

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

4 Alternative Semantics for Truth-Values and Truth-Functions: Numeric Truth-Values and Abstract Algebras

4.1 Numeric Truth-Values for Classical Logic

We’ve been using the letters T and F to stand in for the truth-values true and false. We could just as well use the numerals 1 and 0 to stand for true and false, recasting truth-tables using these two numerals, for example,

P

¬P

1

0

0

1

More interesting, though, is using the integers 1 and 0, rather than the numerals that name these integers, as the truth-values of formulas in classical logic. A propositional truth-value assignment will consist in the assignment of one of these values to each atomic formula, and we can then numerically define the values for complex formulas rather than simply listing these values in truth-tables. Letting V(P) mean the value of P on a (numeric) truth-value assignment V, the following definitions will do the job:

1.V(¬P) = 1 – V(P)

2.V(P Q) = min (V(P), V(Q)) (i.e., the minimum of these two values)

3.V(P Q) = max (V(P), V(Q)) (i.e., the maximum of these two values)

4.V(P Q) = max (1–V(P), V(Q))

5.V(P Q) = min (max (1–V(P), V(Q)), max (1–V(Q), V(P)))

Clause 1 “reverses” the value for a negated formula—since (looking at the right-hand side of the formula) 1 1 is 0 and 1 0 is 1. Clause 2 indicates that a conjunction is only as true as its least true conjunct. By contrast, clause 3 indicates that a disjunction is as true as its most true disjunct. Clause 4 is based on the equivalence of P Q and ¬P Q, using clauses 1 and 3 to define the truth-conditions for the latter. Clause 5 is based on the equivalence of P Q and (P Q) (Q P). If we had instead used the equivalence of P Q and (P Q) (¬P ¬Q) to capture the truth-conditionals for biconditionals, the right-hand side of clause 5 would have been written as max (min (V(P), V(Q)), min (1–V(P), 1–V(Q))). This is equivalent to the right-hand side that we have chosen to use for clause 5 (and is left as an exercise to verify).

57

58

Alternative Semantics for Truth-Values and Truth-Functions

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:

P

 

 

¬P

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

0

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

P Q

 

P Q

 

P Q

 

P Q

 

P Q

 

P Q

 

P Q

 

P Q

 

 

 

 

 

 

 

1 1

 

 

1

1 1

 

1

1 1

 

1

1 1

 

1

1 0

 

 

0

1 0

 

1

1 0

 

0

1 0

 

0

0 1

 

 

0

0 1

 

1

0 1

 

1

0 1

 

0

0 0

 

 

0

0 0

 

0

0 0

 

1

0 0

 

1

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}