8.3 Exercises

8.3 Exercises

SECTION 8.1

1Construct derivations that show that the following can be introduced as derived axiom schemata in L3 A:

a.P(a/x) → ( x)P

b.( x)(P → Q) → (( x)P →( x)Q)

2Show how to derive the following rules in L3 A:

a.Universal Generalization in the Antecedent. From P(a/x) → Q infer ( x)P → Q

where x is any individual variable.

b.Existential Generalization in the Consequent. From P → Q(a/x) infer P → ( x)Q

where x is any individual variable.

c.Existential Generalization in the Antecedent. From P(a/x) → Q infer ( x)P → Q

where x is any individual variable, provided that no assumption contains the constant a and that P does not contain the constant a.

3Construct derivations that show that the following formulas are theorems of L3 A:

a.( x)Qx → ( x)(Pa → Qx)

b.( x)Qx → ( x)(Px → Qx)

c.( x)(Pa → Qx) → (( x)Px → ( x)Qx)

d.(Pa ( x)Qx) → ( x)(Pa Qx)

4Show that the new SUB:

SUB (Substitution). From the universal closure of P → Q, the universal closure of Q → P, and a formula R that contains P as a subformula, infer any formula R* that is the result of replacing one or more occurrences of P in R with Q.

is indeed derivable in L3 A by proving the following:

If formulas P(a/x) → Q(a/x) and Q(a/x) → P(a/x) occur in a derivation, where a is a constant that does not occur in P, Q, or any assumption, then both ( x)P → ( x)Q and ( x)Q → ( x)P can also be derived (this would be Case 4 of the new proof).

This will do the trick because the new SUB applies when we have already derived the universal closures of P → Q and Q → P, say, ( x1) . . . ( xn)(P → Q) and ( x1) . . . ( xn)(Q → P). From these universal closures we can then derive specific instances (P → Q)(a1/x1) . . . (an/xn) and (Q → P)(a1/x1) . . . (an/xn) using L3 6 and MP. Thus as we derive the series of conditionals building up the formulas described in Chapter 6’s justification of SUB we can use constants a1, a2, . . . , an that do not occur in the derivation in place of the free variables x1, x2, . . . , xn in P and Q, reintroducing the free variables along with their universal quantifiers

at the relevant point as the formulas are built up. Case 4 licenses the necessary universal quantification.

SECTION 8.2

5Derive graded versions of the following rules for L3 PA (the first was derived in Section 8.1, and the rest were derived in Exercise 2):

a.Universal Generalization in the Consequent

b.Universal Generalization in the Antecedent

c.Existential Generalization in the Consequent

d.Existential Generalization in the Antecedent

6Construct derivations (without assumptions) of the following graded formulas in L3 PA:

a.[(n → Pa) → (n → ( x)Px), T]

b.[(n → Pa) → ( x)Px, N]

c.[( x)(Px → Qx) → ((n → Pa) → (n → Qa)), T]

d.[( x)(Px → Qx) → ((( x)¬Px → n) → ( x)Qx), N]

e.[(n → ( x)(Px → Qx)) → ((t → ( x)Px) → (n → ( x)Qx)), T]

7Construct derivations of the conclusions of the following graded arguments from the graded premises:

a.[n → ( x)Qx, T] [( x)(n → Qx), T]

b.[( x)(n → Qx), T] [n → ( x)Qx, T]

c.[( x)( y)Rxy, N] [( x)(Rxx → Sx), T] [( x)Sx, N]

d.[( x)(Px → (Qx → n)), T] [( x)(Px → (Qx → f)), N]

For example, we can replace the classical clauses 4 and 5 with

4.V(P → Q) = min (1, 1 – V(P) + V(Q))

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

to obtain the propositional system L3 with the following truth-tables for the conditional and biconditional:

Moreover, these clauses also work for classical propositional logic—as they should, since we know that L3 is a normal system.

To understand the formulas used for the L3 conditional and biconditional recall our comment in Section 4.2 that in classical logic disjunction is almost like addition: V(P Q) = V(P) + V(Q), except when both V(P) and V(Q) are 1, so that classical disjunction can be defined as min(1, V(P) + V(Q)).1 Now, this formula can’t work for weak disjunction in L3: min (1, V(P) + V(Q)) is 1 when V(P) and V(Q) are both 1/2, but L3 s weak disjunction has the value 1/2 in this case. However, the formula does characterize bold disjunction, which produces a true formula when both disjuncts are neither true nor false. But then, if we think of P →L Q as ¬LP L Q (we showed in Chapter 5 that Lukasiewicz’s conditional could be so defined in terms of negation and bold disjunction), we get clause 4 for the conditional. Clause 5 results from defining P ↔L Q as (P →L Q) L (Q →L P).

There’s another way to look at clause 4 for Lukasiewicz’s conditional. The formula 1 – V(P) + V(Q) is equivalent to 1 – (V(P) – V(Q)), and V(P) – V(Q) is a kind of measure of how much “truer” P is than Q. If P is indeed truer than Q, then we subtract this value from 1, so that the truer P is than Q, the less true the conditional is. If P is not truer than Q, that is, if it is the same or less true than Q, then the conditional is simply true.

For Bochvar’s internal connectives we have the numeric clauses

2.V(P BI Q) = 1/2 if V(P) = 1/2 or V(Q) = 1/2

min (V(P), V(Q)) otherwise

3.V(P BI Q) = 1/2 if V(P) = 1/2 or V(Q) = 1/2

max (V(P), V(Q)) otherwise

1 Recall that for classical logic we could also define conjunction in terms of multiplication: V(P Q) = V(P) · V(Q). However, we can’t define conjunction this way in three-valued logi-

cal systems because it would be ill-defined—when P and Q both have the value 1/ , the product

2

of their values is 1/ , which is not a numeric truth-value in our three-valued systems. We’ll revisit

4

this definition of conjunction, however, when we turn to fuzzy logics.

for conjunction and disjunction. The “otherwise” cases give us a normal system— Bochvar’s system, like the others, behaves classically when only classical truthvalues are involved. We leave it as an exercise to develop numeric clauses for Bochvar’s internal conditional and biconditional as well as for his external connectives.

Turning to three-valued first-order logics, we revise numeric clause 1 for classical first-order logic, assigning 1/2 to formulas that are undetermined:

1.Iv(Pt1 . . . tn) = 1 if <I*(t1), . . . , I*(tn)> ext(P), where I*(ti) is I(ti) if ti is a constant

and is v(ti) if ti is a variable, 0 if <I*(t1), . . . , I*(tn)> cxt(P), and 1/2 if <I*(t1), . . . , I*(tn)> fge(P).

If we then use the remaining numeric satisfaction clauses 2–8 for classical first-order logic, we get KS3 . Here are the quantifier clauses:

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}

Clause 7 gives a universal quantification the value 1 if the quantified formula is satisfied by everything, 0 if it is dissatisfied by at least one thing, and 1/2 otherwise. Clause 8 gives an existential quantification the value 1 if the quantified formula is satisfied by at least one thing, 0 if it is dissatisfied by everything, and 1/2 otherwise. Since the quantifiers have the same definitions in both KS3 andL3, the revised clauses 1, 7, and 8 will generate L3 if we add them to the numeric propositional clauses for L3, revised to talk of interpretations and variable assignments.

The following clause suffices for the universal quantifier in the first-order generalization of BI3:

7.Iv(( BI x)P) = 1/2 if Iv (P) = 1/2 for any x-variant v of v

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

Again, the “otherwise” cases give us classically defined values when the third truthvalue is ignored. We leave it as an exercise to develop numeric clauses for Bochvar’s internal existential quantifier and for his external universal and existential quantifiers.

9.2 Abstract Algebras for L3, KS3, BI3, and BE3

We’ll now explore the abstract algebraic structures characterizing the operations of three-valued logical systems, whether those operations are defined for T, N, and F or {1,1/2, 0}. We begin with the algebra induced by Lukasiewicz’s weak disjunction and weak conjunction (and hence by Kleene’s strong operations) as defined on the values {1,1/2, 0}: this algebra is a distributed lattice with domain {1,1/2, 0} in which

based on the Kleene algebra. The following result tells us that the semantics for propositional logic based on any three-valued Kleene algebra give us KS3’s truthfunctions for disjunction, conjunction, and negation, with unit, zero, and the third element, respectively, in place of 1, 0, and 1/2:

Result 9.1: Every three-valued Kleene algebra KA = <{unit, zero, other}, , ∩, , unit, zero> generates the following truth-tables for assignments of unit, zero, or other to each atomic formula of propositional logic when , ∩, and respectively define the disjunction, conjunction, and negation operations (we use u, z, and o to stand for unit, zero, and other):

Proof: Left as an exercise.

Because there are no KS3 tautologies, it follows that no formulas of propositional logic are tautologies of any three-valued Kleene algebra (where the algebraic definition of a tautology is analogous to the definition of BA-tautologies in Chapter 4).

The algebra BIA induced by Bochvar’s internal disjunction and conjunction isn’t a lattice because lattice condition iv, absorption, doesn’t hold—when x is 1 or 0 the expressions x (x ∩ 1/2) and x ∩ (x 1/2) both evaluate to 1/2 in BIA, not x. Conditions i–iii do hold for BIA, as do conditions v and vi earlier specifying distributivity and unit and zero elements. The equations involving the join operation in conditions i–iii define a structure known as a semi-lattice, and so do the equations involving the meet operation. (It is idempotency condition iv that unifies join and meet semi-lattices into a single lattice.) Thus we may simply describe BIA as a distributive dual system of semi-lattices with unit and zero elements. The addition of Bochvar’s internal negation would give us an orthocomplement for the dual system. Finally, the algebra BEA defined by Bochvar’s external disjunction and conjunction is a distributed lattice, but the elements 0 and 1 don’t serve as identity elements for join and meet as specified by condition vi (1/2 0 = 0 in BEA, not 1/2, and 1/2 ∩ 1 = 0, not 1/2). The reader will be asked to discuss these systems, as well as the algebraic operation corresponding to Bochvar’s external negation, in the exercises.