classical propositional logics are all Boolean algebras. Lindenbaum algebras are beyond the scope of this text, so we refer the interested reader to Dunn and Hardegree (2001).4

4.4 Exercises

SECTION 4.1

1Prove that min(max(1–V(P), V(Q)), max(1–V(Q), V(P))), which we used to define the truth-conditions for conditional formulas is equivalent to the alternative definition based on the equivalence of P ↔ Q and (P Q) (¬P ¬Q):

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

SECTION 4.2

2 Complete the definition of clause 5:

5. V(P ↔ Q) = . . .

for a semantics in which the values true and false are replaced by {5} and Ø, respectively, as suggested at the beginning of Section 4.2.

3Complete the proof that <{1,0}, max, min, 1–, 1, 0> forms a Boolean algebra by proving that the second equations for absorption, distribution, and complemention hold in this structure.

4Complete the proof of Result 4.1: show that conditions i, iii, vi, and vi of (twovalued) Boolean algebras generate the following tables when negation and disjunction are defined as algebraic complementation and join:

P ¬P

unit zero zero unit

P Q P Q

unit unit unit unit zero unit zero unit unit zero zero zero

SECTION 4.3

5Prove the dual to unit consumption, which we may call zero consumption: zero ∩ x = zero.

6Show how to derive the second idempotence, absorption, and association conditions for Boolean algebras from conditions i and v–vii of Boolean algebras.

4Lindenbaum algebras are named after the logician Adolf Lindenbaum and are sometimes called Lindenbaum-Tarski algebras after Alfred Tarski as well. These algebras were studied by both logicians.

7 Prove that the equality

(x y) ∩ (x ∩ y ) = zero

which was used to establish the first DeMorgan Law holds in every Boolean algebra.

8Prove that the second DeMorgan Law holds in every Boolean algebra.

9Prove that the following formulas are tautologies of classical propositional logic by showing that they must evaluate to unit under any Boolean algebraic semantics:

a.¬(P ¬P)

b.(P Q) → (P Q)

c.¬P → (P → Q)

10In Section 4.2 we claimed that Boolean algebra conditions ii, iv, and v can be derived from the remaining four conditions for every Boolean algebra in which the domain B contains exactly two elements. Prove this.

Hint: You can derive the the first associativity condition x (y z) = (xy) z, for example, by looking at four cases that among them will cover all possible combinations of values for x, y, and z in a two-valued Boolean algebra:

a.x = unit (y and z can each be either unit or zero)

b.y = unit

c.z = unit

d.x = y = z = zero.

For case a, we have unit (y z) = unit by unit consumption, and (unit y)z = unit z = unit by the same law, thus establishing the first associativity condition for case a. The reader can pick up the proof from here.

11Prove that the lattice ordering relation ≤, defined as

x≤ y if and only if x ∩ y = x,

is transitive.

12Prove that in every lattice, x ∩ y = x if and only if x y = y. Hint: The absorption condition will prove useful.

13Complete the proof of Result 4.3 by showing that the following hold in every Boolean algebra:

a.x (y x) = unit

b.(x (y z) ((x y) (x z)) = unit

c.(x y ) (y x) = unit

d.Modus Ponens preserves tautologousness in any Boolean algebra:

if x = unit and x y = unit then y = unit.

system, to be introduced in Section 5.3). The negation truth-function in KS3 is defined as

P ¬KP

T F

N N

F T

(We will subscript connectives within nonclassical logical systems to make clear which system we’re working in.) Note that when P has one of the values T or F (we’ll call these the classical truth-values), ¬KP is defined as in classical logic. When P has the value N, reflecting a vague predicate’s application to a borderline case, so does its negation. If Mary Middleford is tall is neither true nor false, Mary Middleford is not tall is also neither true nor false.

The truth-functions corresponding to the binary connectives in Kleene’s system

are

In these tables the expression P \ Q means that P has the value in the column listed below while Q has the value in the row listed to the right. So, for example, each row beginning with T covers cases where P has the value T, and each column headed by T covers cases where Q has the value T. The intersection of the T row and the T column represents the case where both P and Q have the value T. In the table for conjunction T is listed at this intersecting point—meaning that the conjunction P K Q has the value T when both P and Q have the value T.

Each of these truth-functions agrees with classical logic when the arguments are both either T or F. For example, restricting attention to the four corners of the truth-table for conjunction:

P K Q

P \ Q T N F

T T N F

N N N F

F F F F

we see that the truth-value of a conjunction in these cases is the same as it would be in classical logic. Connectives with this property are normal:3 a propositional

3This terminology, along with the technical use of the term uniform a few paragraphs hence, is from Rescher (1969, pp. 54–57).

connective in a three-valued logical system is normal if, whenever the connective combines formulas with classical truth-values, the resulting formula has the same truth-value as it does in classical logic (dropping the subscripts on the connectives, of course). We will also say that the truth-function denoted by the connective is normal.

How are the remaining values in the truth-table determined? Let’s continue to look at conjunction. A classical conjunction is false whenever at least one conjunct is false, no matter what the value of the other conjunct. The same is true of Kleene’s conjunction—the row and column representing the falsity of one of the conjuncts P and Q uniformly have the value F:

P K Q

We say that a propositional connective in a three-valued system is uniform if, whenever the truth-value of a formula formed with that connective is uniquely determined by the truth-value of one of its constituent formulas in classical logic, the truth-value of the formula formed with that connective is also uniquely so determined in the three-valued system. (We will also say that the truth-function denoted by the connective is uniform.) In classical logic a false conjunct guarantees the falsehood of a conjunction, and the fact that this is also the case in KS3 means that conjunction is uniform in this system. The other connectives are uniform as well.

Normality and uniformity account for all but three of the values—namely, the three Ns—in the truth-table for conjunction. Generally, the value N appears in KS3 whenever neither normality nor uniformity requires a particular value for one of the connectives. Owing to normality, these cases always involve N as one or both of the arguments to the truth function. Thus a conjunction has the value N when at least one conjunct has this value and neither conjunct has the value F.

Disjunction in KS3 is normal and it is also uniform since the truth of one disjunct is sufficient for the truth of the whole, as is the case in classical logic. The KS3 conditional and biconditional are both normal and uniform. The conditional is uniform because it forms a true formula whenever the antecedent is false or the consequent is true. (Note that in the case of the biconditional, there is no case in classical logic where the truth-value of a compound formula can be determined by the truth-value of only one of its immediate components—so the biconditional is trivially uniform.) Finally, negation in KS3 is also normal and uniform (the latter in an uninteresting way since it is a unary rather than binary connective). Moreover, it is only when neither normality nor uniformity determines a truth-value that one of these functions will assign the value N.

Here are truth-tables for some formulas in KS3:

The first formula, an instance of the Law of Excluded Middle, can have the value N as well as the value T in KS3. It shouldn’t be surprising that this classical tautology can fail to have the value T in KS3, since we noted in Chapter 1 that the Principle of Bivalence (which three-valued logic rejects) and the Law of Excluded Middle are closely (although not necessarily!) related. The third formula is, like the Law of Excluded Middle, a tautology of classical logic, but it can also have the value N in KS3. The second formula, neither a tautology nor a contradiction of classical logic, can have any of the three values T, F, or N in KS3.

We have introduced KS3 as a possible three-valued logic for vagueness, so it is interesting to note that Kleene’s motivation for presenting this three-valued system was altogether different. On Kleene’s interpretation, the value N means not defined rather than simply neither true nor false. Kleene introduced his system in connection with mathematical functions that may be undefined for certain values (just as division by 0 is undefined), and of the atomic formulas, only those in which all functions are defined for their arguments would one of the values be T or F. Nevertheless, athough Kleene’s motivation was different from ours, his system nevertheless turns out to be one reasonable three-valued logic for vagueness. For example, if P is true but Q is neither true nor false because it concerns a borderline case, then P K Q is also neither true nor false—it’s not true, since that would require the truth of both conjuncts, but it’s also not false, since neither conjunct is false.

We may choose to designate some connectives as primitive and introduce the others as defined. In fact, any choice of primitive connectives and accompanying definitions for the others that works in classical logic also works in KS3. So, for example, we can take ¬K and K as primitive connectives and introduce the other ones with the definitions

P K Q = def ¬K (¬KP K ¬KQ)

P →K Q = def ¬K (P K ¬KQ)

P ↔K Q = def ¬K (P K ¬KQ) K ¬K (¬K P K Q)

We leave it as an exercise to explore this claim.

As in classical logic, we define a tautology in a three-valued logical system to be a formula that always has the value T—there is no assignment on which it has

either the value F or the value N. We define a contradiction in a three-valued logical system to be a formula that always has the value F—that is, it never has the value T or N. It turns out that there are no tautologies or contradictions in KS3! We’ll prove this in a moment, but in case the reader is alarmed by this fact we reassure you that later in the chapter we’ll offer variations on the concepts of tautologies and contradictions, variations that will not be trivial for KS3.

Result 5.1: There are no tautologies or contradictions in KS3.

Proof: Examination of the truth-tables shows that whenever all of the atomic components of a compound formula have the value N, so does the compound formula. This means that for any formula, there is at least one truth-value assignment on which it has the value N—so no formula can be either a tautology or a contradiction in KS3.

Thus, the Law of Excluded Middle (as we already knew) is not a tautology in KS3 (nor is its negation a contradiction), but neither are there any other classical tautologies that are tautologies in KS3.

Before turning to entailment and validity in Kleene’s system, we introduce a lemma to which we shall often refer. Let us call a three-valued truth-value assignment classical if it assigns only the classical values T and/or F to atomic formulas— that is, it doesn’t make any assignments of N.

Normality Lemma: In a normal three-valued system, a classical truth-value assignment behaves exactly as it does in classical logic—every formula that is true on that assignment in the three-valued system is also true on that assignment in classical logic, and every formula that is false on that assignment in the three-valued system is also false on that assignment in classical logic.

Proof: The lemma follows from the fact that the connectives in a normal system behave exactly as they do in classical logic whenever they operate on formulas with classical truth-values.

We will say that a set of formulas entails a formula P in three-valued logic if, whenever all of the formulas in are true P is true as well (there is no truth-value assignment on which all the formulas in have the value T while P has the value F or N), and an argument is valid in three-valued logic if the set of premises of the argument entails its conclusion. We will use a standard notation for entailment: where

is a set of formulas, Γ |= P means the set of formulas entails the sentence P. Since entailment is within a system, we’ll use unsubscripted |= to indicate entailment in classical logic and |= K to indicate entailment in KS3.

Result 5.2: If Γ |= K P then Γ |= P (i.e., every entailment in KS3 is also an entailment in classical propositional logic).

Proof: Assume that Γ |= K P. It follows from the definition of entailment that on every classical (and nonclassical) truth-value assignment in KS3 on which the formulas in are all true, P is also true. But then since KS3 is normal, the same is true in classical logic by the Normality Lemma. So Γ |= P as well.