In the opposite direction some, but not all, classical entailments hold in KS3. An example of a classically valid argument that is also valid in KS3 is

P

P → Q

Q

We leave it as an exercise to verify that this argument is indeed valid in KS3. But not all classical entailments carry over:

Result 5.3: Not all entailments of classical propositional logic hold in KS3.

Proof: An example of an argument that is classically valid but not valid in KS3 is ¬ (P ↔ Q)

(P ↔ R) (Q ↔ R)

This is classically valid because, for the premise to be true in classical logic, P and Q must have different truth-values. But then no matter what truth-value R has, it will be equivalent to one or the other of P and Q since there are only two truth-values in classical logic—the validity depends crucially on the fact that classical logic has only two truth-values. So it is not surprising that this argument isn’t valid in KS3, where the premise can have the value T while the conclusion has the value N if P and Q have “opposite” classical values (one has the value T and the other has the value F) but R has the value N.

5.2 Lukasiewicz’s Three-Valued Logic

Now we’ll look at a three-valued system originating with the Polish logician Jan Lukasiewicz (Lukasiewicz 1930). This system, which we will call L3, defines three of the propositional connectives identically to Kleene’s strong connectives, but the conditional and biconditional differ from Kleene’s in one truth-table entry each. This difference, we will see, yields a system that contains both tautologies and contradictions (in the sense defined in the previous section). Here are L3’s truth-tables:

The differences between Lukasiewicz’s conditional and biconditional and Kleene’s are in the center of the tables. Each of these two connectives forms a true formula in

Lukasiewicz’s system when both of its immediate components have the value N. Why didn’t Lukasiewicz assign the compound formula the value N as well in this case? It’s because he reasoned that any conditional whose antecedent and consequent are identical, for instance, A →L A, should be a tautology—as it is in L3, as can be verified by examining the diagonal in the truth-table that travels from the upper left to the lower right:

P →L Q

So even though Mary Middleford is a borderline case of tallness, the sentence If Mary Middleford is tall, then she’s tall turns out true in L3 as do the similar conditionals about Gina Biggerly and Tina Littleton. On the other hand, If Mary Middleford is tall, then so is Tina Littleton is neither true nor false, and If Mary Middleford is tall, then so is Gina Biggerly is true. In a similar vein, Lukasiewicz wanted biconditionals like A ↔L A to be tautologies. Note that even though the truth-tables for the two connectives differ from Kleene’s truth-tables, the L3 connectives are also both normal and uniform.

Here are Lukasiewicz’s truth-tables for the formulas that we examined in Section 4.1:

The Law of Excluded Middle behaves as it did in KS3—not surprising because the connectives in this formula are defined identically in the two systems. The truthtable for the second formula makes a conditional true when its antecedent and consequent both have the value N, unlike Kleene’s tables. And the third formula always has the value T in L3, again unlike the treatment of that formula in KS3.

Having explained Lukasiewicz’s reason for assigning the value T to a conditional whose antecedent and consequent both have the value N, a reason that seems reasonable, we add that formulas that are not tautologies that exemplify this assignment strike some as odd. If the unrelated formulas P and Q both have the value N why should P → Q have the value T rather than the value N as it does in KS3? The

assignment of T would seem to make sense when antecedent and consequent are identical, or related as they are in the formula (P L Q) →L (P L Q) but not when they are completely unrelated. Some of the oddity is dispelled, however, when we consider that in the case where P and Q both have the value N, as well as in the other cases where P → Q has the value T, Q is “at least as true” as P.

Lukasiewicz, like Kleene, was not motivated by vagueness in constructing his system, but his motivation differed from Kleene’s. Lukasiewicz was interested in the truth-values of so-called future contingent sentences (a concern raised by the ancient Greek philosopher Aristotle). A future contingent sentence is a sentence about the future that might turn out to be true and also might turn out to be false— neither its truth nor its falsehood is necessary. Consider, for example, the sentence

The U.S. president in the year 3000 will be a woman. Such a sentence, according to Lukasiewicz’s reasoning, is neither true nor false today—for if it is true, then there will have to be a female U.S. president in 3000, and if it is false, then there cannot be a female U.S. president in 3000. But since there doesn’t have to be a female U.S. president in 3000, although there might be, it follows that the sentence is neither true nor false today. Given the assumption that future contingent sentences are neither true nor false L3 presents a nice logic for such sentences. For example, a conjunction of a true sentence and a future contingent one such as George Bush was the U.S. president in 2004 and the U.S. president in 3000 will be a female is neither true nor false. Despite the contingency of the sentence about the future presidency, however, the sentence If the U.S. president in 3000 will be a female then the U.S. president in 3000 will be a female is true in Lukasiewicz’s system.

Because the truth-tables for the conditional and the biconditional assign T to a formula whose immediate components both have the value N, these connectives cannot be defined in L3 in terms of (any combination of) the other three. The reason is fairly simple. If you construct a formula using only ¬L, L, and L as connectives, then whenever the atomic formulas from which it is constructed all have the value N the compound formula will have the value N as well. But now consider A →L A and A ↔L A. Both formulas have the value T when A has the value N. Since we can’t form a compound formula that has this property from A, ¬L, L, and L, we cannot define either →L or ↔L in terms of the other three connectives.

Lukasiewicz in fact took ¬L and →L as primitive and used them to define the other three connectives:

P L Q = def (P →L Q) →L Q

P L Q = def ¬L(¬L P L ¬LQ)

P ↔L Q = def (P →L Q) L (Q →L P)

Proof that these definitions produce the correct truth-functions is left as an exercise.4

4It is also possible to define different conjunction and disjunction operations in L3 that do support the classical interdefinability of the five connectives; more on this in Section 5.7.

Let us now consider various semantic concepts in L3:

Result 5.4: Every formula that is a tautology in L3 is also a tautology in classical logic, and every formula that is a contradiction in L3 is also a contradiction in classical logic.

Proof: A formula that is a tautology in L3 is true in L3 on every classical truthvalue assignment. Since L3 is normal, it follows from the Normality Lemma that the formula is true on every truth-value assignment in classical logic and is therefore a tautology in classical logic. Similar reasoning holds for contradictions.

Result 5.5: Not every formula that is a tautology in classical logic is also a tautology in L3, and not every formula that is a contradiction in classical logic is also a contradiction in L3.

Proof: Any instance of the Law of the Excluded Middle, for example, A L ¬LA, is an example of a classical tautology that does not always have the value T in L3. Another example is the formula (P →L (Q →L R)) →L ((P →L Q) →L (P →L R)). This formula always has the value T in classical logic, but in L3 it has the value N when P and Q have the value N and R has the value F. The formula A L ¬LA, which is a classical contradiction, is not a contradiction in L3—it has the value N when A has the value N.

Note that Result 5.5 does not claim that all classical tautologies fail to be tautologies of L3 (nor that all classical contradictions fail to be contradictions of L3). For example, A →L A is a tautology in both systems.

Result 5.6: If |=L P then |= P.

Proof: This follows from the Normality Lemma since L3 is normal.

Result 5.7: Not every entailment in classical propositional logic holds in L3.

Proof: The argument and truth-value assignment in Result 5.3 will suffice here as well.

We note that other classically valid arguments are valid in L3. For example, the classically valid

P

P → Q

Q

is valid in L3 as well as in KS3.