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80

Three-Valued Propositional Logics: Semantics

5.3 Bochvar’s Three-Valued Logics

The Russian mathematician Dmitri Bochvar proposed two very different systems of three-valued logic (Bochvar 1937)—different from Kleene’s strong system and Lukasiewicz’s system but also different from each other. Bochvar was concerned with paradoxical sentences like the Liar Paradox, which, in its simplest form, is

This sentence is false.

The paradox begins with the assumption that the sentence is true or false. But now consider: if the sentence is true, then what it says is the case, and so it is false. So it can’t be true since that leads to a contradiction. Is it false, then? Well, if it is false then what it says is not the case and so it must be true. So the sentence can’t be false either. There’s your paradox. The Liar Paradox has been extensively studied,5 so we will only note here that Bochvar’s position was that the sentence is meaningless and hence neither true nor false, since only meaningful sentences can say true or false things. The third truth-value N represents meaninglessness for Bochvar.

Bochvar’s “internal” three-valued system, which we will designate as BI3, has the following truth-tables:

P

 

 

¬BIP

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T

 

 

F

 

 

 

 

 

 

 

 

 

 

 

 

 

N

 

 

N

 

 

 

 

 

 

 

 

 

 

 

 

 

F

 

 

T

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P BI Q

 

 

P BI Q

 

P BI Q

P BI Q

P \ Q

 

T N F

 

P \ Q

 

T N F

P \ Q

 

T N F

P \ Q

 

T N F

 

 

 

 

 

T

 

 

T N F

 

T

 

 

T N T

 

T

 

T N F

 

T

 

T N F

N

 

 

N N N

 

N

 

N N N

 

N

 

N N N

N

 

N N N

F

 

 

F N F

 

F

 

 

T N F

 

F

 

T N T

F

 

F N T

We mentioned earlier that Kleene had a second system of three-valued connectives, which he called the weak connectives. That system is identical to BI3. We shall nevertheless refer to the system as Bochvar’s, as is customary.

Negation in BI3 is identical to negation in the previous systems, but the truth-functions for the other connectives are all different. We might say that the truth-value N is contagious in BI3—whenever a component of a compound formula has the value N, so does the compound formula as a whole—regardless of the value of any other component. If N represents meaninglessness, then it is quite sensible that this value should be contagious. Just as Thiggledy piggledy is meaningless, so is Thiggledy piggledy and grass is green. (The expression as a whole is meaningless, although part of it is meaningful.) Of course, our interest in this text is vagueness and so the “contagiousness” is that of the value N based on borderline cases. So, for

5

See, for example, Martin (1970, 1984).

 

5.3 Bochvar’s Three-Valued Logics

81

example, even if a disjunction has a true disjunct it is nevertheless vague as a whole if the other disjunct is vague.

Here are the BI3 truth-tables for our earlier formulas:

P

 

P BI ¬L P

 

P Q

 

P BI (P BI Q)

 

 

 

T

 

T T F T

 

T T

 

T T T T T

N

 

N N N N

 

T N

 

T N T N N

F

 

F T T F

 

T F

 

T F T F F

 

 

 

 

N T

 

N N N N T

 

 

 

 

N N

 

N N N N N

 

 

 

 

N F

 

N N N N F

 

 

 

 

F T

 

F

T

F

T

T

 

 

 

 

F N

 

F

N

F

N

N

 

 

 

 

F F

 

F

T

F

T

F

P Q (P BI Q) BI (P BI Q)

T T

T

T

T

T

T

T

T

T N

T

N

N

N

T

N

N

T F

T

F

F

T

T

T

F

N T

N

N

T

N

N

N

T

N N

N N N

N N N N

N F

N

N

F

N

N

N

F

F T

F

F

T

T

F

T

T

F N

F

N

N

N

F

N

N

F F

F

F

F

T

F

F

F

Neither of the classical tautologies is a tautology here, and the second formula receives the value N more often than it did in KS3 or L3.

The BI3 connectives are all normal—they agree with the classical tables when their components are either T or F—but of the binary connectives only the biconditional is uniform, and that only trivially so. Uniformity of conjunction, for example, would require that a conjunction be false whenever one of the conjuncts is. But since the value N is contagious, this is not the case. Similar comments show that disjunction and the conditional are also not uniform in this system.

As with KS3, any way of interdefining connectives in classical logic will also work for BI3. This is because not only are the connectives normal—so we will get the desired results for truth-value assignments involving only T and F—but they all agree on what happens when a formula has a component with the value N (namely, the compound formula is also assigned the value N).

Also like KS3, BI3 has no tautologies. Because N is contagious, every formula has the value N on at least one truth-value assignment to its atomic components— namely, on any truth-value assignment that assigns N to at least one atomic component. So no formula is true on every truth-value assignment in BI3. We thus have

Result 5.8: No formula is a tautology in BI3, and no formula is a contradiction in BI3.

Concerning entailment:

Result 5.9: If |=BI P then |= P.

Proof: This follows from the Normality Lemma since BI3 is normal.

Result 5.10: Not every entailment that holds in classical propositional logic holds in BI3 as well.

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Three-Valued Propositional Logics: Semantics

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

There are also significant examples of classically valid arguments that are not valid in BI3 but that are valid in both KS3 and L3. One example is

Q

P Q

In BI3, the premise has the value T but the conclusion has the value N when Q has the value T and P has the value N.

Bochvar introduced a second system of connectives that together constitute his external system of three-valued logic, BE3. In BE3, the value N acts as if it is actually the value F:

P

 

¬BEP

 

T

 

F

N

 

T

F

 

T

P BE Q

P BE Q

 

P BE Q

 

P BE Q

P \ Q

T N F

P \ Q

T N F

P \ Q

T N F

P \ Q

T N F

T

 

T F F

T

T T T

 

T

T F F

 

T

T F F

N

 

F F F

N

T F F

 

N

T T T

 

N

F T T

F

 

F F F

F

T F F

 

F

T T T

 

F

F T T

Bochvar introduced both the internal and external connectives within a single system; in that system the external connectives were defined connectives, using the internal connectives and a special external assertion operator a:

P aP

T T

N F

F F

To define the external version of a connective, we apply the internal version of the connective to externally asserted formulas. Thus, for example, if we apply the internal ¬BI to aP we get the table for external negation:

P ¬BI aP

T F T

N T F

F T F

5.3 Bochvar’s Three-Valued Logics

83

and if we apply the internal BI to aP and aQ we get the table for external conjunction:

P Q

 

aP BI aQ

 

T T

 

T T

T

T N

 

T F

F

T F

 

T

F

F

N T

 

F

F

T

N N

 

F F

F

N F

 

F

F

F

F T

 

F

F

T

F N

 

F

F

F

F F

 

F

F

F

We may read the external assertion connective as asserting truth: aP means P is true. This assertion is true if P is true, and is false otherwise. In particular, if P has the truth-value N then it is false that P is true. Rather than take the external assertion operator as primitive, however, we will define it using external negation:

aP = def ¬BE¬BEP

Henceforth, for simplicity in comparing systems, when we speak of Bochvar’s external connectives we will mean the five connectives other than external assertion. The external connectives are both normal and uniform.

By introducing the external connectives Bochvar created a system with tautologies as well as contradictions; indeed, we have the following results:

Result 5.11: The set of formulas that are tautologies in BE3 is exactly the set of formulas that are tautologies in classical logic, and the set of formulas that are contradictions in BE3 is exactly the set of formulas that are contradictions in classical logic.

Proof: Since BE3 is normal, it follows from the Normality Lemma that every formula that is a tautology in BE3 is a classical tautology, and similarly for contradictions.

Conversely, we note that every classical tautology is a compound formula. Since the connectives in BE3 treat their N components as if they were false, BE3 treats the atomic components of any compound formula on a truth-value assignment where they are N as if they were false—and hence assigns the truthvalue to the formula that classical logic would in that case. So a classical tautology must be a tautology in BE3 as well, and similar reasoning holds for contradictions.

Entailment also behaves classically in BE3:

Result 5.12: If |=BE P then |= P.

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Three-Valued Propositional Logics: Semantics

Proof: This follows from the Normality Lemma, since BE3 is normal.

Result 5.13: If |= P then |=BE P.

Proof: We shall show this by contraposition; that is, we’ll show that if an entailment does not hold in BE3 then it doesn’t hold in classical logic either. So consider a set and formula P such that |=BE P. Then there is some three-valued assignment on which all the formulas in have the value T but on which P has either the value F or the value N. We can convert this to a classical truth-value assignment by keeping the T and F assignments to atomic formulas but turning the N assignments (if any) to atomic formulas into F assignments. This classical truth-value assignment will make the premises of the argument true in classical logic because compound formulas in BE3 behave as if their N-valued atomic components have the value F, and if any of the formulas in are atomic, then, since they have the value T on the BE3 assignment, they will have the value T on the classical assignment as well. But P has the value F on the classical truth-value assignment for similar reasons.

Thus tautologousness, contradictoriness, and entailment all coincide for classical logic and BE3.

5.4 Evaluating Three-Valued Systems; Quasi-Tautologies

and Quasi-Contradictions

There are several ways in which we can measure the adequacy of a three-valued system as a logic of vagueness. First, we note that the Principle of Bivalence fails for three-valued systems, by definition! On this count, all four systems that we have presented are good candidates for such a logic. However, we note that although BE3 rejects the Principle of Bivalence it does so only for atomic formulas, and compound formulas behave exactly as they do in classical logic. So if we believe, for example, that the Law of Excluded Middle fails for vague sentences, that would be a reason for rejecting BE3 as a logic for vagueness.

How else might we evaluate the three-valued systems, and in particular are there significant criteria that will distinguish among the remaining systems KS3, L3, and BI3? One way to evaluate different logical systems is to compare their sets of tautologies and contradictions. We have seen that in each of the three systems, there are classical tautologies that fail to be tautologies in the three-valued system (and that a similar situation holds for classical contradictions). This is desirable, at least if we believe that classical tautologies such as instances of the Law of Excluded Middle should fail in the case of borderline attributions of vague predicates. But there is a difference in the sets of classical tautologies that fail for the three systems. KS3 and BI3 have no tautologies. On the other hand, L3 does have tautologies. If we believe that the simple classical tautologies A A and A A should remain tautologies

5.4 Evaluating Three-Valued Systems

85

within a three-valued system, that would be a reason for preferring L3 to KS3 and BI3.

Butthere is a second way to define a tautology-like concept in three-valued logic. We will say that a formula is a quasi-tautology if it is never false. Note that in classical logic the concepts of being a tautology and being a quasi-tautology coincide, since a formula that is never false in classical logic is always true, and vice versa. But the concepts do not coincide in three-valued systems. For example, although KS3 and BI3 have no tautologies they both have quasi-tautologies; the formula A ¬A is a quasi-tautology in each of the two systems (as well as in L3).

It is common to talk of designated truth-values in connection with tautologies and their kin: the designated truth-values include T and any other truth-values that we wish to count as “good” or at least as “not bad.” We can then define tautologies in terms of designated truth-values: a formula is a tautology if it has a designated truth-value on every truth-value assignment. If only the value T is designated, we end up with the definition of tautology that we’ve been using: a formula that always has the value T. If both the values T and N are designated, we end up with the definition of quasi-tautologies instead.

Why might we be interested in quasi-tautologies? For one thing, we might be interested in avoiding falsehood as much as we are interested in embracing truth. If the former is the case, the set of quasi-tautologies should be of interest. But practical interests aside, the concept of a quasi-tautology is a second way of generalizing the classical notion of a tautology—as a formula that is never false rather than as a formula that is always true—and the concept therefore also has purely theoretical interest.

As we just noted, there are quasi-tautologies in KS3 and BI3 even though there are no tautologies in either of the systems. In fact, every classical tautology is a quasi-tautology in both systems, and vice versa. We’ll prove this first for Bochvar’s system, since that is the simpler of the two proofs.

Result 5.14: The set of BI3 quasi-tautologies is exactly the set of classical tautologies.

Proof: Let P be a BI3 quasi-tautology. Then P does not have the value F on any truth-value assignment. Since BI3 is normal, it follows from the Normality Lemma that P does not have the value F on any classical truth-value assignment in classical logic and is therefore a tautology of classical logic. So every BI3 quasitautology is a classical tautology.

Conversely, assume that a formula P is not a BI3 quasi-tautology. Then P has the value F on some truth-value assignment in BI3. This truth-value assignment must be a classical assignment, since the value N is contagious in BI3. It follows from the Normality Lemma that P has the value F on this assignment in classical logic and therefore is not a classical tautology.

86

Three-Valued Propositional Logics: Semantics

Although we have the same result for KS3, the proof in the second direction is somewhat different:

Result 5.15: The set of KS3 quasi-tautologies is exactly the set of classical tautologies.

Proof: The proof that every KS3 quasi-tautology is a classical tautology follows from the Normality Lemma.

The converse claim, that a formula P that is not a KS3 quasi-tautology is also not a classical tautology, is equivalent to saying that a formula P that has the value F on some truth-value assignment in KS3 will also have the value F on some classical assignment. The restated claim holds trivially if the assignment on which P has the value F in KS3 is a classical assignment. So we need to establish that if a formula P has the value F on some nonclassical truth-value assignment in KS3, P will also have the value F on some classical assignment in KS3.

In order for P to have the value F in KS3 on an assignment on which one or more of its atomic components have the value N, uniformity must have kicked in at some point to override the Ns in favor of classical truth-values. And at each point where uniformity kicked in, the same classical value would have resulted if the N had been a T or an F instead. So if we replace all of the Ns that the three-valued assignment assigns with either Ts or Fs (it doesn’t matter which), P will end up having the same value on the resulting classical assignment as it did on the three-valued assignment.

On the other hand, the quasi-tautologies of L3 do not coincide with the tautologies of classical logic:

Result 5.16: Every L3 quasi-tautology is a classical tautology; every classical tautology that contains only negation, conjunction, and disjunction is an L3 tautology; but some classical tautologies containing the conditional or the biconditional are not L3 quasi-tautologies.

Proof: The proof that every L3 quasi-tautology is a classical tautology follows from the Normality Lemma.

It follows from Result 5.15 that every classical tautology that contains only negation, conjunction, and disjunction is an L3 tautology because L3 negation, conjunction, and disjunction are defined the same as in KS3.

An example of a classical tautology containing a conditional that is not a quasi-tautology in L3 is ¬(A → ¬A) ¬(¬A A). When A has the truth-value N, this formula is false in L3 and therefore is not a quasi-tautology. An example of a classical tautology containing a biconditional that is not a quasi-tautology in L3 is ¬(A ↔ ¬A); this formula is also false when A has the value N. (It is easily verified that these formulas are both classical tautologies.)

5.4 Evaluating Three-Valued Systems

87

For BE3, we have

Result 5.17: The set of BE3 quasi-tautologies coincides with the set of classical tautologies.

Proof: The only formulas that are not tautologies in BE3 but might be quasitautologies are atomic formulas, for no other formulas can ever have the truthvalue N in this system. But these formulas are neither classical tautologies nor quasi-tautologies in BE3, since they can have the value F. So quasi-tautologies and tautologies coincide in BE3, and we have already established that the BE3 tautologies coincide with the set of classical tautologies.

As a dual to the concept of quasi-tautology, we introduce quasi-contradictions: a formula is a quasi-contradiction if it is never true; that means that in a three-valued system it always has the value T or the value N.6 The results concerning quasitautologies in the three-valued systems also hold for quasi-contradictions: namely, in each of BI3, BE3, and KS3 the set of quasi-contradictions coincides with the set of classical contradictions; every L3 quasi-contradiction is a classical contradiction; and some classical contradictions are not L3 quasi-contradictions. Proofs of these claims are left as an exercise.

We also introduce the concept of a quasi-entailment: a set of formulas quasientails a formula P if there is no truth-value assignment on which each of the formulas in has the value T or N while P has the value F; that is, whenever each formula in has one of the values T or N so does P. An argument is quasi-valid if the set consisting of its premises quasi-entails its conclusion. We have

Result 5.18: Every quasi-entailment in each of KS3, L3, BI3, and BE3 is a classical entailment.

Proof: If a set of formulas quasi-entails a formula P in any of the four systems then there is no classical truth-value assignment in any of these systems on which all of the formulas in have the value T (none will have the value N on a classical truth-value assignment) and P has the value F in that system. Since these systems are all normal, it follows from the Normality Lemma that the entailment holds in classical logic.

The converse does not generally hold:

Result 5.19: Not every classical entailment is a quasi-entailment in KS3, and ditto for the other three systems L3, BI3, and BE3.

Proof: The argument

A ¬A

B

6We may also define contradictions via anti-designated truth-values—truth-values including F and other values that we wish to single out. The sets of designated and anti-designated truthvalues may overlap—so N can be included in both sets.

88

Three-Valued Propositional Logics: Semantics

is valid in classical logic. But it fails to be quasi-valid in any of KS3, L3, or BI3 (although it is valid in all three systems). This is because the premise has the value N and the conclusion has the value F in each of these three systems when A has the value N and B has the value F.

The classically valid argument

A

A A

isn’t valid in BE3. When the premise has the value N, the conclusion is false.

Of course, some classically valid arguments are also quasi-valid in more than one of the four systems. The argument

P Q

P

is quasi-valid in KS3, L3, and BE3, and the argument

P

P Q

is quasi-valid in KS3, L3, and BI3.

There is a third interesting version of entailment (and hence of validity) in three-valued systems: rather than simply preserving truth (as in entailment proper) or preserving non-falsehood (as in quasi-entailment), we can rank the three truthvalues and require that the value of P be at least as great as the value of the lowestranked formula in . We rank the three truth-values as T N F. We will say that a set of formulas degree-entails7 a formula P if P’s value can never be less than the least value of the formulas in . If all of the formulas in have the value T, then P must also have the value T, and if each of the formulas in has either the value T or the value N, then P must have either the value T or the value N as well. An argument is degree-valid in a three-valued system if the set of its premises degree-entails its conclusion. Not surprisingly, we have

Result 5.20: Every degree-entailment that holds in BI3, BE3, KS3, or L3 is a classical entailment.

Proof: Left as an exercise.

In fact, degree-entailment is equivalent to entailment proper plus quasi-entailment (to be proved in the exercises). So if fails to entail or to quasi-entail P in any threevalued system then will also fail to degree-entail P in that system. Consequently, we have

7The name degree-entailment anticipates the concept of degrees of truth to be introduced when we turn to fuzzy logic.