Many-Valued Logics

Bivalence is an assumption, not a discovery. Nothing in the notion of a truth function requires exactly two arguments, and once a third value is admitted the tables can be completed in several ways, each answering to a different reason for having admitted it.

The reasons are what distinguish the systems. A third value may mark a sentence that is neither true nor false, or one that is both, or one whose truth is undetermined. The tables come out nearly identical and the logics come out very different, because what separates them is which values count as good.

Designation

This is the hinge of the whole subject. Two systems can share every truth table and differ on which arguments are valid, purely through the designation. The point recurs below and is the reason the article treats designation before the systems.

The systems

Łukasiewicz introduced a third value in 1920, motivated by Aristotle's sea battle: the statement that there will be a sea battle tomorrow is, he held, not yet true and not yet false. His tables treat the third value as a genuine indeterminacy, and the distinctive choice is the conditional, which takes the value true when antecedent and consequent are both indeterminate. That choice keeps $\varphi \rightarrow \varphi$ valid, and keeping it is why the system has a conditional worth the name.

Kleene's motivation was computational rather than metaphysical. A partial recursive predicate may fail to return on some input, and the strong tables say what can be concluded before knowing whether it returns: a search that has already succeeded on one branch need not wait for the other. This is short-circuit evaluation, and the tables are those a programming language implements.

The strong scheme with only truth designated is the system usually called K3, and it has no tautologies at all, since assigning every atom the third value makes every formula take it. Excluded middle fails, the liar being the intended application: it is gappy, so neither it nor its negation is true.

The pair K3 and LP is the clearest illustration of the designation point. Identical tables; opposite properties. K3 has no valid formulas and is explosive, since a contradiction cannot have both conjuncts designated. LP validates every classical tautology and is not explosive, since the third value makes $\varphi$ and $\neg \varphi$ both designated while $\psi$ is false. What LP loses is modus ponens, which fails when $\varphi$ is glutty and $\psi$ is false, and the loss is serious enough that the system is usually supplemented with a further conditional.

The last row is Belnap's four-valued system, which separates the gap from the glut instead of using one value for both. Its four values are the subsets of the classical pair, ordered by information, and Belnap's 1977 presentation motivated it by a computer receiving reports from several sources: a proposition may be reported true by one and false by another, or by neither. The system is the algebraic heart of first-degree entailment, so the route through many-valued semantics arrives at the relevance logic of the previous article from a different direction.

Next

Degrees of truth were offered above as a treatment of vagueness, and vagueness has a paradox of its own that is older and more resistant than the liar. The next article takes it up, along with the three main responses and the reason none of them is comfortable.