many-valued logic a logic that rejects the principle of bivalence: every proposition is true or false. However, there are two forms of rejection: the truth-functional mode (many-valued logic proper), where propositions may take many values beyond simple truth and falsity, values functionally determined by the values of their components; and the truth-value gap mode, in which the only values are truth and falsity, but propositions may have neither. What value they do or do not have is not determined by the values or lack of values of their constituents.
Many-valued logic has its origins in the work of Lukasiewicz and (independently) Post around 1920, in the first development of truth tables and semantic methods. Lukasiewicz’s philosophical motivation for his three-valued calculus was to deal with propositions whose truth-value was open or ‘possible’ – e.g., propositions about the future. He proposed they might take a third value. Let 1 represent truth, 0 falsity, and the third value be, say, ½. We take Ý (not) and P (implication) as primitive, letting v(ÝA) % 1 † v(A) and v(A P B) % min(1,1 † v(A)!v(B)). These valuations may be displayed: Lukasiewicz generalized the idea in 1922, to allow first any finite number of values, and finally infinitely, even continuum-many values (between 0 and 1). One can then no longer represent the functionality by a matrix; however, the formulas given above can still be applied. Wajsberg axiomatized Lukasiewicz’s calculus in 1931. In 1953 Lukasiewicz published a four-valued extensional modal logic. In 1921, Post presented an m-valued calculus, with values 0 (truth), . . . , m † 1 (falsity), and matrices defined on Ý and v (or): v(ÝA) % 1 ! v(A) (modulo m) and v(AvB) % min (v(A),v(B)). Translating this for comparison into the same framework as above, we obtain the matrices (with 1 for truth and 0 for falsity): The strange cyclic character of Ý makes Post’s system difficult to interpret – though he did give one in terms of sequences of classical propositions. A different motivation led to a system with three values developed by Bochvar in 1939, namely, to find a solution to the logical paradoxes. (Lukasiewicz had noted that his three-valued system was free of antinomies.) The third value is indeterminate (so arguably Bochvar’s system is actually one of gaps), and any combination of values one of which is indeterminate is indeterminate; otherwise, on the determinate values, the matrices are classical. Thus we obtain for Ý and P, using 1, ½, and 0 as above: In order to develop a logic of many values, one needs to characterize the notion of a thesis, or logical truth. The standard way to do this in manyvalued logic is to separate the values into designated and undesignated. Effectively, this is to reintroduce bivalence, now in the form: Every proposition is either designated or undesignated. Thus in Lukasiewicz’s scheme, 1 (truth) is the only designated value; in Post’s, any initial segment 0, . . . , n † 1, where n‹m (0 as truth). In general, one can think of the various designated values as types of truth, or ways a proposition may be true, and the undesignated ones as ways it can be false. Then a proposition is a thesis if and only if it takes only designated values. For example, p P p is, but p 7 Ýp is not, a Lukasiewicz thesis.
However, certain matrices may generate no logical truths by this method, e.g., the Bochvar matrices give ½ for every formula any of whose variables is indeterminate. If both 1 and ½ were designated, all theses of classical logic would be theses; if only 1, no theses result. So the distinction from classical logic is lost. Bochvar’s solution was to add an external assertion and negation. But this in turn runs the risk of undercutting the whole philosophical motivation, if the external negation is used in a Russell-type paradox.
One alternative is to concentrate on consequence: A is a consequence of a set of formulas X if for every assignment of values either no member of X is designated or A is. Bochvar’s consequence relation (with only 1 designated) results from restricting classical consequence so that every variable in A occurs in some member of X.
There is little technical difficulty in extending many-valued logic to the logic of predicates and quantifiers. For example, in Lukasiewicz’s logic, v(E xA) % min {v(A(a/x)): a 1. D}, where D is, say, some set of constants whose assignments exhaust the domain. This interprets the universal quantifier as an ‘infinite’ conjunction.
In 1965, Zadeh introduced the idea of fuzzy sets, whose membership relation allows indeterminacies: it is a function into the unit interval [0,1], where 1 means definitely in, 0 definitely out. One philosophical application is to the sorites paradox, that of the heap. Instead of insisting that there be a sharp cutoff in number of grains between a heap and a non-heap, or between red and, say, yellow, one can introduce a spectrum of indeterminacy, as definite applications of a concept shade off into less clear ones.
Nonetheless, many have found the idea of assigning further definite values, beyond truth and falsity, unintuitive, and have instead looked to develop a scheme that encompasses truthvalue gaps. One application of this idea is found in Kleene’s strong and weak matrices of 1938. Kleene’s motivation was to develop a logic of partial functions. For certain arguments, these give no definite value; but the function may later be extended so that in such cases a definite value is given. Kleene’s constraint, therefore, was that the matrices be regular: no combination is given a definite value that might later be changed; moreover, on the definite values the matrices must be classical. The weak matrices are as for Bochvar. The strong matrices yield (1 for truth, 0 for falsity, and u for indeterminacy): An alternative approach to truth-value gaps was presented by Bas van Fraassen in the 1960s. Suppose v(A) is undefined if v(B) is undefined for any subformula B of A. Let a classical extension of a truth-value assignment v be any assignment that matches v on 0 and 1 and assigns either 0 or 1 whenever v assigns no value. Then we can define a supervaluation w over v: w(A) % 1 if the value of A on all classical extensions of v is 1, 0 if it is 0 and undefined otherwise. A is valid if w(A) % 1 for all supervaluations w (over arbitrary valuations). By this method, excluded middle, e.g., comes out valid, since it takes 1 in all classical extensions of any partial valuation. Van Fraassen presented several applications of the supervaluation technique. One is to free logic, logic in which empty terms are admitted. See also FREE LOGIC, VAGUENES. S.L.R.