Many-valued logics, philosophical issues in
Authors: Humberstone Lloyd 
Published: September 2018

The first philosophically-motivated use of many-valued truth tables arose with Jan Łukasiewicz in the 1920s. What exercised Łukasiewicz was a worry that the principle of bivalence, ‘every statement is either true or false’, involves an undesirable commitment to fatalism. Should not statements about the future whose eventual truth or falsity depends on the actions of free agents be given some third status – ‘indeterminate’, say – as opposed to being (now) regarded as determinately true or determinately false? To implement this idea in the context of the language of sentential logic (with conjunction, disjunction, implication and negation), we need to show – if the usual style of treatment of such connectives in a bivalent setting is to be followed – how the status of a compound formula is determined by the status of its components.

Łukasiewicz’s decision as to how the appropriate three-valued truth-functions should look is recorded in truth tables in which (determinate) truth and falsity are represented by ‘1’ and ‘3’ respectively, with ‘2’ for indeterminacy (see tables in the main body of the entry). Consider the formula A∨B (‘A or B’), for example, when A has the value 2 and B has the value 1. The value of A∨B is 1, reasonably enough, since if A’s eventual truth or falsity depends on how people freely act, but B is determinately true already, then A∨B is already true independently of such free action. There are no constraints as to which values may be assigned to propositional variables. The law of excluded middle is invalidated in the case of indeterminacy: if p is assigned the value 2, then p∨ ¬p also has the value 2. This reflects Łukasiewicz’s idea that such disjunctions as ‘Either I shall die in a plane crash on January 1, 2030 or I shall not die in a plane crash on January 1, 2030’ should not be counted as logical truths, on pain of incurring the fatalistic commitments already alluded to.

Together with the choice of designated elements (which play the role in determining validity played by truth in the bivalent setting), Łukasiewicz’s tables constitute a (logical) matrix. An alternative three-element matrix, the 1-Kleene matrix, involves putting 2→2=2, leaving everything else unchanged. And a third such matrix, the 1,2-Kleene matrix, differs from this in taking as designated the set of values {1,2} rather than {1}. The 1-Kleene matrix has been proposed for the semantics of vagueness. In the case of a sentence applying a vague predicate, such as ‘young’, to an individual, the idea is that if the individual is a borderline case of the predicate (not definitely young, and not definitely not young, to use our example) then the value 2 is appropriate, while 1 and 3 are reserved for definite truths and falsehoods, respectively. Łukasiewicz also explored, as a technical curiosity, n-valued tables constructed on the same model, for higher values of n, as well as certain infinitely many-valued tables. Variations on this theme have included acknowledging as many values as there are real numbers, with similar applications to vagueness and approximation in mind.