ABSTRACT

A many-valued (aka multiple-or multi-valued) semantics, in the strict sense, is one which employs more than two truth values; in the loose sense it is one which countenances more than two truth statuses. So if, for example, we say that there are only two truth values-True and False-but allow that as well as possessing the value True and possessing the value False, propositions may also have a third truth status-possessing neither truth value-then we have a many-valued semantics in the loose but not the strict sense. A many-valued logic is one which arises from a many-valued semantics and does not also arise from any two-valued semantics [Malinowski, 1993, 30]. By a ‘logic’ here we mean either a set of tautologies, or a consequence relation. We can best explain these ideas by considering the case of classical propositional logic. The language contains the usual basic symbols (propositional constants p, q, r, . . .; connectives ™, š, ›ol; and parentheses) and well-formed formulas are defined in the standard way. With the language thus specified-as a set of well-formed formulas-its semantics is then given in three parts. (i) A model of a logical language consists in a free assignment of semantic values to basic items of the non-logical vocabulary. Here the basic items of the non-logical vocabulary are the propositional constants. The appropriate kind of semantic value for a proposition is a truth value, and so a model of the language consists in a free assignment of truth values to basic propositions. Two truth values are countenanced: 1 (representing truth) and 0 (representing falsity). (ii) Rules are presented which determine a truth value for every proposition of the language, given a model. The most common way of presenting these rules is via truth tables (see Figure 5.6.1). Another way of stating such rules-which will be useful below-is first to introduce functions on the truth val-

Figure 5.6.1 Classical truth tables

ues themselves: a unary function ™* and four binary functions š*, ›*o* and l* (see Figure 5.6.2). Representing the truth value of D(on a given model) as [D], we then specify the truth values of compound formulas as in Figure 5.6.3. Once one becomes familiar with the distinction between connectives and truth functions, it is customary to use the same symbols for both and to let context disambiguate. As it generally increases readability, I shall mostly follow this practice below (i.e. omit the *’s on truth functions). (iii) Denitions of tautology and logical consequence are introduced. In this case, a tautology is a proposition which gets the value 1 on every model (e.g. p ›™p, p o p), and a proposition Dis a logical consequence of the set of propositions *(written *_ D) if, on every model on which every proposition in * has the value 1, Dhas the value 1 (e.g. {p, p o q} _ q, {p} _ p ›q). Classical logic is then the language just introduced together with either the set of tautologies, or the consequence relation, just dened. The definition of a logic in terms of a consequence relation is more powerful, in that once we have the consequence relation, we can reconstruct the set of tautologies as the set of propositions D such that À _ D. However sometimes we are interested only in tautologies-hence we allow that a logic may be specified just by giving a set of tautologies, without a consequence relation.