truth-value gaps See MANY-VALUED LOGIC, PRESUP-. POSITIO. truth-value semantics, interpretations of formal systems in which the truth-value of a formula rests ultimately only on truth-values that are assigned to its atomic subformulas (where ‘subformula’ is suitably defined). The label is due to Hugues Leblanc. On a truth-value interpretation for first-order predicate logic, for example, the formula atomic ExFx is true in a model if and only if all its instances Fm, Fn, . . . are true, where the truth-value of these formulas is simply assigned by the model. On the standard Tarskian or objectual interpretation, by contrast, ExFx is true in a model if and only if every object in the domain of the model is an element of the set that interprets F in the model. Thus a truth-value semantics for predicate logic comprises a substitutional interpretation of the quantifiers and a ‘non-denotational’ interpretation of terms and predicates. If t1, t2, . . . are all the terms of some first-order language, then there are objectual models that satisfy the set {Dx-Fx, Ft1, Ft2 . . . .}, but no truth-value interpretations that do. One can ensure that truth-value semantics delivers the standard logic, however, by suitable modifications in the definitions of consistency and consequence. A set G of formulas of language L is said to be consistent, for example, if there is some G’ obtained from G by relettering terms such that G’ is satisfied by some truth-value assignment, or, alternatively, if there is some language L+ obtained by adding terms to L such that G is satisfied by some truth-value assignment to the atoms of L+. Truth-value semantics is of both technical and philosophical interest. Technically, it allows the completeness of first-order predicate logic and a variety of other formal systems to be obtained in a natural way from that of propositional logic. Philosophically, it dramatizes the fact that the formulas in one’s theories about the world do not, in themselves, determine one’s ontological commitments. It is at least possible to interpret first-order formulas without reference to special domains of objects, and higher-order formulas without reference to special domains of relations and properties.
The idea of truth-value semantics dates at least to the writings of E. W. Beth on first-order predicate logic in 1959 and of K. Schütte on simple type theory in 1960. In more recent years similar semantics have been suggested for secondorder logics, modal and tense logics, intuitionistic logic, and set theory.
See also FORMAL SEMANTICS , MEANING, QUANTIFICATION , TRUTH TABL. S.T.K.