In formal semantics, truth-value semantics is an alternative to Tarskian semantics. It has been primarily championed by Ruth Barcan Marcus, H. Leblanc, and M. Dunn and N. Belnap. It is also called the substitution interpretation or substitutional quantification. The idea of these semantics is that universal quantifier may be read as a conjunction of formulas in which constants replace the variables in the scope of the quantifier. E.g. ∀xPx may be read where a,b,c are individual constants replacing all occurrences of x in Px. The main difference between truth-value semantics and the standard semantics for predicate logic is that there are no domains for truth-value semantics. Only the truth clauses for atomic and for quantificational formulas differ from those of the standard semantics. Whereas in standard semantics atomic formulas like Pb or Rca are true if and only if b is a member of the extension of the predicate P, respectively, if and only if the pair is a member of the extension of R, in truth-value semantics the truth-values of atomic formulas are basic. A universal formula is true if and only if all substitution instances of it are true. Compare this with the standard semantics, which says that a universal formula is true if and only if for all members of the domain, the formula holds for all of them; e.g. ∀xA is true if and only if for all k in the domain D, A is true. Truth-value semantics is not without its problems. First, the strong completeness theorem and compactness fail. To see this consider the set. Clearly the formula ∀xF is a logical consequence of the set, but it is not a consequence of any finite subset of it. It follows immediately that both compactness and the strongcompleteness theorem fail for truth-value semantics. This is rectified by a modified definition of logical consequence as given in Dunn and Belnap 1968. Another problem occurs in free logic. Consider a language with one individual constant c that is nondesignating and a predicate F standing for 'does not exist'. Then ∃xFx is false even though a substitution instance of it is true. To solve this problem we simply add the proviso that an existentially quantified statement is true under an interpretation for at least one substitution instance in which the constant designates something that exists.
