In formal semantics, truth-value semantics is an alternative to Tarskian semantics. It has been primarily championed by Ruth Barcan Marcus, H. Leblanc, and J. Michael Dunn and Nuel Belnap. It is also called the ''substitution interpretation'' (of the quantifiers) or substitutional quantification. The idea of these semantics is that a universal (respectively, existential) quantifier may be read as a conjunction (respectively, disjunction) of formulas in which constants replace the variables in the scope of the quantifier. For example, ∀''xPx'' may be read (''Pa'' & ''Pb'' & ''Pc'' &...) 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 Standard may refer to: Symbols * Colours, standards and guidons, kinds of military signs * Standard (emblem), a type of a large symbol or emblem used for identification Norms, conventions or requirements * Standard (metrology), an object th ...

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 (the referent of) ''b'' is a member of the extension of the predicate ''P'', respectively, if and only if the pair (''c'', ''a'') is a member of the extension of ''R'', in truth-value semantics the truth-values of atomic formulas are basic. A universal (existential) formula is true if and only if all (some) substitution instances of it are true. Compare this with the standard semantics, which says that a universal (existential) formula is true if and only if for all (some) members of the domain, the formula holds for all (some) of them; for example,

\forall x A

is true (under an interpretation) if and only if for all ''k'' in the domain ''D'', ''A''(''k''/''x'') is true (where A(k/x) is the result of substituting ''k'' for all occurrences of ''x'' in ''A''). (Here we are assuming that constants are names for themselves—i.e. they are also members of the domain.) Truth-value semantics is not without its problems. First, the strong completeness theorem and

compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...

fail. To see this consider the set . Clearly the formula

\forall x F(x)

is a

logical consequence Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is on ...

of the set, but it is not a consequence of any finite subset of it (and hence it is not deducible from it). It follows immediately that both compactness and the strong completeness 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 A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain. A free logic with the latter propert ...

. Consider a language with one individual constant ''c'' that is nondesignating and a predicate ''F'' standing for 'does not exist'. Then

\exists x F x

is false even though a substitution instance (in fact ''every'' such instance under this interpretation) 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.

References

{{DEFAULTSORT:Truth-Value Semantics Mathematical logic Semantics

See also

References