In formal semantics, truth-value semantics is an alternative to Tarskian semantics. It has been primarily championed by

Ruth Barcan Marcus Ruth Barcan Marcus (; born Ruth Charlotte Barcan; 2 August 1921 – 19 February 2012) was an American academic philosopher and logician best known for her work in modal and philosophical logic. She developed the first formal systems of quant ...

, H. Leblanc, and J. Michael Dunn and

Nuel Belnap Nuel Dinsmore Belnap Jr. (; born 1930) is an American logician and philosopher who has made contributions to the philosophy of logic, temporal logic, and structural proof theory. He taught at the University of Pittsburgh from 1963 until his reti ...

. It is also called the ''substitution interpretation'' (of the quantifiers) or substitutional quantification. The idea of these semantics is that a

universal Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a ...

(respectively, existential) quantifier may be read as a

conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction, a rule of inference of propositional logic * Conjunction (astronomy), in which two astronomical bodies ...

(respectively,

disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...

) 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 for

predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

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 formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformu ...

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

\forall x F(x)

is a logical consequence 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. 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