Truth-value semantics
Encyclopedia
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 (of the quantifiers) or substitutional quantification.
The idea of these semantics is that universal (existential) quantifier may be read as a conjunction (disjunction) of formulas in which constants replace the variables in the scope of the quantifier. E.g. ∀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
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
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, resp., 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; e.g. ∀xA 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 {F(1), F(2),...}. Clearly the formula ∀xF(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 ∃xFx 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.
Semantic theory of truth
A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences.-Origin:The semantic conception of truth, which is related in different ways to both the correspondence and deflationary conceptions, is due to work published by Polish...
. It has been primarily championed by Ruth Barcan Marcus
Ruth Barcan Marcus
Ruth Barcan Marcus is the American philosopher and logician after whom the Barcan formula is named. She is a pioneering figure in the quantification of modal logic and the theory of direct reference...
, H. Leblanc, and M. Dunn and N. Belnap. It is also called the substitution interpretation (of the quantifiers) or substitutional quantification.
The idea of these semantics is that universal (existential) quantifier may be read as a conjunction (disjunction) of formulas in which constants replace the variables in the scope of the quantifier. E.g. ∀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
Predicate logic
In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...
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
Atomic formula
In mathematical logic, an atomic 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 subformulas. Atoms are thus the simplest well-formed formulas of the logic...
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, resp., 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; e.g. ∀xA 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
Compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model...
fail. To see this consider the set {F(1), F(2),...}. Clearly the formula ∀xF(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
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...
. 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 (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.
See also
- Game semanticsGame semanticsGame semantics is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player, somewhat resembling Socratic dialogues or medieval theory of Obligationes. In the late 1950s Paul Lorenzen was the...
- Kripke semanticsKripke semanticsKripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
- Model-theoretic semantics
- Proof-theoretic semanticsProof-theoretic semanticsProof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within the system...
- Truth-conditional semanticsTruth-conditional semanticsTruth-conditional semantics is an approach to semantics of natural language that sees the meaning of assertions as being the same as, or reducible to, their truth conditions...