In formal semantics, truth-value semantics is an alternative to

Tarskian semantics First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...

. 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. (; May 1, 1930 – June 12, 2024) was 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 ...

. 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 that is a subsidiary of Comcast ** Universal Animation Studios, an American Animation studio, and a subsidiary of N ...

(respectively,

existential Existentialism is a family of philosophical views and inquiry that explore the human individual's struggle to lead an authentic life despite the apparent absurdity or incomprehensibility of existence. In examining meaning, purpose, and value ...

) quantifier may be read as a conjunction (respectively,

disjunction In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is ...

) of formulas in which constants replace the variables in the scope of the quantifier. For example,

\forall xPx

may be read (

Pa \and Pb \and Pc \and\dots

) where

a, b, c

are individual constants replacing all occurrences of

x

Px

. The main difference between truth-value semantics and the standard semantics for

predicate logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables ove ...

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

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) ground substitution instances of the unquantified subformula 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

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. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...

fail. To see this consider the set

\

. Clearly the formula

\forall x F(x)

is a

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

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