Semantics

in mathematical logic

The investigation of interpretations of a logical calculus (a formal axiomatic theory), of the study of the sense and meaning of constructions in formal language theory, and of the methods of understanding its logical connectives and formulas. Semantics studies the precise description and definition of such concepts as "truth" , "definability" , "denotation" , at least in the context of a formal language. In a slightly narrower sense, by the semantics of a formalized language one means a system of agreements that determine the understanding of the formulas of the language, and that define the conditions for these formulas to be true.

The semantics of logical connectives in classical and intuitionistic logic has an extensional nature: that is, the truth of a complex statement is determined only by the truth character of the expressions that form it. In other classical logics — for example, relevance logics — the meaningful content of concepts can be taken into account (such logics are called intensional). E.g., in logics of this kind not all true expressions are necessarily equivalent.

The construction of a precise semantics for fairly complex formal languages, such as the languages of axiomatic set theory, is a difficult problem. This is essentially connected with the fact that the process of abstraction in mathematics is very complicated and multi-levelled and involves such deep and non-obvious abstractions as the abstraction of actual infinity or the abstraction of potential realizability. As a result, the range of objects to be investigated in mathematics, the methods of treating them, and the methods for proving assertions about them, as a set of an arbitrary nature, becomes very indeterminate. With an imprudent treatment of the principles of proof within the limits of a theory there arise contradictions (cf. Antinomy); for example, Russell's paradox in set theory. In this situation it is necessary to abstain from the construction of exhaustive and intuitively convincing semantics of a language, and to restrict oneself to the formulation of semantic agreements. In formalizing a theory, one strives for the derivation rules of a calculus to be sound with respect to these agreements; that is, when applied to true formulas they again give true formulas. The resulting formal system can be studied within the framework of a certain meta-theory with a clearer semantics.

Often the semantic concepts for a certain language can be formulated exactly within the framework of a richer language, which plays the role of a meta-language for the first. For example, by means of set theory one can give a strict mathematical definition of the (classical) truth of the formulas of a given first-order language for an algebraic system. This concept is fundamental in model theory. On the other hand, as A. Tarski has shown (1936), the truth of a sufficiently rich theory cannot be expressed in the language of the same theory.

The semantics of non-classical theories — for example, mathematical theories developed within the framework of intuitionism — have been widely studied. In such investigations, the role of models is played by algebraic structures that take into account the non-classical nature of the understanding of the logical connectives. Examples are Kripke models, Kleene realizability and A.A. Markov's stepwise semantic system.

References

[1]	R. Carnap, "Meaning and necessity" , Univ. Chicago Press (1947)
[2]	A. Church, "Introduction to mathematical logic" , 1 , Princeton Univ. Press (1956)
[3]	S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)
[4]	A.G. Dragalin, "Mathematical intuitionism. Introduction to proof theory" , Amer. Math. Soc. (1988) (Translated from Russian)
[5]	R. Feys, "Modal logics" , Gauthier-Villars (1965)

Comments

For general semantics (i.e. model theory) of first-order languages see [a1]. For models of set theory see [a2], [a3]. For semantics of intuitionistic logics see [a4].

References

[a1]	C.C. Chang, H.J. Keisler, "Model theory" , North-Holland (1973)
[a2]	T.J. Jech, "Set theory" , Acad. Press (1978) pp. 523ff (Translated from German)
[a3]	K. Kunen, "Set theory" , North-Holland (1980)
[a4]	A.S. Troelstra, D. van Dalen, "Constructivism in mathematics, an introduction" , 1–2 , North-Holland (1988)