×

Formalized token models and duality in semantics: an algebraic approach. (English) Zbl 1069.03031

Introduction: The purpose of this paper is to introduce the concept of a token model and to demonstrate some of its potential. An interpretation is a Birkhoff polarity connecting the domain of a model and the wwfs describing it, and leads to a duality relationship between the elements of a model and the predicates of its theory. Models on one hand and wffs on the other, together with the interpretation connecting these, will all be inductively defined, leading to a formal development of semantics. Token models are simplifying for some proofs in model theory. The examples in this paper include an extension of the existence theorem for atomic models to all powers, together with necessary and sufficient algebraic conditions for categoricity in all language powers, yielding, as simple corollaries, the Morley theorem and its extension to all powers.
Section 2 is a brief survey of Birkhoff polarities. Sections 3 and 4 develop inductively the lattice structures related to polarities for formulas and models respectively. Section 5 relates the two structures by the Birkhoff polarity of an interpretation. Section 6 is the pivotal section and develops the concept of a dual interpretation, which is a dual method of generating models of a given complete theory. Section 7 contains four applications including those mentioned above. I have tried, wherever possible, to adhere to the definitions and notations of C. C. Chang and H. J. Keisler’s book [Model theory. 3rd rev. ed. Amsterdam: North Holland (1990; Zbl 0697.03022)].

MSC:

03C95 Abstract model theory
03G10 Logical aspects of lattices and related structures
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)
08C05 Categories of algebras

Citations:

Zbl 0697.03022
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Foundations of the Theory of Signs (1938)
[2] Introduction to Mathematical Logic (1963)
[3] Lectures on Boolean Algebras (1963) · Zbl 0114.01603
[4] Lattice Theory (1967)
[5] The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory (1951)
[6] Model Theory (1991)
[7] Lattice Theory: First concepts and distributive lattices (1971) · Zbl 0232.06001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.