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CONCEPTUAL DISTANCE AND ALGEBRAS OF CONCEPTS

Part of: General logic Algebraic logic Philosophical aspects of logic and foundations Model theory

Published online by Cambridge University Press:  22 February 2024

MOHAMED KHALED  and
GERGELY SZÉKELY
MOHAMED KHALED
Affiliation:
SCHOOL OF ENGINEERING AND NATURAL SCIENCES ISTANBUL MEDIPOL UNIVERSITY ISTANBUL, TURKEY and SCHOOL OF MATHEMATICS AND COMPUTATIONAL SCIENCES UNIVERSITY OF PRINCE EDWARD ISLAND CHARLOTTETOWN, PE CANADA E-mail: rutmohamed@yahoo.com
GERGELY SZÉKELY*
Affiliation:
HUN-REN ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS BUDAPEST, HUNGARY and UNIVERSITY OF PUBLIC SERVICE BUDAPEST, HUNGARY
*
E-mail: szekely.gergely@renyi.hu
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Abstract

We show that the conceptual distance between any two theories of first-order logic is the same as the generator distance between their Lindenbaum–Tarski algebras of concepts. As a consequence of this, we show that, for any two arbitrary mathematical structures, the generator distance between their meaning algebras (also known as cylindric set algebras) is the same as the conceptual distance between their first-order logic theories. As applications, we give a complete description for the distances between meaning algebras corresponding to structures having at most three elements and show that this small network represents all the possible conceptual distances between complete theories. As a corollary of this, we will see that there are only two non-trivial structures definable on three-element sets up to conceptual equivalence (i.e., up to elementary plus definitional equivalence).

Keywords

distance between theoriesalgebras of conceptsequivalence of structuresfirst-order logicdefinitional equivalence

MSC classification

Primary: 03G15: Cylindric and polyadic algebras; relation algebras 03B10: Classical first-order logic
Secondary: 03C40: Interpolation, preservation, definability 03A10: Logic in the philosophy of science
Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 16
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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