CONCEPTUAL DISTANCE and ALGEBRAS of CONCEPTS


KHALIFA M. K. M. I., Székely G.

Review of Symbolic Logic, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Publication Date: 2024
  • Doi Number: 10.1017/s1755020324000029
  • Journal Name: Review of Symbolic Logic
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Arts and Humanities Citation Index (AHCI), Scopus, Academic Search Premier, MathSciNet, Philosopher's Index, zbMATH
  • Keywords: algebras of concepts, definitional equivalence., distance between theories, equivalence of structures, firstorder logic
  • Istanbul Medipol University Affiliated: Yes

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 3 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).