
TL;DR
This paper explores how category theory provides a mathematical framework for understanding the relationship between syntax and semantics in formal theories, offering new philosophical insights into the nature of meaning.
Contribution
It introduces a categorical perspective on the syntax-semantics dualism, linking it to algebra-geometry duality and analyzing its philosophical implications.
Findings
Syntax-semantics dualism corresponds to algebra-geometry duality.
Adjoint functors characterize the interaction between syntax and semantics.
The approach offers a new criterion to distinguish formal from empirical theories.
Abstract
One of the greatest problems in philosophy is that of meaning. The turning point in thinking on meaning was Tarski's definition of truth, and the rapid development of logical semantics and model theory was a consequence of this achievement. Perhaps less well-known among classical logicians and philosophers is that it is category theory that provides adequate mathematical tools to study the relationship between the syntax of formalized theories and their semantics. The aim of this article is to change this situation and make a preliminary philosophical analysis of the results obtained so far. They concern formalized algebraic theories with axioms in the form of equational laws, theories based on propositional logic and coherent Boolean logic, as well as decidable logic which is not necessarily Boolean. The syntactic-semantics relation for these theories takes the form of dualisms between…
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Taxonomy
TopicsLogic, Reasoning, and Knowledge · Advanced Algebra and Logic · Logic, programming, and type systems
