First-order Logic: Modality and Intensionality

TL;DR

This paper proposes a minimal intensional semantic enrichment for first-order logic by unifying extensional and modal semantics, avoiding the need for intensional abstraction, and establishing a new algebraic framework consistent with Tarski's semantics.

Contribution

It introduces a minimal intensional enrichment of FOL using PRP theory, unifies extensional and modal semantics, and constructs a new algebraic framework that aligns with Tarski's semantics.

Findings

Not all modal predicate logics are inherently intensional.

A pure extensional modal predicate logic can be derived from modal interpretation of FOL.

The proposed framework avoids the need for intensional abstraction in FOL.

Abstract

Contemporary use of the term 'intension' derives from the traditional logical Frege-Russell's doctrine that an idea (logic formula) has both an extension and an intension. From the Montague's point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In this paper we analyze the minimal intensional semantic enrichment of the syntax of the FOL language, by unification of different views: Tarskian extensional semantics of the FOL, modal interpretation of quantifiers, and a derivation of the Tarskian theory of truth from unified semantic theory based on a single meaning relation. We show that not all modal predicate logics are intensional, and that an equivalent modal Kripke's interpretation of logic quantifiers in FOL results in a particular pure extensional modal predicate logic (as is the standard Tarskian semantics of the FOL). This minimal intensional enrichment is obtained by adopting the theory of properties, relations and propositions (PRP) as the universe or domain of the FOL, composed by particulars and universals (or concepts), with the two-step interpretation of the FOL that eliminates the weak points of the Montague's intensional semantics. Differently from the Bealer's intensional FOL, we show that it is not necessary the introduction of the intensional abstraction in order to obtain the full intensional properties of the FOL. Final result of this paper is represented by the commutative homomorphic diagram that holds in each given possible world of this new intensional FOL, from the free algebra of the FOL syntax, toward its intensional algebra of concepts, and, successively, to the new extensional relational algebra (different from Cylindric algebras), and we show that it corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world.