Intensionality and Two-steps Interpretations

TL;DR

This paper extends First-order Logic with Bealer's intensional abstraction operator, exploring the relationship between intensional and extensional semantics through algebraic structures and possible worlds.

Contribution

It introduces a novel algebraic framework connecting intensional and extensional semantics in FOL using possible worlds and algebraic diagrams.

Findings

Established a commutative homomorphic diagram in intensional FOL

Connected intensional algebra of concepts to Tarski's extensional interpretation

Differentiated the new extensional algebra from Cylindric algebras

Abstract

In this paper we considered the extension of the First-order Logic (FOL) by Bealer's intensional abstraction operator. 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. Although there is divergence in formulation, it is accepted that the extension of an idea consists of the subjects to which the idea applies, and the intension consists of the attributes implied by the idea. From the Montague's point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In the case of the pure FOL we obtain commutative homomorphic diagram that holds in each given possible world of the 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). Then we show that it corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world.