Two Treatments of Definite Descriptions in Intuitionist Negative Free Logic

TL;DR

This paper compares two formal approaches to representing definite descriptions in intuitionist negative free logic, showing their equivalence and discussing advantages of the binary quantifier method over traditional term-forming operators.

Contribution

It introduces and analyzes a binary quantifier approach for definite descriptions in intuitionist negative free logic, demonstrating its equivalence to other formal systems and highlighting its advantages.

Findings

The binary quantifier approach is equivalent to systems with identity restrictions.

The approach is also equivalent to an intuitionist version of Lambert's system with predicate abstraction.

Advantages of the binary quantifier method are discussed in comparison to alternatives.

Abstract

Sentences containing definite descriptions, expressions of the form `The $F$', can be formalised using a binary quantifier $\iota$ that forms a formula out of two predicates, where $\iota x[F, G]$ is read as `The $F$ is $G$'. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system $\mathbf{INF}^\iota$ of intuitionist negative free logic extended by such a quantifier, which was presented in \citep{kurbisiotaI}, $\mathbf{INF}^\iota$ is first compared to a system of Tennant's and an axiomatic treatment of a term forming $\iota$ operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of $\mathbf{INF}^\iota$ in which the $G$ of $\iota x[F, G]$ is restricted to identity. $\mathbf{INF}^\iota$ is then compared to an intuitionist version of a system of Lambert's which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion.