Puzzles of Existential Generalisation from Type-theoretic Perspective

TL;DR

This paper examines the Rule of Existential Generalization in simple type theory, clarifying its relation to a modified rule and proposing a natural deduction system suitable for systems with total and partial functions.

Contribution

It distinguishes between (EG) and a modified existential introduction rule, showing (EG) is derivable from the latter, and develops a simpler natural deduction system for such logic.

Findings

(EG) is derivable from a modified existential introduction rule.

A natural deduction system is proposed for systems with total and partial functions.

The system handles higher-order quantification and (hyper)intensional contexts.

Abstract

The present paper addresses several puzzles related to the Rule of Existential Generalization, (EG). In solution to these puzzles from the viewpoint of simple type theory, I distinguish (EG) from a modified Rule of Existential Quantifier Introduction which is derivable from (EG). Both these rules are often confused and both are considered as primitive but I show that (EG) itself is derivable from the proper Rule of Existential Quantifier Introduction. Moreover, the latter rule must be primitive in logical systems that treat both total and partial functions, for the universal and the existential quantifiers are not interdefinable in them. An appropriate natural deduction for such a system is deployed. The present logical system is simpler than the system recently proposed and applied by the present author. It utilises an adequate definition of substitution which is capable of handling not only a higher-order quantification, but also (hyper)intensional contexts.