Abstract

This paper is divided into three parts. In the first part, we review the historical background of a system of logic devised by Henry S. Leonard to allow for reasoning using existence as a predicate. In the second part, we consider various directions in which his logic could be further developed, syntactically, semantically, and as an adjunct to quantifier elimination and set theory. In the third and final part, we develop proofs of some underlying results of his logic, using modern notation but retaining his axioms and rules of inference.

Highlights

  • Mathematicians frequently use names without knowing whether a corresponding ostensible object exists

  • We confine ourselves here to those further developments that can be described within first-order model theory

  • We describe a first-order model theory for a logic of existence

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Summary

Introduction

Mathematicians frequently use names without knowing whether a corresponding ostensible object exists. Perhaps the most fundamental way in which model theory for first-order logic with an existence predicate differs from conventional model theory is that this assignment function is partial for argument constants. We wish to discuss one of these authors in detail as an example of the adaptation of their proofs, using quantifier elimination, in a way compatible with our first-order logic of existence. He specifies constants that must be chosen to refer to elements of the model. We examine the adaptations of Presburger’s proof that are required by the logic of the existence predicate, due to Leonard, that we are here proposing. We refer to the axioms for ZF given in Jech (Jech, 2006)

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