The Distinction between Predicate Intension and Extension

I. Introduction

In the lengthy Introduction to the second edition of Principia Mathematica, Whitehead and Russell offer a remark which aptly sets the stage for the considerations of the present paper:

It is an old dispute whether formal logic should concern itself mainly with intensions or extensions. In general, logicians whose training was mainly philosophical have decided for intensions, while those whose training was mainly mathematical have decided for extensions (1).

I propose here to give some indications why the distinction between intension and extension, although of no critical significance for mathematics, not only is useful, but quite important for philosophical purposes. The present discussion will confine itself to a consideration of predicates, and will accordingly be limited to concern with the distinction between predicates in extension (which are tantamount to sets or classes) and predicates in intension (i. e., attributes or properties). My task is thus a twofold one, it being my aim both to show why the attribute versus class distinction is important for philosophical purposes, and to demonstrate how this distinction may be articulated within the framework of modern formal logic.

II. Attributes vs. Classes

Two predicates are said to have the same extension when the class of objects to which one can be ascribed is identical with the