How universal generalization works according to natural reason

Kyle Hodge

doi:10.32995/cogency.v13i2.366

Abstract

Universal Generalization, if it is not the most poorly understood inference rule in natural deduction, then it is the least well explained or justified. The inference rule is, prima facie, quite ambitious: on the basis of a fact established of one thing, I may infer that the fact holds of every thing in the class to which the one belongs—a class which may contain indefinitely many things. How can such an inference be made with any confidence as to its validity or ability to preserve truth from premise to conclusion? My goal in this paper is to explain how Universal Generalization works in a way that makes sense of its ability to preserve truth. In doing so, I shall review common accounts of Universal Generalization and explain why they are inadequate or are explanatorily unsatisfying. Happily, my account makes no ontological or epistemological presumptions and therefore should be compatible with whichever ontological or epistemological schemes the reader prefers.

Highlights

I note that the conclusion of this demonstration on the particular—the singular proposition that c in S has F—is itself the premise from which the universal proposition will be validly inferred on the basis of the fact that c is a generalized particular
Universalizing a Generalized Particular: Validly Inferring a Universal Proposition Once we have demonstrated of c that it has F in the manner described above, c is such that it cannot differ from any other particular in S with respect to the properties involved in the demonstration because all differentiating properties were omitted
Since Euclid proves of the generalized particular triangle ABC that the sum of its internal angles are equal to the sum of two right angles, he may validly conclude the universal proposition that this fact holds of all triangles

Summary

Introduction

The extant accounts of Universal Generalization which aim to render the rule intuitive to students of logic are all correct, but they do not do the explanatory work that they should do and which they purport to do. 2.1—“It Could Have Been Any x” and the Appeal to Arbitrariety Perhaps the most common way of explaining why Universal Generalization is valid is because it is evident that the inference from “any x” to “every x” is valid (since they are logically equivalent). This approach is common in logic textbooks when it comes time to explain Universal Generalization independently of formal rules.4 There are usually two ways in which a particular is said to be arbitrary: 1.

Results

Conclusion