Extensionalism without Logicism: Ambrose and Extensional Logic

TL;DR

This paper explores Alice Ambrose's finitist extensionalism, emphasizing her approach to logic that avoids metaphysical commitments and employs only finite, extensional methods to interpret existential claims.

Contribution

It highlights Ambrose's role as a transitional figure between Russell's formalism and Brouwer's intuitionism, emphasizing her finitist, extensional reformulation of logic.

Findings

Ambrose's approach preserves extensional logic without metaphysical commitments.

She reformulates existential claims as finite disjunctions with explicit stopping rules.

Her methods demonstrate a finitist perspective in foundational logic.

Abstract

Drawing primarily on her early work (1931-1934), I argue that Alice Ambrose develops a philosophical project centered on preserving the rigor of extensional logic while rejecting the metaphysical and epistemological endorsements of logicism because of its commitment to the notion of material infinity. Positioning Ambrose as a transitional figure between formalism (Russell) and the constructivist turn represented by intuitionism (Brouwer), I demonstrate how Ambrose offers a practice oriented statement of finitist extensionalism. Employing only extensional methods (considering classes, relations, and propositions by reference to their members and truth values instead of mental processes), Ambrose reformulates an existential claim about pi as an explicit infinite disjunction of concrete instances insisting, against intensional projects, that such claims gain meaning only through a finite stopping rule that produces a witness.