Bennett's Conjecture in Lean 4: Counter-Models for the PSR-Reducibility of Spinoza's Propositions V and XIV

TL;DR

This paper formalizes and machine-checks Spinoza's propositions V and XIV in Lean 4, providing counter-models that challenge the derivability of these propositions under certain philosophical assumptions.

Contribution

It introduces the first formal, machine-verified analysis of Spinoza's propositions, demonstrating the limits of derivability with specific axioms and providing counter-models.

Findings

Counter-models show propositions cannot be derived under certain axioms.

Formalization in Lean 4 captures Bennett's and Della Rocca's interpretations.

The approach provides the first kernel-checked evidence in this philosophical debate.

Abstract

In A Study of Spinoza's Ethics (1984, {\S}17), Jonathan Bennett argues that the demonstration of Proposition V of Spinoza's Ethica contains identifiable invalid moves and that, even granted those moves, "cannot yield more than the conclusion that two substances could not have all their attributes in common" -- while Spinoza concludes that they cannot share any. Bennett doubts that any valid reconstruction is available from Spinoza's stated resources without importing further commitments. Michael Della Rocca (Spinoza, 2008, ch. 2) responds that the proposition can be derived if the Principle of Sufficient Reason (PSR) is committed substantively. The debate has remained at the level of prose argument for forty years. This paper provides the first machine-checked evidence in the debate. We formalise Ethica Pars I in Lean 4, encoding Bennett's reading of Spinoza's stated axioms as a typeclass and Della Rocca's substantive PSR as an extension class. The derivation attempt yields a partial result -- substances sharing all attributes are identical -- but cannot reach the full "sharing-any-attribute -> identity" content of Proposition V, mechanically tracking Bennett's own all-attributes ceiling. A four-element counter-model satisfying both axiom sets while falsifying Proposition V's content establishes the irreducibility against this specific augmentation. A second counter-model establishes the analogous result for axiom A15, a load-bearing universality clause for Spinoza's Proposition XIV. Bennett's diagnosis receives its first kernel-checked counter-model against the Della-Rocca PSR-substance reconstruction (the non-derivability claim itself a meta-logical consequence of kernel consistency); stronger PSR variants and the broader narrative claim against the full Section I + II + A1--A7 register remain open as future mechanical projects.