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

Abstract: In A Study of Spinoza's Ethics (1984, §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-level confirmation against the Della-Rocca PSR-substance reconstruction; stronger PSR variants remain open as future mechanical projects.

• The paper establishes that PSR augmentation partially reduces Spinoza’s Proposition V by demonstrating that only full attribute-sharing guarantees substance identity.
• Kernel-level counter-models in Lean 4 expose irreducibility in the derivation of universality clauses, contesting Della Rocca’s reconstructions.
• The formalisation differentiates between equal-strength translations and non-derivable axioms, providing a framework for mechanised philosophical inquiry.

Machine-Checked Counter-Models for PSR-Reducibility in Spinoza’s Ethica Pars I

Context and Motivation

This paper addresses a longstanding debate in Spinoza scholarship regarding Proposition V of Ethica Pars I, which asserts the impossibility of two substances sharing the same attribute. Jonathan Bennett’s skeptical assessment in "A Study of Spinoza’s Ethics" underscores gaps and invalid moves in Spinoza’s argument, suggesting that the only demonstrable conclusion is that distinct substances cannot share all attributes, not merely any attribute. Contrastingly, Michael Della Rocca and Don Garrett have advanced reconstructions, with Della Rocca, in particular, arguing that the Principle of Sufficient Reason (PSR) as operatively implicit in Spinoza’s framework suffices to derive Proposition V.

The dispute has persisted at the level of prose argumentation for decades, lacking precise adjudication through formal methods. This paper supplies the first machine-checked evidence in the debate, formalizing Ethica Pars I in Lean 4 to fix axiomatic commitments and enable mechanised tests of derivability—particularly focusing on whether substantive PSR commitments can reduce Proposition V (and the related universality clause of Proposition XIV) to base axioms.

Formalisation in Lean 4

The authors rigorously encode Spinoza’s ontology into Lean 4, leveraging its dependent type theory and typeclass abstractions. The universe of discourse is an abstract Thing : Type u, and primitive predicates match Spinoza's ontological (inItself, inAnother, limitedBy), conceptual, existence, modal, and ethical distinctions. Definitions for Substance, Attribute, Mode, and God are rendered as higher-order types conformant with Spinozistic interpretations, with particular attention to translation issues (e.g., "same nature" as "shared attribute", identifying vs. disjunctive readings of sive).

Crucially, the formalisation distinguishes three classes of axiomatic resources:

• Section I-II: Spinoza’s explicit axioms (A1–A7) plus auxiliary bridges and definitional connections required for demonstration consistency.
• Section III: Substantive metaphysical commitments, not provable from base axioms alone and typically flagged as contested by Bennett. These include:
  • A12: Two substances sharing an attribute are identical (Proposition V).
  • A15: Every realised substance attribute is also a God’s attribute (universality clause of Proposition XIV).
  • A13: Substance involves existence (Proposition VII).
  • A14: Substance has at least one attribute.

PSR-augmentations by Della Rocca are encoded as additional Section III axioms (e.g., PSRSubstance as A22: distinct substances must differ in at least one attribute).

Methodology: Demote Experiments and Counter-Models

The demote experiment schema tests whether Section III axioms are reducible to weaker, PSR-flavoured commitment families (Σ). Outcomes are classified as full reduction, equal-strength translation, partial reduction, decomposition-only, or irreducibility.

Kernel-level counter-models are constructed—finite universes with explicit typeclass instances and predicate graphs—to rigorously establish non-derivability by providing concrete witnesses falsifying the target axiom while satisfying the weaker PSR commitments.

Main Results

A12 (Proposition V): Partial Reduction and Kernel-Level Irreducibility

• Partial Reduction:

PSRSubstance (A22) yields that substances sharing all attributes are identical. Mechanically, the Lean 4 script proves that under the all-shared-attribute hypothesis, Spinoza’s proposition holds. This mirrors Bennett’s assessment that the demonstration can only recover the all-attributes case.

• Full Irreducibility:

The full content (any shared attribute ⇒ identity) cannot be derived from PSRSubstance and base axioms. A four-element counter-model satisfies all axioms but falsifies A12, confirming at kernel level (type checker) Bennett's skepticism against the Della Rocca PSR reconstruction.

A15 (Proposition XIV): Decomposition and Non-derivability

• Decomposition Only:

Universality (all God’s attributes encompass all substance attributes) cannot be derived from plenitude alone. The demote requires both plenitude (every realised attribute belongs to some God) and uniqueness (all Gods are identical), the latter itself a Section III substantive claim and not strictly weaker than A15.

• Counter-Model:

With plenitude but without uniqueness, a three-element model is constructed where universality fails, confirming irreducibility.

Existence Clauses (A13, A14): Equal-Strength Translations

• For clauses asserting the existence of substance or attributes, PSR-flavoured modalities (self-causation, essence-perception) produce equal-strength reformulations, amounting to restatements rather than reductions.

Structural Typology and Implications

The paper systematises the four outcome patterns for reductive attempts:

• Universality clauses (A12, A15): Resist reduction to existence-explanatory PSR, requiring either partial reduction (A12) or decomposition with additional universality commitments (A15).
• Existence clauses (A13, A14): Translate trivially under PSR-flavoured axioms.

This structural typology links the shape of quantifier-prefixes (universal vs. existential) to demote-tractability and is hypothesized to generalise beyond Spinoza’s axiomatic system, potentially applicable to rationalist systems seeking universality from existence-explanatory principles.

For practical philosophical projects, the machine-checked counter-model methodology enables precise, decidable adjudication of derivability, moving interpretive debates out of prose and into formal discipline. However, the results remain bounded—alternative augmentations (thoroughgoing PSR, Garrett’s strong-Definition-III), as well as more Spinoza-faithful counter-models, remain open targets for future mechanised experiments.

Limits and Open Directions

• Bounded Claims:

The irreducibility results are established only against specific PSR augmentations; thoroughly stated and stronger PSR variants are not refuted.

• Philosophical Fidelity:

Some model construction choices (uniformly applying expressesEternalEssence, collapsing the three-category ontology) are deliberate approximations, not affecting the meta-logical claim but warranting refinement in further work.

• Extensibility:

Formalisation is currently limited to Pars I; extending to Pars II (mind-body parallelism) and Pars III (conatus doctrine) will test the reducibility typology more broadly.

Conclusion

Machine-checked formalisation in Lean 4 makes explicit the structural limits of PSR-driven reductions within Spinoza's Ethica Pars I. Existence clauses translate directly, but universality clauses resist reduction—requiring either irreducible commitments or distribution across multiple axioms. Kernel-level counter-models convert a longstanding interpretive debate into a discipline accessible for further mechanical exploration. The consequences are methodological: interpretive disagreements about Spinoza's metaphysics can now be subjected to formal, even transparent, adjudication, catalysing new research programs in the mechanisation of philosophical reasoning.

For further experiments, code and formalisation resources are available at (2605.02331).