Lean verification registry · Authored proofs · The Logos Architecture
Written formal sequents verified
Axiom-free: true · 2 checked theorem statements.
Lean establishes the listed formal statements and dependencies. The exact authored source, Ground, known premise meanings, application validation and empirical evidence remain separately typed. No unencoded section is certified by a neighboring theorem.
Original source author: Maclain Hunter. Machine formalization: Replit Agent. No original wording was rewritten.
Source registered: 2026-10-02T21:22:35.838Z (not original authorship).
Formalized: 2026-10-03T10:37:05.082Z.
Checked: 2026-10-03T10:40:45.029Z.
Independent timestamp: 2026-10-03T10:42:55.000Z.
Authored source SHA-256: 537ab5eeea3e01df462a1c0697f8767ad89195410f7adb98bef180d67a9289c9
Lean source · Verification JSON · Exact archived source JSON · Compiler report · Current authored source page · Signed release and verification instructions
Lean (version 4.7.0, commit v4.7.0, Release) exitCode=0 'HunterAuthored.inference_1' does not depend on any axioms 'HunterAuthored.source_identity_rule' does not depend on any axioms
{P, P & Q, Q -> R, P <-> S} |= (S -> (P & Q)) -> R
Exact source sequent under finite two-valued propositional semantics. Source proposition meanings and Logic/Logos/God Ground are preserved separately; this checks the inference, not unencoded prose.
Law of Identity
The explicitly named Identity rule, over every type and element. The remaining prose is not replaced by this witness.
On Spinoza’s Axioms by Maclain Hunter
Premises:
P: Axioms 1, 2, 3, 4, and 6 are valid and possible to prove through reductio ad absurdum as the assumption of the negation of these axioms leads to incoherence and violation of the Law of Non-Contradiction. I shall demonstrate this axiom by axiom for each:
Axiom 1: Everything which exists, exists either in itself or in something else. This distinguishes substances (self-existent) from modes (dependent) but in any case a thing cannot exist outside of itself and anything/all else. It must be true that things exist either/both in themselves and/or in something else.
Axiom 2: That which cannot be conceived through anything else must be conceived through itself. This supports the self-sufficiency of substance and the contingency of what is not self-sufficient. To negate this is absurd as it means that something can come from nothing and/or that something (ie - Logic) that can’t logically come from an external source is also not possibly derived from itself (as Logic necessarily is).
Axiom 3: From a given definite cause an effect necessarily follows; and, on the other hand, if no definite cause be granted, it is impossible that an effect can follow. This establishes strict causality. To negate cause (universal and finite) is to negate effect, as an effect is by definition what follows a cause, thus the negation of the axiom leads to absurdity.
Axiom 4: The knowledge of an effect depends on and involves the knowledge of a cause. This ties epistemology to causation and its negation is absurd for reasons similar to the negation of axiom 3, since an effect implies a cause and loses meaning if cause is negated, ignored, or not required.
Axiom 6: A true idea must agree with its object. This grounds truth in correspondence and its negation is absurd as it opens the door to “Truth’s that aren’t aligned with themselves and the concept(s) in question” which violates the Law of Non-Contradiction and the Law of the Excluded Middle which both demand coherent Identity in any Logical Truth (or Logic).
Q: Spinoza’s fifth and seventh axioms are rather absurd when analyzed, as they assume impossible things to be possible, but understanding this we can salvage them (in a similar sense that if-then reasoning has an “if not then not” that is not absurd even when the proposition itself is for some reason absurd). I will demonstrate this as I did with the valid premises above:
Axiom 5: Things which have nothing in common cannot be one the cause of the other. This limits causal relations in a nonsensical way as even the identity of two things being what they are constitutes having “logical identity” in common. There is universal interconnection that follows from Logic (or the Logos) and the Law of Identity that makes A be A and all equal all (A = A when A = all) so there is no feasible way for things to have “nothing in common”. However IF there were two things that had nothing in common THEN it is true that these things could not be the cause or God of one another in any logically valid manner, so the axiom is not completely false, even while it considers an absurd hypothetical.
Axiom 7: If a thing can be conceived as not existing, its essence does not involve existence. This is meant to distinguish contingent from necessary existence but it fails in the sense that there is no way for something that is possible to conceive to also “not exist in any way” as the very existence of the conception as a conception is a form of existence and the denial of this violates the Law of Non-Contradiction and the Law of the Excluded Middle - since it cannot be that something is truly a concept and not truly a concept at once, and that a concept can be reasoned with while it also doesn’t exist. Even nonexistence is an existing concept that can be conceived of and reasoned with - which paradoxically helps keep nothing nonexistent since even rational concepts of nothingness are something still (and not actually nonexistent as “nothing” must be). However IF a thing did not exist THEN it is true that it’s Essence and existence could not even be conceived of - to avoid the contradiction of “not existing” while existing as a mental reality or conceived concept.
R: This shows how Spinoza’s Axioms are not wrong and can be proven in several cases but are also absurd and only valid in a hypothetical sense in others. The demonstration/clarification of this can allow us to reinforce and better understand Spinoza’s original work - moving us further towards a perfect understanding of God as Reason and of all as one. All exists in a logically grounded pan-theism (where Logic and all logical concepts are God) that doesn’t involve logical jumps or unproven axioms and premises.
S: Assume the negation of the premises P, Q, R. This leads to logical contradiction where axioms that are possible to prove through Logic and axioms that are possible to demonstrate as absurd through Logic are accepted or denied arbitrarily, which Logic (as a system of universal rationality and non-arbitrariness) cannot ignore or deem valid.
Formal Reasoning:
{P, P & Q, Q -> R, P <-> S} |= (S -> (P & Q)) -> R
Conclusion: {P, Q, R, S}:
Spinoza’s proof for God and his axioms are quite solid. This proof demonstrates that each axiom is true, even if it is true under absurdist conditions and in a scenario that can’t occur. This helps us understand God as Logic (better) and to better apply Logic to these matters dispassionately. It also allows us to be sure of Spinoza’s proof since the axioms are not false (upon which the premises rest).
Truth Table Analysis:
PQRS⎪P P&Q Q→R P↔S╞(S→(P&Q))→R
TTTT⎪T TTT TTT TTT TT TTT TT
TTTF⎪T TTT TTT TFF FT TTT TT
TTFT⎪T TTT TFF TTT TT TTT FF
TTFF⎪T TTT TFF TFF FT TTT FF
TFTT⎪T TFF FTT TTT TF TFF TT
TFTF⎪T TFF FTT TFF FT TFF TT
TFFT⎪T TFF FTF TTT TF TFF TF
TFFF⎪T TFF FTF TFF FT TFF FF
FTTT⎪F FFT TTT FFT TF FFT TT
FTTF⎪F FFT TTT FTF FT FFT TT
FTFT⎪F FFT TFF FFT TF FFT TF
FTFF⎪F FFT TFF FTF FT FFT FF
FFTT⎪F FFF FTT FFT TF FFF TT
FFTF⎪F FFF FTT FTF FT FFF TT
FFFT⎪F FFF FTF FFT TF FFF TF
FFFF⎪F FFF FTF FTF FT FFF FF
(Valid)
Q.E.D.