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Modal Proof of Logos and the Tautological Argument

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Axiom-free: not established · 0 checked theorem statements.

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Original source author: Maclain Hunter. Machine formalization: Replit Agent. No original wording was rewritten.

Source registered: 2026-10-03T01:36:08.067Z (not original authorship).
Formalized: 2026-10-03T10:37:05.082Z.
Checked: 2026-10-03T10:40:11.522Z.
Independent timestamp: 2026-10-03T10:42:55.000Z.

Authored source SHA-256: 7a45b01009b146831ec0924f2e112f44c04e16232f231f0ed5fd887ba325eadf

Lean source · Verification JSON · Exact archived source JSON · Compiler report · Current authored source page · Signed release and verification instructions

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Lean (version 4.7.0, commit v4.7.0, Release)
Not executed: no complete licensed formal sequent recovered.

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Exact authored source in the timestamped release

Modal Proof: 

Definitions:
L = Logos (the fundamental Order of Existence)
T = Tautological Argument
P = Premise/Term of the Tautological Argument
C = Conclusion of the Tautological Argument

Axioms:
1. □(L → ∀P) - It is necessarily the case that the Logos entails all the premises/terms of the Tautological Argument.
2. ◇(∃P) - It is possibly the case that at least one premise/term of the Tautological Argument exists.

Proof:
1. ◇(∃P) (Axiom 2)
2. ∃P (From 1, by the rule of possibility introduction)
3. □(L → ∀P) (Axiom 1)
4. L (From 2 and 3, by the rule of necessitation)
5. ∀P (From 3 and 4, by the rule of universal instantiation)
6. T (From 4 and 5, by the definition of Tautological Argument)
7. □T (From 4 and 6, by the rule of necessitation)

The key steps are:

1. Establishing the possibility of at least one premise/term of the Tautological Argument existing (Axiom 2).
2. Deriving the necessity of the Logos from the existence of a single premise (Steps 3-4).
3. Deducing the necessity of the entire Tautological Argument from the necessity of the Logos (Steps 5-7).

This proof demonstrates that by acknowledging even the possibility of a single premise or term within the Tautological Argument, one is logically committed to the necessity of the Logos and the Tautological Argument as a whole. To deny the Logos or the Tautological Argument at this point would be to engage in self-refutation, a logical contradiction.

The modal logical framework employed here captures the inescapable, tautological nature of the Logos-based argument, as discussed in the previous exchange. This proof cannot be rationally argued with, as it follows directly from the acknowledged premises and the rules of modal logic.