Grounded FOL and FCL · Maclain Hunter

Authored by Maclain Hunter · first-party-authored · authored-foundational-record

Stored 4P source diagnostic: unresolved · finding unverified_concern · admission admitted_provisionally. This intake result is separate from the theorem checks of the exact propositional sequent.

Grounded FOL and FCL by Maclain Hunter Terms: FOL (First Order Logic): Logic formally dealing in quantified existence, predicates/relations, and identity, built on top of propositional Logic. FCL (Fusion Core Logic): a dynamic set of universal symbols and principles (identity, non-contradiction, binary truth values, derivability (⊢) as the carrier of necessity, etc.) — that hold across major formal systems. It is held together by coherent grounding (or Coherence) with roots in propositional and first order Logic. FCL also explicitly allows the self-reference of a constant as its own predicate whereas that is ambiguous in FOL, as Coherence itself is both the Universal Constant (Ground) and its own defining characteristic and predicate, without contradiction. A formal system that won’t allow that “self-predicating constant” is incoherent with Logic as (logically predicated) Logic. Let: P = The Ground (of Logic); the Logos; Logical “God” or Foundation. Note: P is intentionally used below as both a constant and a predicate. How and where P is used in a given formalism determines what role it is playing, the meaning of the term doesn’t actually shift. The Ground of Logic has the property of grounding itself and Logic, so P and “P(P)” both make objective sense. G = Grounded (in Coherence); is grounded; has grounding (in the). L = Logic and logics (or formal systems of Logic); Logos and logoi. C = Objective Coherence, which has no valid and coherent divergence/multiplicity in itself, (is the property of) and is unique/coherent/one. F = First Order Logic (FOL) and Fusion Core Logic (FCL). Proof: L(F) ∧ ∀L (C(L)) ⊢ G(F) ∧ C(P) Ergo: The Logic underlying (and a property of) First Order Logic (FOL)— and the fact that all Logic is (and all systems of Logic are) necessarily coherent — prove the Ground of Logic is coherent and that valid Logic, logics, and FOL are coherent/grounded. Coherentized: axiom1: ∀x (Lg(x) → C(x)) axiom2: Lg(f) axiom3: ∀x (C(x) → G(x)) axiom4: ∀x (G(x) → ∃y (C(y) ∧ Ground(y,x))) axiom5: ∀x∀y ((C(x) ∧ Ground(x) ∧ C(y) ∧ Ground(y)) → x=y) step1: C(f) ⊢ axiom1, axiom2 step2: G(f) ⊢ axiom3, step1 step3: ∃y(C(y) ∧ Ground(y,f)) ⊢ axiom4, step2 identify: y=p ⊢ axiom5, step3, [C(p) ∧ Ground(p)] conclusion: C(p) ∧ G(f) Propositional Argument: Premises: P: “step1” (C(f) ⊢ axiom1, axiom2) from the coherentization, as well as P as P has been defined and used. Q: “step2” (G(f) ⊢ axiom3, step1). R: “step3” (∃y(C(y) ∧ Ground(y,f)) ⊢ axiom4, step2). S: Identification (y=P ⊢ axiom5, step3, [C(P) ∧ P(P)]): The requirement for Coherence (C) and “some coherent ground” (y) is tantamount to a demand for the Ground of Logic (P). T: Conclusion (C(p) ∧ G(f)): First Order Logic is a formal system of Logic grounded in the Ground of Logic, which is ultimately coherent itself. Proof: {P, P → Q, Q → R, R → S, S → T} ⊢ T Truth Table Analysis: TTTTT⎪T TTT TTT TTT TTT T TTTTF⎪T TTT TTT TTT TFF F TTTFT⎪T TTT TTT TFF FTT T TTTFF⎪T TTT TTT TFF FTF F TTFTT⎪T TTT TFF FTT TTT T TTFTF⎪T TTT TFF FTT TFF F TTFFT⎪T TTT TFF FTF FTT T TTFFF⎪T TTT TFF FTF FTF F TFTTT⎪T TFF FTT TTT TTT T TFTTF⎪T TFF FTT TTT TFF F TFTFT⎪T TFF FTT TFF FTT T TFTFF⎪T TFF FTT TFF FTF F TFFTT⎪T TFF FTF FTT TTT T TFFTF⎪T TFF FTF FTT TFF F TFFFT⎪T TFF FTF FTF FTT T TFFFF⎪T TFF FTF FTF FTF F FTTTT⎪F FTT TTT TTT TTT T FTTTF⎪F FTT TTT TTT TFF F FTTFT⎪F FTT TTT TFF FTT T FTTFF⎪F FTT TTT TFF FTF F FTFTT⎪F FTT TFF FTT TTT T FTFTF⎪F FTT TFF FTT TFF F FTFFT⎪F FTT TFF FTF FTT T FTFFF⎪F FTT TFF FTF FTF F FFTTT⎪F FTF FTT TTT TTT T FFTTF⎪F FTF FTT TTT TFF F FFTFT⎪F FTF FTT TFF FTT T FFTFF⎪F FTF FTT TFF FTF F FFFTT⎪F FTF FTF FTT TTT T FFFTF⎪F FTF FTF FTT TFF F FFFFT⎪F FTF FTF FTF FTT T FFFFF⎪F FTF FTF FTF FTF F (Valid) QED

Lean 4 verification and timestamps

The machine formalization date is separate from the original authored work. See the verification record for exact checked scope and any unresolved parts.

Logic-centered continuity

This authored proof is connected to Logic, Logos, God, and the self-evident Ground of Reality, and to the distinct systems 4P Engine, MCB Core, 4P-MCB, and the source-first Canon. Formal proof, empirical evidence, and system-generated records remain separately typed.