# Philosophical Algebra: Expanded Proof

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

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

# Philosophical Algebra: Expanded Proof ## By Maclain Hunter ### Symbols, Terms, and Definitions - **P**: Premise - The basis of an argument. - **Q**: Logic - The method and standard by which arguments are validated. - **R**: God - The origin and ruler of existence; the rational standard of moral authority. - **S**: Structure - A construct or arrangement that provides organization according to a plan. - **T**: Coherence - Logical alignment of concepts; unity that results in logical equivalence. - **U**: Incoherence - Lack of logical alignment; divergence or contradiction indicating invalidity. - **Logician**: An expert in the study and use of logic, reasoning formally and consistently. - **Existence**: Reality; the universe; nature; that which exists. - **Evidence**: Data supporting a theory or conclusion in a logically valid manner. - **Proof**: A formal argument synthesizing evidence to prove conclusions with logical certainty. - **Law**: A universal rule with consequences for violations. - **Science**: A method of drawing inferences from empirical observation using experimentation. - **Objective Truth**: That which aligns with reality and enjoys validity through coherence with truth's standards. ### Logical Proof (Formal Reasoning) \[ \{ P \leftrightarrow Q, R \leftrightarrow Q, S \leftrightarrow Q, T \leftrightarrow \neg U \} \models (((P \leftrightarrow Q) \leftrightarrow (R \leftrightarrow Q) \leftrightarrow S) \leftrightarrow T) \leftrightarrow \neg U) \] ### Conclusion Valid premises must cohere with and be logically equivalent to logic, which must also be unified with the origin of existence (God). This structure ensures coherence, contrasting with incoherence. Thus, the conclusion stands true based on definitions and logical structure. ### Coherence in Sciences and Logic To be considered genuine, sciences and logicians must align with logic, demonstrating coherence without contradiction. Reality (existence) logically aligns with logic and God as the self-consistent source of reality. ### Proof and Evidence Proof, evidence, and law must cohere with logic universally. Validity in these areas indicates a coherent relationship with rational law, suggesting that existence necessitates proof and vice versa. ### Truth Table (Validity) | P | Q | R | S | T | U | P↔Q | R↔Q | S↔Q | T↔¬U | Conclusion | |---|---|---|---|---|---|------|------|------|-------|------------| | T | T | T | T | T | F | T | T | T | T | Valid | | T | T | T | T | F | T | T | T | T | F | Invalid | | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | | F | F | F | F | T | T | F | F | F | F | Invalid | ### Summary Through coherent premises, logic, structure, and the negation of incoherence, the argument's validity is established and confirmed. The interconnectedness of these elements underlines the logical structure of existence and the pursuit of objective truth.

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.