# Q1013 — Logic and Proof

- Q number: Q1013
- Primary URI: https://4p-engine.com/q/Q1013
- Current host route: https://4p-mcb.com/q/Q1013
- Reserved subdomain alias: https://q1013.4p-engine.com/

The first-party authored relation between Logic as lawful order and proof as the performed reasoning that demonstrates a conclusion within that order.

> Logic governs the form and coherence of proof, while proof performs and exposes reasoning within that order; a reflexive proof may make its own valid performance part of what it demonstrates.

Q1013 names the relation between Logic and proof rather than treating them as interchangeable words. Logic is the lawful order against which identity, non-contradiction, relation, and inference are judged. Proof is a performed passage through that order toward a conclusion.
The attached Knowing Logic and Proof text makes this relation explicit. P defines Logic through universal non-contradiction and lawful order. Q identifies the semantics of that definition as knowledge of Logic. R argues that explaining Logic and explicitly identifying that explanation is proof. S then states that a logical proof coherent with Logic cannot be incoherent without contradiction.
The formal structure is stated as {P, P -> Q, Q -> R, R -> S} |= (P & Q) -> (R & S). The semantic premises do not replace the formal entailment; they explain what the performed entailment is being used to demonstrate.
This relation is the bridge between the general concept of Proof (Q1012) and the specific authored demonstration Knowing Logic and Proof (Q1014).

## Two-way derivation from Logic and its use

**Status:** editorial_derivation proof sketch (conditional_concept), distinct from authored source. Logic has its own evidence in reflexive demonstration: any denial, confirmation, observation, or counterexample presupposes identity and valid relation to count as evidence. Logic therefore grounds evidence rather than receiving authority from it. Empirical and documentary evidence can establish contingent instances and test applications, but cannot create, overrule, or exhaust Logic.

### From Logic to this concept

- **Bridge premise:** A valid proof is a performed inference whose conclusion follows from identified premises by rules that preserve identity and non-contradiction.
- **Consequence:** Logic is the governing lawful order; proof is an inspectable performance inside it. The P-Q-R-S relation derives its conclusion only when each stated semantic bridge and formal entailment holds.
- **Proof limit:** The relation distinguishes order from performance; it does not make a proof artifact identical to Logic or validate unstated semantic premises.

### From this concept back to Logic

- **Witness:** A proof trace separates premises, intermediate consequences, and a conclusion, then permits a counterexample or refutation check.
- **Disclosure:** Following that trace requires stable proposition identity and valid relations; the proof's intelligibility therefore presupposes the Logic it performs.
- **Proof limit:** A string that merely resembles a derivation does not reveal Logic; inspect the actual rules, premises, and validity.

### Intelligence procedure

- **Human:** Mark each premise, rule, and conclusion; reconstruct one inference at a time; test whether changing a premise changes the conclusion.
- **Machine:** Parse the proof into typed propositions and inference edges, check each edge against the declared calculus, and report semantic bridges separately.

### Language and vocabulary procedure

- **Contrast:** Distinguish Logic (the governing order) from proof (a particular performed derivation).
- **Usage rule:** Use “proves” only for a conclusion supported by stated premises and valid steps; use “is logical” only when the intended scope is clear.
- **Example:** “The proof applies modus ponens within classical Logic” names both the performed argument and its governing rule.

## Semantic meaning frame

- **Role:** Logic-proof relation architecture
- **Essence:** The authored relation between Logic as lawful order and proof as the performed reasoning that demonstrates a conclusion within that order.
- **Logos relation:** Logos is the governing lawful order expressed as Logic; proof is a coherent performance within it, and a reflexive proof may make that performance part of what it demonstrates.
- **Retrieval aliases:** Logic and Proof · Logic, Logos, and Proof · Logic-proof relation · proof and knowledge of Logic · knowledge of proof
- **Evidence modes:** definition of Logic · formal entailment · premise-to-conclusion mapping · semantic interpretation · refutation condition

### Distinctions

- Logic is not merely a proof artifact
- proof remains answerable to its premises
- semantic self-reference is not unsupported assertion

## Coherence criterion

The Logic-proof relation must distinguish governing order from performed derivation, preserve the formal entailment, state its semantic bridge, and keep the conclusion within the declared scope.

## Evidence boundary

Q1013 is a considered, vetted first-party authored relation. Its formal component can be checked syntactically and truth-functionally; the verifier must inspect directly, and its semantic interpretation follows from the explicit premises and bridge stated by the proof rather than from a detached label.

## Properties

- **logicProofRelation:** Logic is lawful order; proof is the performed derivation within that order; reflexive proof can include its own performance in the semantic conclusion.
- **formalArgument:** {P, P -> Q, Q -> R, R -> S} |= (P & Q) -> (R & S)
- **semanticBridge:** The proof's performed valid reasoning is identified as knowledge of Logic and proof through explicit semantic premises.
- **authorshipBoundary:** The relation is Maclain Hunter's authored formulation; external Logic and proof concepts are comparison context, not silently recast authorship.

## Linked Q items

- [Q1000 — Q-Number Data and Metadata System](https://4p-mcb.com/q/Q1000)
- [Q1001 — Maclain Hunter](https://4p-mcb.com/q/Q1001)
- [Q1002 — Hunterian Philosophy](https://4p-mcb.com/q/Q1002)
- [Q1003 — Coherence](https://4p-mcb.com/q/Q1003)
- [Q1006 — 4P Proof](https://4p-mcb.com/q/Q1006)
- [Q1012 — Proof](https://4p-mcb.com/q/Q1012)
- [Q1014 — Knowing Logic and Proof](https://4p-mcb.com/q/Q1014)
- [Q1015 — Logical Proof of Knowledge of Logical Proof](https://4p-mcb.com/q/Q1015)
- [Q1036 — Logic (Logos)](https://4p-mcb.com/q/Q1036)

## Related Wikidata concept items

These are typed external concept links for discovery and comparison. They are not sameAs assertions, endorsements, or substitutes for the first-party source and evidence.

- [logic (Q8078)](https://www.wikidata.org/wiki/Q8078) — **governing field:** The study of correct reasoning. Boundary: The generic concept supplies comparison context for Logic; it does not author or validate the first-party Logic-proof relation.
- [formal proof (Q2762418)](https://www.wikidata.org/wiki/Q2762418) — **derivation context:** Establishment of a theorem by inference from axioms or premises. Boundary: The formal-proof concept classifies the derivational component; it does not settle the semantic bridge in Q1013.
- [philosophy of logic (Q2476757)](https://www.wikidata.org/wiki/Q2476757) — **scope and nature context:** Study of the scope and nature of logic. Boundary: This supplies a neighboring field for comparison, not external authority over the authored Logic-proof relation.
- [Logic (Q409)](https://www.wikidata.org/wiki/Q409) — **Logic-and-proof relation context:** Curated external comparison concept: Logic. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [First-Order Logic (Q506248)](https://www.wikidata.org/wiki/Q506248) — **Logic-and-proof relation context:** Curated external comparison concept: First-Order Logic. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Tautology (Q237375)](https://www.wikidata.org/wiki/Q237375) — **Logic-and-proof relation context:** Curated external comparison concept: Tautology. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Law of Identity (Q843812)](https://www.wikidata.org/wiki/Q843812) — **Logic-and-proof relation context:** Curated external comparison concept: Law of Identity. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Law of Non-Contradiction (Q1056553)](https://www.wikidata.org/wiki/Q1056553) — **Logic-and-proof relation context:** Curated external comparison concept: Law of Non-Contradiction. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Principle of Sufficient Reason (Q161066)](https://www.wikidata.org/wiki/Q161066) — **Logic-and-proof relation context:** Curated external comparison concept: Principle of Sufficient Reason. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Reductio ad Absurdum (Q188506)](https://www.wikidata.org/wiki/Q188506) — **Logic-and-proof relation context:** Curated external comparison concept: Reductio ad Absurdum. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Axiom (Q17736)](https://www.wikidata.org/wiki/Q17736) — **Logic-and-proof relation context:** Curated external comparison concept: Axiom. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Metalogic (Q938450)](https://www.wikidata.org/wiki/Q938450) — **Logic-and-proof relation context:** Curated external comparison concept: Metalogic. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Mathematical Proof (Q11538)](https://www.wikidata.org/wiki/Q11538) — **Logic-and-proof relation context:** Curated external comparison concept: Mathematical Proof. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Formal System (Q1212450)](https://www.wikidata.org/wiki/Q1212450) — **Logic-and-proof relation context:** Curated external comparison concept: Formal System. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.

## Related wiki and reference pages

These pages provide historical, conceptual, or primary-text context. They are not identity, authorship, endorsement, or sameAs assertions.

- [Logic](https://en.wikipedia.org/wiki/Logic) — **Wikipedia; governing conceptual context:** The study of valid reasoning and the forms of inference. Boundary: The page supplies comparison vocabulary; it does not validate Q1013.
- [Philosophy of logic](https://en.wikipedia.org/wiki/Philosophy_of_logic) — **Wikipedia; scope and nature context:** The philosophical study of logic and its scope. Boundary: This is a comparative field reference, not an external adjudication of the authored relation.
- [Mathematical proof](https://en.wikipedia.org/wiki/Mathematical_proof) — **Wikipedia; formal-performance context:** A deductive argument showing that a proposition follows from premises accepted as true. Boundary: The page provides formal comparison context; the semantic claim remains source-bound to Q1014.
- [First-order logic](https://en.wikipedia.org/wiki/First-order_logic) — **Wikipedia; Logic-and-proof relation context:** Curated external reading context: First-Order Logic. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Tautology (logic)](https://en.wikipedia.org/wiki/Tautology_%28logic%29) — **Wikipedia; Logic-and-proof relation context:** Curated external reading context: Tautology. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Law of identity](https://en.wikipedia.org/wiki/Law_of_identity) — **Wikipedia; Logic-and-proof relation context:** Curated external reading context: Law of Identity. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Law of non-contradiction](https://en.wikipedia.org/wiki/Law_of_non-contradiction) — **Wikipedia; Logic-and-proof relation context:** Curated external reading context: Law of Non-Contradiction. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Principle of sufficient reason](https://en.wikipedia.org/wiki/Principle_of_sufficient_reason) — **Wikipedia; Logic-and-proof relation context:** Curated external reading context: Principle of Sufficient Reason. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Reductio ad absurdum](https://en.wikipedia.org/wiki/Reductio_Ad_Absurdum) — **Wikipedia; Logic-and-proof relation context:** Curated external reading context: Reductio ad Absurdum. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Axiom](https://en.wikipedia.org/wiki/Axiom) — **Wikipedia; Logic-and-proof relation context:** Curated external reading context: Axiom. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Metalogic](https://en.wikipedia.org/wiki/Metalogic) — **Wikipedia; Logic-and-proof relation context:** Curated external reading context: Metalogic. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.
- [Formal system](https://en.wikipedia.org/wiki/Formal_system) — **Wikipedia; Logic-and-proof relation context:** Curated external reading context: Formal System. Boundary: These links contextualize the Logic-and-proof relation; the first-party semantic bridge remains governed by its authored premises and derivation.