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→Logical proof: new section
'''Proof''' is a firmly attested and evident objective fact, or a coherent set of facts, which cannot be refuted, often an inescapable conclusion based on undeniable [[evidence]]. Proofs have been set aside by [[logical fallacies]] and [[prejudice]].
'''Proof''' is a firmly attested and evident objective fact, or a coherent set of facts, which cannot be refuted, often an inescapable conclusion based on undeniable [[evidence]]. Proofs have been set aside by [[logical fallacies]] and [[prejudice]].
==Logical proof==
The founder or earliest-known expositor of the field of logic, [[Aristotle]], set down some requirements he observed for rigorous proofs (in addition to the ordinary rules of deduction used for less rigorous proofs by means of the well-known [[syllogism]]).
The form of thinking productive of knowledge that we call "deduction", Aristotle called "analytics" (from leading away, loosening out—subtracting a middle term in the argument, and arranging premises so this is possible by taking into account the preservation of the truth-bearing chain of a series of syllogisms), and in his book ''Posterior Analytics'' written in the mid-fourth century B.C. ("posterior" meaning "after" as in ''after'' insuring the logical patterns by which the different qualities and quantities of propositions are correctly accounted), he exposits upon what makes a sound, and not merely valid, argument. This extends from the bearing the coordination and quality of the premises has upon it down to the conclusion itself.
Aristotle used the subject-predicate type of propositional relationships, the kind usually learned about today beginning in elementary school which may differ from more modern attempts at organized logic.
Aristotle declares the premises of demonstrated knowledge must be:
#True;
#Primary;
#Immediate;
#Better known and prior to the conclusion;
#Related to the conclusion as cause to effect.
1. The truth of the premises is what insures the argument is grounded in ''reality'', however difficult that truth may be to obtain.
2. That the premises are called primary means they are the true premises that are the most ''relevant'' to the conclusion that is sought.
3. That the premises are immediate means they ultimately bear ''directly'' on the conclusion.
4. A premise that is better known than the conclusion has the ''[[genus]]'' (or relevant class to which the subject belongs) of the conclusion's subject for its own subject, and to be prior to the conclusion means the [[genus]] in the premise is ''not somehow determined'' by the subject of the conclusion.
5. Since premises are to be related to their conclusion through cause and effect, they must be as ''necessarily'' true as the desired necessary truth of the conclusion would demand: either ''rigorously proven'' themselves or ''self-evident'', without the truth or its necessity narrowing as it goes down the syllogistic chain to the point where it could no longer apply to the conclusion.
However a state of "conditional necessity" may exist in the premises, so long as the condition is not determined by the subject of the conclusion as per requirement 4, but is rather an "assumption for a conditional proof", an assumption which would appear prior to the validly co-ordinated and soundly applied series of premises.
==Mathematical proof==
==Mathematical proof==
A mathematical proof is a step-by-step demonstration of the truth of a mathematical theorem. Proofs build on [[axiom]]s, which are statements that are assumed to be true without proof, as well as previously-proved [[theorem]]s.
A mathematical proof is a step-by-step demonstration of the truth of a mathematical theorem. Proofs build on [[axiom]]s, which are statements that are assumed to be true without proof, as well as previously-proved [[theorem]]s.