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.
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 this 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:
- 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 or appropriately in its predicate, and to be prior to the conclusion means the genus in the premise is not somehow determined by the subject of the conclusion that belongs to it.
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, self-evident or what is asserted to be common sense (by definition antecedent to philosophical distinctions), 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 somewhere in the validly co-ordinated and soundly applied series of premises.
As an example, a conditional proof or demonstration would fail its "better-known" soundness criterion if one assumed as a condition "eating nuts make me allergic" and yet conclude in a simple way, "Walnuts are nuts with hard shells; I can eat walnuts and not get allergic." The would-be conclusion would need to attempt to redetermine the thought-to-be "better-known" genus of "nuts" in the condition "eating nuts make me allergic" with the implied, qualified subject of the dubious conclusion, "soft-shelled nuts" or else itself be corrected.
So that, until that happens, the conditional premise, whatever use it eventually has, containing the genus "nuts" would no longer be better-known than the conclusion due to that conditional premise having (implied) notions about the species "hard-shelled nuts" that contradict information, or perhaps don't allow for an exception, alleged in the conclusion.
A mathematical proof is a step-by-step demonstration of the truth of a mathematical theorem. Proofs build on axioms, which are statements that are assumed to be true without proof, as well as previously-proved theorems.
Main article: Scientific evidence
Unlike the theorems of mathematics, science does not seek to prove that its theories are true. Instead, the scientific method seeks to check whether the predictions implied by a theory are observed in nature. Therefore, as philosopher of science Karl Popper argued, science can only hope to show that a theory is false. But scientists recognize that science can never prove that a theory is true in the same sense that a mathematical theorem is true. Therefore, scientists never claim that their theories are facts. Instead, science searches for theories that are not disproved by currently-known experimental observations. Insofar as theories are consistent with nature, they may serve as a guide to improve technology for example and can be considered as true in laymen's terms.
A notable exception may be found in the field of biology, where educators and other proponents frequently contend that, "Evolution is a fact."
- John L. Synge:
"...when one examines some proofs in the Neo-cartesian spirit, too often they seem to dissolve completely away, leaving one in a state of wonder as to whether the author really thought he had proved something. Or is the reader stupid? It is hard to say."
In American courts, crimes are proved "beyond reasonable doubt" to a jury, based on the jury's own analysis of the admissible evidence. Other legal issues may be decided by clear and convincing evidence or by a preponderance of the evidence (more likely than not).
- Marco M. Capria, Aubert Daigneaut et al. (2005). "5. General Relativity: Gravitation as Geometry and the Machian Programme", Physics Before and After Einstein. IOS Press, 97, 114. ISBN 1-58603-462-6. “John L. Synge, who was the author of one of the classic reference books on relativity , wrote half a century after Einstein’s first formulation of general relativity: [...] when one examines some proofs in the Neo-cartesian spirit, too often they seem to dissolve completely away, leaving one in a state of wonder as to whether the author really thought he had proved something. Or is the reader stupid? It is hard to say. In any case I am still waiting for a rational treatment of the dynamics of the solar system according to Einstein’s theory [100, p. 14].”