The concept is applied in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition. In any area of mathematics defined by its assumptions or axioms, a proof is an argument establishing a theorem of that area via accepted rules of inference starting from those axioms and other previously established theorems. The subject of logic, in particular proof theory, formalizes and studies the notion of formal proof. In the areas of epistemology and theology, the notion of justification plays approximately the role of proof, while in jurisprudence the corresponding term is evidence, with burden of proof as a concept common to both philosophy and law.
In most disciplines, evidence is required to prove something. Evidence is drawn from experience of the world around us, with science obtaining its evidence from nature, law obtaining its evidence from witnesses and forensic investigation, and so on. A notable exception is mathematics, whose proofs are drawn from a mathematical world begun with axioms and further developed and enriched by theorems proved earlier.
Exactly what evidence is sufficient to prove something is also strongly area-dependent, usually with no absolute threshold of sufficiency at which evidence becomes proof. In law, the same evidence that may convince one jury may not persuade another. Formal proof provides the main exception, where the criteria for proofhood are ironclad and it is impermissible to defend any step in the reasoning as "obvious"; for a well-formed formula to qualify as part of a formal proof, it must be the result of applying a rule of the deductive apparatus of some formal system to the previous well-formed formulae in the proof sequence.
Proofs have been presented since antiquity. Aristotle used the observation that patterns of nature never display the machine-like uniformity of determinism as proof that chance is an inherent part of nature. On the other hand, Thomas Aquinas used the observation of the existence of rich patterns in nature as proof that nature is not ruled by chance.
Proofs need not be verbal. Before Galileo, people took the apparent motion of the Sun across the sky as proof that the Sun went round the Earth. Suitably incriminating evidence left at the scene of a crime may serve as proof of the identity of the perpetrator. Conversely, a verbal entity need not assert a proposition to constitute a proof of that proposition. For example, a signature constitutes direct proof of authorship; less directly, handwriting analysis may be submitted as proof of authorship of a document. Privileged information in a document can serve as proof that the document's author had access to that information; such access might in turn establish the location of the author at certain time, which might then provide the author with an alibi.
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- Mathematical proof
- Proof theory
- Proof of concept
- Provability logic
- Evidence, information which tends to determine or demonstrate the truth of a proposition
- Proof procedure
- Proof complexity
- Standard of proof
- Proof and other dilemmas: mathematics and philosophy by Bonnie Gold, Roger A. Simons 2008 ISBN 0883855674 pages 12–20
- Philosophical Papers, Volume 2 by Imre Lakatos, John Worrall, Gregory Currie, ISBN Philosophical Papers, Volume 2 by Imre Lakatos, John Worrall, Gregory Currie 1980 ISBN 0521280303 pages 60–63
- Evidence, proof, and facts: a book of sources by Peter Murphy 2003 ISBN 0199261954 pages 1–2
- Logic in Theology – And Other Essays by Isaac Taylor 2010 ISBN 1445530139 pages 5–15
- John Langshaw Austin: How to Do Things With Words. Cambridge (Mass.) 1962 – Paperback: Harvard University Press, 2nd edition, 2005, ISBN 0-674-41152-8.
- Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic Press, 2001. Page 3.
- Alfred Tarski, Introduction to Logic and to the Methodology of the Deductive Sciences (ed. Jan Tarski). 4th Edition. Oxford Logic Guides, No. 24. New York and Oxford: Oxford University Press, 1994, xxiv + 229 pp. ISBN 0-19-504472-X
- Reference Manual on Scientific Evidence, 2nd Ed. (2000), p. 71. Accessed May 13, 2007.
- John Henry Wigmore, A Treatise on the System of Evidence in Trials at Common Law, 2nd ed., Little, Brown, and Co., Boston, 1915
- Simon, Rita James, and Mahan, Linda. (1971). "Quantifying Burdens of Proof—A View from the Bench, the Jury, and the Classroom". Law and Society Review 5 (3): 319–330. doi:10.2307/3052837. JSTOR 3052837.
- Katie Evans, David Osthus, Ryan G. Spurrier. "Distributions of Interest for Quantifying Reasonable Doubt and Their Applications" (PDF). Retrieved 2007-01-14.
- The Principle of Sufficient Reason: A Reassessment by Alexander R. Pruss
- A. S. Troelstra, H. Schwichtenberg (1996). Basic Proof Theory. In series Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, ISBN 0-521-77911-1.
- Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971
- Aristotle's Physics: a Guided Study, Joe Sachs, 1995 ISBN 0813521920 p. 70
- The treatise on the divine nature: Summa theologiae I, 1–13, by Saint Thomas Aquinas, Brian J. Shanley, 2006 ISBN 0872208052 p. 198
- Thomas S. Kuhn, The Copernican Revolution, pp. 5–20
- Trial tactics by Stephen A. Saltzburg, 2007 ISBN 159031767X page 47