Proof (truth)

Proof (truth)

A proof is <a href="/en/Necessity_and_sufficiency" title="Necessity and sufficiency" class="wl">sufficient</a> <a href="/en/Evidence" title="Evidence" class="wl">evidence</a> or a sufficient <a href="/en/Argument" title="Argument" class="wl">argument</a> for the <a href="/en/Truth" title="Truth" class="wl">truth</a> of a <a href="/en/Proposition" title="Proposition" class="wl">proposition</a>.<a href="#cite_note-1" style="counter-reset: mw-Ref 1;">[1]</a><a href="#cite_note-2" style="counter-reset: mw-Ref 2;">[2]</a><a href="#cite_note-3" style="counter-reset: mw-Ref 3;">[3]</a><a href="#cite_note-4" style="counter-reset: mw-Ref 4;">[4]</a>

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The concept applies in a variety of disciplines,<a href="#cite_note-5" style="counter-reset: mw-Ref 5;">[5]</a>
with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written <a href="/en/Communication" title="Communication" class="wl">communication</a> such as <a href="/en/Conversation" title="Conversation" class="wl">conversation</a>, <a href="/en/Dialogue" title="Dialogue" class="wl">dialog</a>, <a href="/en/Rhetoric" title="Rhetoric" class="wl">rhetoric</a>, etc., a proof is a <a href="/en/Persuasive" title="Persuasive" class="mw-redirect wl">persuasive</a> <a href="/en/Perlocutionary_act" title="Perlocutionary act" class="wl">perlocutionary</a> <a href="/en/Speech_act" title="Speech act" class="wl">speech act</a>, which demonstrates the truth of a proposition.<a href="#cite_note-6" style="counter-reset: mw-Ref 6;">[6]</a> In any area of <a href="/en/Mathematics" title="Mathematics" class="wl">mathematics</a> defined by its assumptions or <a href="/en/Axioms" title="Axioms" class="mw-redirect wl">axioms</a>, a proof is an argument establishing a <a href="/en/Theorem" title="Theorem" class="wl">theorem</a> of that area via accepted <a href="/en/Rules_of_inference" title="Rules of inference" class="mw-redirect wl">rules of inference</a> starting from those axioms and from other previously established theorems.<a href="#cite_note-7" style="counter-reset: mw-Ref 7;">[7]</a> The subject of <a href="/en/Logic" title="Logic" class="wl">logic</a>, in particular <a href="/en/Proof_theory" title="Proof theory" class="wl">proof theory</a>, formalizes and studies the notion of <a href="/en/Formal_proof" title="Formal proof" class="wl">formal proof</a>.<a href="#cite_note-8" style="counter-reset: mw-Ref 8;">[8]</a> In some areas of <a href="/en/Theory_of_justification" title="Theory of justification" class="mw-redirect wl">epistemology</a> and <a href="/en/Existence_of_god" title="Existence of god" class="mw-redirect wl">theology</a>, the notion of <a href="/en/Theory_of_justification" title="Theory of justification" class="mw-redirect wl">justification</a> plays approximately the role of proof,<a href="#cite_note-9" style="counter-reset: mw-Ref 9;">[9]</a> while in <a href="/en/Jurisprudence" title="Jurisprudence" class="wl">jurisprudence</a> the corresponding term is <a href="/en/Evidence_(law)" title="Evidence (law)" class="wl">evidence</a>,<a href="#cite_note-10" style="counter-reset: mw-Ref 10;">[10]</a>
with "burden of proof" as a concept common to both <a href="/en/Philosophic_burden_of_proof" title="Philosophic burden of proof" class="mw-redirect wl">philosophy</a> and <a href="/en/Legal_burden_of_proof" title="Legal burden of proof" class="mw-redirect wl">law</a>.

In most disciplines, evidence is required to prove something. Evidence is drawn from the experience of the world around us, with <a href="/en/Science" title="Science" class="wl">science</a> obtaining its evidence from <a href="/en/Nature_(science)" title="Nature (science)" class="mw-redirect wl">nature</a>,<a href="#cite_note-11" style="counter-reset: mw-Ref 11;">[11]</a> law obtaining its evidence from <a href="/en/Witness" title="Witness" class="wl">witnesses</a> and <a href="/en/Forensics" title="Forensics" class="mw-redirect wl">forensic investigation</a>,<a href="#cite_note-12" style="counter-reset: mw-Ref 12;">[12]</a> 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.<a href="#cite_note-13" style="counter-reset: mw-Ref 13;">[13]</a><a href="#cite_note-14" style="counter-reset: mw-Ref 14;">[14]</a><a href="#cite_note-15" style="counter-reset: mw-Ref 15;">[15]</a> In law, the same evidence that may convince one <a href="/en/Jury" title="Jury" class="wl">jury</a> may not persuade another. <a href="/en/Formal_proof" title="Formal proof" class="wl">Formal proof</a> provides the main exception, where the criteria for proofhood are ironclad and it is impermissible to defend any step in the reasoning as "obvious" (except for the necessary ability of the one proving and the one being proven to, to correctly identify any symbol used in the proof.);<a href="#cite_note-16" style="counter-reset: mw-Ref 16;">[16]</a> for a <a href="/en/Well-formed_formula" title="Well-formed formula" class="wl">well-formed formula</a> to qualify as part of a formal proof, it must be the result of applying a rule of the deductive apparatus of some <a href="/en/Formal_system" title="Formal system" class="wl">formal system</a> to the previous well-formed formulae in the proof sequence.<a href="#cite_note-17" style="counter-reset: mw-Ref 17;">[17]</a>

Proofs have been presented since antiquity. <a href="/en/Aristotle" title="Aristotle" class="wl">Aristotle</a> used the observation that patterns of nature never display the machine-like uniformity of <a href="/en/Determinism" title="Determinism" class="wl">determinism</a> as proof that <a href="/en/Chance_(Ancient_Greek_concept)" title="Chance (Ancient Greek concept)" class="mw-redirect wl">chance</a> is an inherent part of nature.<a href="#cite_note-18" style="counter-reset: mw-Ref 18;">[18]</a> On the other hand, <a href="/en/Thomas_Aquinas" title="Thomas Aquinas" class="wl">Thomas Aquinas</a> used the observation of the existence of <a href="/en/Law_of_nature_(science)" title="Law of nature (science)" class="mw-redirect wl">rich patterns in nature</a> as proof that nature is not ruled by chance.<a href="#cite_note-19" style="counter-reset: mw-Ref 19;">[19]</a>

Proofs need not be verbal. Before <a href="/en/Copernicus" title="Copernicus" class="mw-redirect wl">Copernicus</a>, people took the apparent motion of the <a href="/en/Sun" title="Sun" class="wl">Sun</a> across the sky as proof that <a href="/en/Geocentric_model" title="Geocentric model" class="wl">the Sun went round the Earth</a>.<a href="#cite_note-20" style="counter-reset: mw-Ref 20;">[20]</a> Suitably incriminating evidence left at the scene of a <a href="/en/Crime" title="Crime" class="wl">crime</a> 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 <a href="/en/Signature" title="Signature" class="wl">signature</a> constitutes direct proof of <a href="/en/Authorship" title="Authorship" class="mw-redirect wl">authorship</a>; less directly, <a href="/en/Handwriting_analysis" title="Handwriting analysis" class="mw-redirect wl">handwriting analysis</a> may be submitted as proof of authorship of a document.<a href="#cite_note-21" style="counter-reset: mw-Ref 21;">[21]</a> <a href="/en/Secret" title="Secret" class="mw-redirect wl">Privileged information</a> 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 <a href="/en/Alibi" title="Alibi" class="wl">alibi</a>.

A proof is  sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines,
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 the experience of the world around us, with science obtaining its evidence from nature, 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. 