Proof (truth)

A proof is sufficient evidence or a sufficient argument for the truth of a proposition.

The concept applies 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 from other previously established theorems. The subject of logic, in particular proof theory, formalizes and studies the notion of formal proof. In some 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 the 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" (except for the necessary ability of the one proving and the one being proven to, to correctly identify any symbol used in the proof.); 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.

Differences between proof and evidence
Proof and evidence are two related but distinct concepts.

Proof is the evidence that is necessary to provide a convincing argument or justification for a conclusion. It is usually expressed in the form of a logical argument or proof. Proof is usually found in written documents, such as legal documents or scientific papers. It is also often used in mathematics and computer science to prove the correctness of an algorithm or a program. Evidence, on the other hand, is any material that can be used to support an argument.

Evidence can be physical, such as a fingerprint or a piece of clothing, or it can be verbal, such as testimony from a witness. Evidence is used to support a conclusion, but it does not provide proof. It is not the same as proof, but it can be used to help build an argument and support a conclusion.