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In logic, a '''proof''' is an [[argument]] that establishes the validity of a [[proposition]]. Although proofs may be based on inductive [[logic]], in general the term proof connotes a rigorous deduction. In [[formal]] [[axiomatic]] [[systems]] of [[logic]] and [[mathematics]], a proof is a [[finite]] sequence of well developed [[formulas]] (generated in accordance with accepted formation rules) in which: (1) each formula is either an [[axiom]] or is derived from some previous formula or formulas by a valid inference; and (2) the last formula is that which is to be proved.

In logic, a '''proof''' is an [[argument]] that establishes the validity of a [[proposition]]. Although proofs may be based on inductive [[logic]], in general the term proof connotes a rigorous deduction. In [[formal]] [[axiomatic]] [[systems]] of [[logic]] and [[mathematics]], a proof is a [[finite]] sequence of well developed [[formulas]] (generated in accordance with accepted formation rules) in which: (1) each formula is either an [[axiom]] or is derived from some previous formula or formulas by a valid inference; and (2) the last formula is that which is to be proved.

==Definition==

==Definition==

*I. Senses relating to the establishment or demonstration of truth or validity (cf. PROVE v. I.).

*I. [[Senses]] relating to the establishment or demonstration of [[truth]] or validity (cf. PROVE v. I.).

:1. a. Something that proves a statement; evidence or argument establishing a fact or the truth of anything, or belief in the certainty of something; an instance of this.

:1. a. Something that proves a statement; [[evidence]] or [[argument]] establishing a [[fact]] or the truth of anything, or [[belief]] in the certainty of something; an instance of this. to make proof: to carry weight as evidence (obs.). proof positive n. definite, [[absolute]], or incontrovertible proof: see POSITIVE adj. 2.

to make proof: to carry weight as evidence (obs.). proof positive n. definite, absolute, or incontrovertible proof: see POSITIVE adj. 2.

::b. Law. Evidence determining the judgment of a tribunal. Also spec. (a) a document or documents so attested as to form legal evidence; (b) a written statement of what a witness is prepared to swear to; (c) the evidence given and recorded in a particular case. See also sense 4.

::b. [[Law]]. Evidence determining the [[judgment]] of a tribunal. Also spec. (a) a document or documents so attested as to form legal evidence; (b) a written statement of what a witness is prepared to swear to; (c) the evidence given and recorded in a particular case. See also sense 4.

::c. A person who gives evidence; a witness. Cf. EVIDENCE n. 7. Obs. (after 1500 only in Sc. use).

::c. A [[person]] who gives evidence; a witness. Cf. EVIDENCE n. 7. Obs. (after 1500 only in Sc. use).

my god in proof: ‘as my god is my witness’.

my god in proof: ‘as my [[god]] is my witness’.

::d. Sc. A piece of scriptural text cited to support a particular doctrine adopted in a Presbyterian catechism, esp. the Shorter Catechism (cf. quot. 1725). Cf. proof-text n. at Compounds 2. In later use hist.

::d. Sc. A piece of scriptural [[text]] cited to support a particular doctrine adopted in a Presbyterian catechism, esp. the Shorter Catechism (cf. quot. 1725). Cf. proof-text n. at Compounds 2. In later use hist.

:2. The action, process, or fact of proving or establishing the truth or validity of a statement; the action of evidence in convincing the mind; demonstration. in proof: as proof, by way of demonstration. burden of proof: see BURDEN n. 2b.

:2. The action, process, or fact of proving or establishing the truth or validity of a statement; the action of evidence in convincing the mind; demonstration. in proof: as proof, by way of demonstration. burden of proof: see BURDEN n. 2b.

:3. Math. and Logic. A sequence of steps by which a theorem or other statement is derived from given premises.

:3. Math. and [[Logic]]. A sequence of steps by which a theorem or other statement is derived from given premises.

:4. Sc. Law. In a civil case: evidence given before a judge, or a commissioner acting as a judge's representative, in determining what is at issue in a trial or establishing the disputed facts; the taking of such evidence. Hence: trial of a civil case before a judge without a: This distinctive development of sense has gradually taken place since the introduction of trial by jury into Scotland in 1815.

:4. Sc. Law. In a civil case: evidence given before a judge, or a commissioner acting as a judge's representative, in determining what is at issue in a trial or establishing the disputed; the taking of such evidence. Hence: trial of a civil case before a judge without a: This distinctive development of sense has gradually taken place since the introduction of trial by jury into Scotland in 1815.

*II. Senses relating to the trying or testing of something (cf. PROVE v. II

*II. Senses relating to the trying or testing of something (cf. PROVE v. II

:5. a. That which anything proves or turns out to be; the issue, outcome, result, effect, or fulfilment of something; esp. in phrase to come to proof. Obs.

:5. a. That which anything proves or turns out to be; the issue, outcome, result, effect, or fulfilment of something; esp. in phrase to come to proof. Obs.

::b. The fact, condition, or quality of proving good, turning out well, or producing good results; thriving; goodness, substance. Eng. regional (chiefly south-west.) in later use. Now rare.

::b. The fact, condition, or [[quality]] of proving [[good]], turning out well, or producing good results; thriving; goodness, substance. Eng. regional (chiefly south-west.) in later use. Now rare.

:6. The action or fact of experiencing or having experience of something; knowledge derived from this; experience. Obs.

:6. The action or fact of experiencing or having experience of something; [[knowledge]] derived from this; [[experience]]. Obs.

:7. a. The action or an act of testing or making trial of something; the condition of being tested; examination, experiment; test, trial; (also) an instance of this. Often in phrases, as to bring (set, etc.) (a thing) in (also on, to) (the, a) proof. Now usually as to put to (the) proof.

:7. a. The [[action]] or an act of testing or making trial of something; the condition of being tested; examination, experiment; test, trial; (also) an instance of this. Often in phrases, as to bring (set, etc.) (a thing) in (also on, to) (the, a) proof. Now usually as to put to (the) proof.

the proof of the pudding is in the eating: see PUDDING n. Phrases 2.

the proof of the pudding is in the eating: see PUDDING n. Phrases 2.

:18. Bookbinding. The rough uncut edges of the shorter or narrower leaves of a book, left in trimming it to show that it has not been cut down. Now rare.

:18. Bookbinding. The rough uncut edges of the shorter or narrower leaves of a book, left in trimming it to show that it has not been cut down. Now rare.