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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. | |
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− | ::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. |
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− | ::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’. |
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− | ::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. |
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| :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. |
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− | :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. |
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− | :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. |
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| *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. |
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− | ::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. |
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− | :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. |
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− | :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. |
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| :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. |
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| + | [[Category: Logic]] |
| + | [[Category: Philosophy]] |
| + | [[Category: Mathematics]] |