# Difference between revisions of "Proof"

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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.). | |

: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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::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. | + | :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. |

− | 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. |

## Revision as of 15:35, 18 August 2009

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

- 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.

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.

- 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’.

- 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.

- 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 jury.

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.).

- 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.

- 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.

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

- b. Math. An operation to check the correctness of an arithmetical calculation. Now hist.

Sometimes understood as sense 2.

- 8. Striving, effort; an attempt or endeavour. Obs.

- 9. a. The condition of having successfully stood a test, or the capability of doing so; proven or tested power, strength, etc. (originally and chiefly of armour and arms). Hence fig. and in extended use: impenetrability, invulnerability. Now rare (but cf. PROOF adj. 1).

armour (etc.) of proof: impenetrable armour, proof armour. at the proof: so as to be proof. to the proof: to the utmost, in the highest degree. proof of lead (also shot): the quality of being impenetrable by lead bullets (or shot).

- b. concr. Proof armour (see PROOF adj. 1a). Also fig. Now arch. and hist. N.E.D. (1908), misreading quot. a16252 as proof-arm, postulated a nonce-verb to proof-arm ‘to arm in or as in armour of proof’. Subsequent checking shows no hyphen, suggesting that the quot. belongs at this sense.

- c. The process of stiffening hats and rendering them waterproof. Cf. PROOF v. 2. rare.

- 10. a. A standard of strength of distilled alcoholic liquors; applied to or denoting liquor of this strength; the relative strength (usually measured in degrees) of alcoholic liquor compared to this standard as 100°. In the United Kingdom spirits are at proof if the alcohol contained in them, made up to the volume of the spirits with water, has a weight equal to that of twelve-thirteenths of the same volume of water, at 51°F (10.6°C); this is said to be equivalent to about 57 per cent of alcohol by volume. In the United States proof is twice the percentage of the alcohol content measured by volume at 60°F (15.6°C).

In the United Kingdom proof has been largely superseded in commercial use by the use of % vol, introduced by the International Organization of Legal Metrology in 1973.

- b. fig. (orig. U.S.). With prefixed numeral or percentage, forming an adjectival phrase with the sense ‘pure, genuine; very strong or powerful’.

- 11. a. The testing of ordnance, gunpowder, or firearms by firing, hydraulic pressure, or some other process.

proof of powder: the testing of the propulsive force of gunpowder (obs.).

- b. A place for testing firearms or explosives. Obs. rare (but cf. proof-ground n. at Compounds 1b, proof house n. at Compounds 1b).

- 12. The degree of concentration at which the syrup formed by boiling sugar will successfully crystallize. Also attrib., designating the concentrated sugar syrup or product. Obs.

- 13. The aeration of dough by a raising agent before baking. Cf. PROVE v. 6h.

- 14. Agric. Evidence of the breeding quality of a male animal, esp. a bull, as expressed in terms of quantifiable traits found in its female progeny such as fertility, milk production, etc.

- III. Senses denoting something produced as a test, or which is a means of or instrument for testing.

- 15. a. A coin struck as a pattern or test piece. Originally: a coin or medal struck in order to test a die (obs.). Later: a preliminary impression of a coin struck as a specimen. These often have their edges left plain and not milled; they may also be executed in a metal different from that used for the actual coin.

- b. Printing. A trial or preliminary impression of a printed text, taken to be checked for errors and marked for correction before subsequent revision or final printing. in proof: as a proof; at the stage in the publication process of printing, reading, or correcting proofs. Cf. earlier PROBE n. 7. Applied esp. to the first or early such impression of a document (cf. galley-proof n. at GALLEY n. Compounds 2, page proof n. at PAGE n.2 Compounds 2); a second or later one, incorporating some revision, is frequently called a revise: see REVISE n. 3.

- c. Printmaking. Originally: an impression taken from an engraved block or plate, a lithographic stone, a silk-screen, etc., so that the artist or printmaker can examine the state of a work in progress. Now also: each of a number of impressions from an engraved plate, especially (in commercial printing) of a limited number before the ordinary issue is printed and before an inscription or signature is added (in full proof before letters). Hence more broadly: any impression of an original print on an unbound sheet.

artist's (also engraver's) proofn. (originally) an impression made for examination or alteration by the artist or engraver; (now) an impression made before or outside the numbered edition of a print. letter (also lettered) proofn. a signed and captioned proof. signed proof n. a proof impression signed by the artist. marked, remarque, touched, trial, wax proof: see the first elements.

- d. Photogr. A photographic print, esp. a first or trial print from a particular negative or transparency.

- 16. An instrument, vessel, etc., used in testing or examining something.

- a. A surgical probe. Obs. rare.

- b. A glass vessel used in testing something. Obs.

- c. An apparatus for testing the strength of gunpowder; = EPROUVETTE n. 1. Cf. sense 11. Obs.

- 17. Printing. A number of ems placed in a composing-stick as a guide to the required length of the line of type (the width of a page being a particular number of ems). Obs.

- 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.