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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.

::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.