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A proof is a valid argument whose premises are all true.

Hence a valid argument that has one or more false premises is not a proof.

Suppose $P$ is a proposition whose truth or falsehood is to be determined.

Constructing a valid argument upon a set of premises, all of which have previously been established as being true, is called proving $P$.

Formal Proof

Let $\mathscr P$ be a proof system for a formal language $\LL$.

Let $\phi$ be a WFF of $\LL$.

A formal proof of $\phi$ in $\mathscr P$ is a collection of axioms and rules of inference of $\mathscr P$ that leads to the conclusion that $\phi$ is a theorem of $\mathscr P$.

The term formal proof is also used to refer to specific presentations of such collections.

For example, the term applies to tableau proofs in natural deduction.

Also known as

Some authors use the term sound argument as a synonym for what is defined here as a proof.

However, as some use sound argument to mean the same thing that is defined here as a valid argument, it is recommended that this term not be used.

Some authors refer to a proof as a derivation, but that term already has connotations from calculus, so it is preferred not to be used.

Also see

  • Results about proofs can be found here.

Historical Note

The first one to realise that a proof needs to follow as a result of logical steps from a series of assumptions appears to have been Pythagoras of Samos.