# Definition:Proof

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## Definition

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.