Definition:Proof

Definition
A proof is another name for a valid argument, but in this context the assumption is made that the premises are all true.

That is, 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 $\LL$.

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.