# 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 $\LL$.

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**.

## 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.

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S 1.1$: Constants and variables - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $2$ Conditionals and Negation - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic - 1995: Merrilee H. Salmon:
*Introduction to Logic and Critical Thinking*: $\S 3.1$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**proof** - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...: Definition $1.1.3$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**proof** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic of Shape: Euclid - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**proof**