# Is that a good proof

## formal proof

Proof of a mathematical statement - usually formulated in an elementary language - solely with the help of fixed formal inference rules.

A formal language is required to specify the mathematical concept of proof *L.* (elementary language), a system of logical axioms formulated in the language Ax (logical axiom), which may be used without restriction as logical presuppositions in every proof and a system of logical inference rules suitable for the intended applications (: = rules of proof; logical rules of derivation). Is Σ any set of expressions or statements *L.* and *φ* an expression then is *φ* formally provable (or deducible) from Σ if there is a finite sequence (*φ*_{1},…,*φ _{n}*) from expressions

*L.*so there that

*φ*=

_{n}*φ,*and for each

*φ*

_{i}With

*i*= 1,…,

*n*one of the following conditions is met:

*φ*∈ Ax (_{i}*φ*is a logical axiom), or*φ*, ∈ Σ (_{i}*φ*belongs to the set of requirements from which_{i}*φ*should be proven), or*φ*, is a direct consequence of the preceding elements, according to the admissible rules of evidence._{i}

(*φ*_{1},…, *φ _{i}*) is then called the sequence of derivatives or formal proof of

*φ*off Σ. In classical two-valued logic (with a suitable system of axioms Ax) the following two rules of evidence are sufficient:

Reasonable rules of evidence are to be chosen in such a way that they lead from true assumptions to true assertions. The axiom system Ax and the rules of proof should be designed in such a way that all statements that follow from Σ in terms of content can also be formally proven (proof methods).

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