# 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 (φ1,…,φn) from expressions L. so there that φn = φ, and for each φi With i = 1,…, n one of the following conditions is met:

1. φi ∈ Ax (φ is a logical axiom), or
2. φi, ∈ Σ (φi belongs to the set of requirements from which φ should be proven), or
3. φi, is a direct consequence of the preceding elements, according to the admissible rules of evidence.

(φ1,…, φi) 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).