Argument-deduction-proof distinctions

Argument-deduction-proof distinctions originated with logic itself. Naturally, the terminology evolved.

An argument, more fully a premise-conclusion argument, is a two-part system composed of premises and conclusion. An argument is valid if and only if its conclusion is a consequence of its premises. Every premise set has infinitely many consequences each giving rise to a valid argument. Some consequences are obviously so but most are not: most are hidden consequences. Most valid arguments are not yet known to be valid. To determine validity in non-obvious cases deductive reasoning is required. There is no deductive reasoning in an argument per se; such must come from the outside.

Every argument's premises are conclusions of other arguments. Every argument's conclusion is a premise of other arguments. The word constituent may be used for either a premise or conclusion.In the context of this article and in most classical contexts, all candidates for consideration as argument constituents fall under the category of truth-bearer: propositions, statements, sentences, judgments, etc.

A deduction is a three-part system composed of premises, a conclusion, and chain of intermediates — steps of reasoning showing that its conclusion is a consequence of its premises. The reasoning in a deduction is by definition cogent. Such reasoning itself, or the chain of intermediates representing it, has also been called an argument, more fully a deductive argument. In many cases, an argument can be known to be valid by means of a deduction of its conclusion from its premises but non-deductive methods such as Venn diagrams and other graphic procedures have been proposed.

A proof is a deduction whose premises are known truths. A proof of the Pythagorean theorem is a deduction that might use several premises — axioms, postulates, and definitions — and contain dozens of intermediate steps. As Alfred Tarski famously emphasized in accord with Aristotle, truths can be known by proof but proofs presuppose truths not known by proof.

...
Wikipedia