In mathematics and logic, premises are the formulas on which a step of a logical argument depends to obtain a consequence of those premises. Premises may be justified either as instances of axioms of an axiomatic system, or as consequences of previous steps of the argument, or as theorems, lemmas, or corollaries that have been established as part of a larger theory, or as assumptions to be discharged later as in natural deduction. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... In logic, an argument is an attempt to demonstrate the truth of an assertion called a conclusion, based on the truth of a set of assertions called premises. ... Consequence can be: Consequences is a game. ... For the algebra software named Axiom, see Axiom computer algebra system. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... In mathematics, theory is used informally to refer to a body of knowledge about mathematics. ... In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... In mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it naturally occurs. ...