Blackwell Reference Online

Extract

L ogic, philosophy of mathematics A system in which a series of propositions are derived from an initial set of propositions in accordance with a set of formation rules and transformation rules. The members of the initial set of propositions are called axioms . They are independent, that is, not derivable from within the system. The derived series of propositions are called theorems. The formulation rules specify what symbols are used and what combinations of the symbols are to count as axioms and propositions directly derived from axioms. It is thus a system in which all axioms and theorems are ordered in a hierarchical arrangement and the relations between them are necessarily deductive. All propositions conforming to formation rules are called well-formed formulae (wff). The transformation rules determine how theorems are proved. If there is a decision procedure with respect to which all theorems of the system are provable, the system is said to be sound. If all provable formulae are theorems of that system, the system is said to be complete with respect to that decision procedure. If a system does not involve contradiction, it is said to be consistent . Soundness, completeness, and consistency are the characteristics required of an axiomatic system. “In an axiomatic system a change anywhere ramifies into a change everywhere – the entire structure is affected when one ... log in or subscribe to read full text