A deductive system (also called a deductive apparatus of a formal system) consists of the axioms (or axiom schemata) and rules of inference that can be used to derive the theorems of the system.
Such a deductive system is intended to preserve deductive qualities in the formulas that are expressed in the system. Usually the quality we are concerned with is truth as opposed to falsehood. However, other modalities, such as justification or belief may be preserved instead.
In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any intended interpretation of the language. The aim is to ensure that each line of a derivation is merely a syntactic consequence of the lines that precede it. There should be no element of any interpretation of the language that gets involved with the deductive nature of the system.
Famous quotes containing the word system:
“If mothers are to be successful in achieving their child-rearing goals, they must have the inner freedom to find their own value system and within that system to find what is acceptable to them and what is not. This means leaving behind the anxiety, but also the security, of simplistic good-bad formulations and deciding for themselves what they want to teach their children.”
—Elaine Heffner (20th century)