Deductive system

A deductive apparatus is a system of syntax rules which when applied to a formal language together become a formal system. It 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.

A deductive apparatus may consist of a set of axioms alone or a set of transformation rules alone or it may have both. It can be defined by laying down by fiat:

• a) that certain formulas are axioms and/or
• b) a set of transformation rules that determine which relations between formulas of the formal language are relations of immediate consequence in the resulting formal system.