Axiomatic system (logic)

In logic, especially mathematical logic, an axiomatic system, sometimes called a "Hilbert-style" deductive system, is a type of system of formal deduction developed by Gottlob Frege,^[1] Jan Łukasiewicz,^[2] Russell and Whitehead,^[3] and David Hilbert.^[3] These deductive systems are most often studied for first-order logic, but are of interest for other logics as well.

Most variants of axiomatic systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference.^[1] Axiomatic systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference. Systems of natural deduction take the opposite tack, including many deduction rules but very few or no axiom schemata. The most commonly studied axiomatic systems have either just one rule of inference – modus ponens, for propositional logics – or two – with generalisation, to handle predicate logics, as well – and several infinite axiom schemes. Axiomatic systems for alethic modal logics, sometimes called Hilbert-Lewis systems, additionally require the necessitation rule. Some systems use a finite list of concrete formulas as axioms instead of an infinite set of formulas via axiom schemes, in which case the uniform substitution rule is required.

A characteristic feature of the many variants of axiomatic systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if one is interested only in the derivability of tautologies, no hypothetical judgments, then one can formalize the axiomatic system in such a way that its rules of inference contain only judgments of a rather simple form. The same cannot be done with the other two deductions systems:^{[citation needed]} as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided – not even if we want to use them just for proving derivability of tautologies.