# Hilbert system

[https://en.wikipedia.org/wiki/Hilbert\_system](https://en.wikipedia.org/wiki/Hilbert_system)

A Hilbert system, also called Hilbert-style deductive system, is a type of system of formal deduction attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well.

Most variants of Hilbert systems take a characteristic approach to balancing a trade-off between logical axioms and rules of inference.

Hilbert systems are characterised by preferring many axiom schemes to a small set of rules of inference. Systems of natural deduction take the opposite approach, including many deduction rules but very few or no axiom schemes.

The most commonly studied Hilbert systems have either just one rule of inference, modus ponens for propositional logics, or two, generalisation along with MP in order to handle predicate logics. They also feature several infinite axiom schemes.

Hilbert systems for propositional modal logics, sometimes called Hilbert-Lewis systems, are generally axiomatised with two additional rules, the _necessitation rule_ and the _uniform substitution rule_.

A characteristic feature of the many variants of Hilbert 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 Hilbert 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 - since context gets 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.
