Hilbert System

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus or Hilbert-Ackermann 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 tack in the way they balance a trade-off between logical axioms and rules of inference. Hilbert systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference. The most commonly studied Hilbert 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. 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.