Hilbert system
System of formal deduction in logic / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Hilbert system?
Summarize this article for a 10 year old
In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege[1] and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well.
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)
|
- In mathematical physics, Hilbert system is an infrequently used term for a physical system described by a C*-algebra.
Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference.[1] 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. Systems of natural deduction take the opposite tack, including many deduction rules but very few or no axiom schemata. 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 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 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:[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.