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In proof complexity, a Frege system is a propositional proof system whose proofs are sequences of formulas derived using a finite set of sound and implicationally complete inference rules. [1] Frege systems (more often known as Hilbert systems in general proof theory ) are named after Gottlob Frege .
The use of "Hilbert-style" and similar terms to describe axiomatic proof systems in logic is due to the influence of Hilbert and Ackermann's Principles of Mathematical Logic (1928). [2] Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference.
More exotic proof calculi such as Jean-Yves Girard's proof nets also support a notion of analytic proof. A particular family of analytic proofs arising in reductive logic are focused proofs which characterise a large family of goal-directed proof-search procedures. The ability to transform a proof system into a focused form is a good indication ...
In general, a proof system for a language L is a polynomial-time function whose range is L. Thus, a propositional proof system is a proof system for TAUT. Sometimes the following alternative definition is considered: a pps is given as a proof-verification algorithm P(A,x) with two inputs. If P accepts the pair (A,x) we say that x is a P-proof of A.
Download as PDF; Printable version; ... Pages in category "Systems of formal logic" ... Propositional proof system; Q. Quantum logic; S.
A proof system includes the components: [1] [2] Formal language: The set L of formulas admitted by the system, for example, propositional logic or first-order logic. Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems. Axioms: Formulas in L assumed to be valid. All theorems are derived from axioms.
Fitch notation, also known as Fitch diagrams (named after Frederic Fitch), is a notational system for constructing formal proofs used in sentential logics and predicate logics. Fitch-style proofs arrange the sequence of sentences that make up the proof into rows.
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi systems, LK and LJ, were introduced in 1934/1935 by Gerhard Gentzen [1] as a tool for studying natural deduction in first-order logic (in classical and intuitionistic versions, respectively).