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Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics. For example, a paradigmatic case is the sequent calculus , which can be used to express the consequence relations of both intuitionistic logic and relevance logic .
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.
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence, according to the rule of inference.
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.
A graphical representation of a partially built propositional tableau. In proof theory, the semantic tableau [1] (/ t æ ˈ b l oʊ, ˈ t æ b l oʊ /; plural: tableaux), also called an analytic tableau, [2] truth tree, [1] or simply tree, [2] is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. [1]
In proof by contradiction, also known by the Latin phrase reductio ad absurdum (by reduction to the absurd), it is shown that if some statement is assumed true, a logical contradiction occurs, hence the statement must be false. A famous example involves the proof that is an irrational number:
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 ...
Proof systems in propositional logic can be broadly classified into semantic proof systems and syntactic proof systems, [86] [87] [88] according to the kind of logical consequence that they rely on: semantic proof systems rely on semantic consequence (), [89] whereas syntactic proof systems rely on syntactic consequence (). [90]