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Sequent calculus
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==Variants== The above rules can be modified in various ways: ===Minor structural alternatives=== There is some freedom of choice regarding the technical details of how sequents and structural rules are formalized without changing what sequents the system derives. First of all, as mentioned above, the sequents can be viewed to consist of sets or [[multiset]]s. In this case, the rules for permuting and (when using sets) contracting formulas are unnecessary. The rule of weakening becomes [[admissible rule|admissible]] if the axiom (I) is changed to derive any sequent of the form <math>\Gamma , A \vdash A , \Delta</math>. Any weakening that appears in a derivation can then be moved to the beginning of the proof. This may be a convenient change when constructing proofs bottom-up. One may also change whether rules with more than one premise share the same context for each of those premises or split their contexts between them: For example, <math>({\lor}L)</math> may be instead formulated as :<math> \cfrac{\Gamma, A \vdash \Delta \qquad \Sigma, B \vdash \Pi}{\Gamma, \Sigma, A \lor B \vdash \Delta, \Pi}. </math> Contraction and weakening make this version of the rule interderivable with the version above, although in their absence, as in [[linear logic]], these rules define different connectives. ===Absurdity=== One can introduce <math>\bot</math>, the [[principle of explosion|absurdity constant]] representing ''false'', with the axiom: :<math> \cfrac{}{\bot \vdash \quad } </math> Or if, as described above, weakening is to be an admissible rule, then with the axiom: :<math> \cfrac{}{\Gamma, \bot \vdash \Delta} </math> With <math>\bot</math>, negation can be subsumed as a special case of implication, via the definition <math>(\neg A) \iff (A \to \bot)</math>. ===Substructural logics=== {{main article|Substructural logic}} Alternatively, one may restrict or forbid the use of some of the structural rules. This yields a variety of [[substructural logic]] systems. They are generally weaker than LK (''i.e.'', they have fewer theorems), and thus not complete with respect to the standard semantics of first-order logic. However, they have other interesting properties that have led to applications in theoretical [[computer science]] and [[artificial intelligence]]. ===Intuitionistic sequent calculus: System LJ=== Surprisingly, some small changes in the rules of LK suffice to turn it into a proof system for [[intuitionistic logic]].<ref>{{harvnb|Gentzen|1934|p=194}}, wrote: "Der Unterschied zwischen ''intuitionistischer'' und ''klassischer'' Logik ist bei den Kalkülen ''LJ'' und ''LK'' äußerlich ganz anderer Art als bei ''NJ'' und ''NK''. Dort bestand er in Weglassung bzw. Hinzunahme des Satzes vom ausgeschlossenen Dritten, während er hier durch die Sukzedensbedingung ausgedrückt wird." English translation: "The difference between ''intuitionistic'' and ''classical'' logic is in the case of the calculi ''LJ'' and ''LK'' of an extremely, totally different kind to the case of ''NJ'' and ''NK''. In the latter case, it consisted of the removal or addition respectively of the excluded middle rule, whereas in the former case, it is expressed through the succedent conditions."</ref> To this end, one has to restrict to sequents with at most one formula on the right-hand side,{{sfn|Tiomkin|1988}} and modify the rules to maintain this invariant. For example, <math>({\lor}L)</math> is reformulated as follows (where C is an arbitrary formula): :<math> \cfrac{\Gamma, A \vdash C \qquad \Gamma, B \vdash C }{\Gamma, A \lor B \vdash C} \quad ({\lor}L) </math> The resulting system is called LJ. It is sound and complete with respect to intuitionistic logic and admits a similar cut-elimination proof. This can be used in proving [[disjunction and existence properties]]. In fact, the only rules in LK that need to be restricted to single-formula consequents are <math>({\to}R)</math>, <math>(\neg R)</math> (which can be seen as a special case of <math>{\to}R</math>, as described above) and <math>({\forall}R)</math>. When multi-formula consequents are interpreted as disjunctions, all of the other inference rules of LK are derivable in LJ, while the rules <math>({\to}R)</math> and <math>({\forall}R)</math> become :<math> \cfrac{\Gamma, A \vdash B \lor C}{\Gamma \vdash (A \to B) \lor C} </math> and (when <math>y</math> does not occur free in the bottom sequent) :<math> \cfrac{\Gamma \vdash A[y/x] \lor C}{\Gamma \vdash (\forall x A) \lor C}. </math> These rules are not intuitionistically valid.
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