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sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed may 2024 learn how and when to remove this message for simplicity the logics presented so far have been intuitionistic classical logic extends intuitionistic logic with an additional axiom or principle of excluded middle for any proposition p the proposition p p is true this statement is not obviously either an introduction or an elimination indeed it involves two distinct connectives gentzen s original treatment of excluded middle prescribed one of the following three equivalent formulations which were already present in analogous forms in the systems of hilbert and heyting xm 1 a a a xm 2 a u a p xm 3 u p a xm 3 is merely xm 2 expressed in terms of e this treatment of excluded middle in addition to being objectionable from a purist s standpoint introduces additional complications in the definition of normal forms a comparatively more satisfactory treatment of classical natural deduction in terms of introduction and elimination rules alone was first proposed by parigot in 1992 in the form of a classical lambda calculus called λμ the key insight of his approach was to replace a truth centric judgment a with a more classical notion reminiscent of the sequent calculus in localised form instead of γ a he used γ δ with δ a collection of propositions similar to γ γ was treated as a conjunction and δ as a disjunction this structure is essentially lifted directly from classical sequent calculi but the innovation in λμ was to give a computational meaning to classical natural deduction proofs in terms of a callcc or a throw catch mechanism seen in lisp and its descendants see also first class control another important extension was for modal and other logics that need more than just the basic judgment of truth these were first described for the alethic modal logics s4 and s5 in a natural deduction style by prawitz in 1965 5 and have since accumulated a large body of related work to give a simple example the modal logic s4 requires one new judgment a valid that is categorical with respect to truth if a is true under no assumption that b is true then a valid this categorical judgment is internalised as a unary connective a read necessarily a with the following introduction and elimination rules a valid i a a e a note that the premise a valid has no defining rules instead the categorical definition of validity is used in its place this mode becomes clearer in the localised form when the hypotheses are explicit we write ω γ a where γ contains the true hypotheses as before and ω contains valid hypotheses on the right there is just a single judgment a validity is not needed here since ω a valid is by definition the same as ω a the introduction and elimination forms are then ω π a i ω box π a ω γ π a e ω γ unbox π a the modal hypotheses have their own version of the hypothesis rule and substitution theorem valid hyp ω u a valid γ u a modal substitution theorem edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed may 2024 learn how and when to remove this message if ω π 1 a and ω u a valid γ π 2 c then ω γ π 1 u π 2 c this framework of separating judgments into distinct collections of hypotheses also known as multi zoned or polyadic contexts is very powerful and extensible it has been applied for many different modal logics and also for linear and other substructural logics to give a few examples however relatively few systems of modal logic can be formalised directly in natural deduction to give proof theoretic characterisations of these systems extensions such as labelling or systems of deep inference the addition of labels to formulae permits much finer control of the conditions under which rules apply allowing the more flexible techniques of analytic tableaux to be applied as has been done in the case of labelled deduction labels also allow the naming of worlds in kripke semantics simpson 1994 presents an influential technique for converting frame conditions of modal logics in kripke semantics into inference rules in a natural deduction formalisation of hybrid logic stouppa 2004 surveys the application of many proof theories such as avron and pottinger s hypersequents and belnap s display logic to such modal logics as s5 and b comparison with sequent calculus edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed may 2024 learn how and when to remove this message main article sequent calculus the sequent calculus is the chief alternative to natural deduction as a foundation of mathematical logic in natural deduction the flow of information is bi directional elimination rules flow information downwards by deconstruction and introduction rules flow information upwards by assembly thus a natural deduction proof does not have a purely bottom up or top down reading making it unsuitable for automation in proof search to address this fact gentzen in 1935 proposed his sequent calculus though he initially intended it as a technical device for clarifying the consistency of predicate logic kleene in his seminal 1952 book introduction to metamathematics gave the first formulation of the sequent calculus in the modern style 44 in the sequent calculus all inference rules have a purely bottom up reading inference rules can apply to elements on both sides of the turnstile to differentiate from natural deduction this article uses a double arrow instead of the right tack for sequents the introduction rules of natural deduction are viewed as right rules in the sequent calculus and are structurally very similar the elimination rules on the other hand turn into left rules in the sequent calculus to give an example consider disjunction the right rules are familiar γ a r 1 γ a b γ b r 2 γ a b on the left γ u a c γ v b c l γ w a b c recall the e rule of natural deduction in localised form γ a b γ u a c γ v b c e γ c the proposition a b which is the succedent of a premise in e turns into a hypothesis of the conclusion in the left rule l thus left rules can be seen as a sort of inverted elimination rule this observation can be illustrated as follows natural deduction sequent calculus hyp elim rules meet intro rules conclusion init left rules right rules conclusion in the sequent calculus the left and right rules are performed in lock step until one reaches the initial sequent which corresponds to the meeting point of elimination and introduction rules in natural deduction these initial rules are superficially similar to the hypothesis rule of natural deduction but in the sequent calculus they describe a transposition or a handshake of a left and a right proposition init γ u a a the correspondence between the sequent calculus and natural deduction is a pair of soundness and completeness theorems which are both provable by means of an inductive argument soundness of wrt if γ a then γ a completeness of wrt if γ a then γ a it is clear by these theorems that the sequent calculus does not change the notion of truth because the same collection of propositions remain true thus one can use the same proof objects as before in sequent calculus derivations as an example consider the conjunctions the right rule is virtually identical to the introduction rule sequent calculus natural deduction γ π 1 a γ π 2 b r γ π 1 π 2 a b γ π 1 a γ π 2 b i γ π 1 π 2 a b the left rule however performs some additional substitutions that are not performed in the corresponding elimination rules sequent calculus natural deduction γ u a π c l 1 γ v a b fst v u π c γ π a b e 1 γ fst π a γ u b π c l 2 γ v a b snd v u π c γ π a b e 2 γ snd π b the kinds of proofs generated in the sequent calculus are therefore rather different from those of natural deduction the sequent calculus produces proofs in what is known as the β normal η long form which corresponds to a canonical representation of the normal form of the natural deduction proof if one attempts to describe these proofs using natural deduction itself one obtains what is called the intercalation calculus first described by john byrnes which can be used to formally define the notion of a normal form for natural deduction the substitution theorem of natural deduction takes the form of a structural rule or structural theorem known as cut in the sequent calculus cut substitution edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed may 2024 learn how and when to remove this message if γ π 1 a and γ u a π 2 c then γ π 1 u π 2 c in most well behaved logics cut is unnecessary as an inference rule though it remains provable as a meta theorem the superfluousness of the cut rule is usually presented as a computational process known as cut elimination this has an interesting application for natural deduction usually it is extremely tedious to prove certain properties directly in natural deduction because of an unbounded number of cases for example consider showing that a given proposition is not provable in natural deduction a simple inductive argument fails because of rules like e or e which can introduce arbitrary propositions however we know that the sequent calculus is complete with respect to natural deduction so it is enough to show this unprovability in the sequent calculus now if cut is not available as an inference rule then all sequent rules either introduce a connective on the right or the left so the depth of a sequent derivation is fully bounded by the connectives in the final conclusion thus showing unprovability is much easier because there are only a finite number of cases to consider and each case is composed entirely of sub propositions of the conclusion a simple instance of this is the global consistency theorem is not provable in the sequent calculus version this is manifestly true because there is no rule that can have as a conclusion proof theorists often prefer to work on cut free sequent calculus formulations because of such properties see also edit philosophy portal mathematical logic sequent calculus gerhard gentzen system l tabular natural deduction argument map the general term for tree like logic notation notes edit indrzejczak jaśkowski 1934 gentzen 1935a gentzen 1935b gentzen 1935a p 176 1 2 prawitz 1965 harvnb error no target citerefprawitz1965 help prawitz 2006 harvnb error no target citerefprawitz2006 help martin löf 1996 1 2 3 4 5 pelletier hazen 2024 quine 1981 see particularly pages 91 93 for quine s line number notation for antecedent dependencies a particular advantage of kleene s tabular natural deduction systems is that he proves the validity of the inference rules for both propositional calculus and predicate calculus see kleene 2002 pp 44 45 118 119 von plato 2013 p 9 weisstein 1 2 3 von plato 2013 pp 9 32 121 sutcliffe 1 2 restall 2018 magnus et al 2023 p 142 chapter 20 proof theoretic concepts 1 2 3 paseau leek paseau pregel 2023 magnus et al 2023 p 82 12 5 the double turnstile 1 2 3 4 5 6 7 8 9 allen hand 2022 allen hand 2022 p 12 kleene 2002 prawitz 1965 sfn error no target citerefprawitz1965 help von plato 2013 p 18 1 2 van dalen 2013 hansson hendricks 2018 p 179 1 2 3 ayala rincón de moura 2017 pp 2 20 hansson hendricks 2018 p 38 bostock 1997 p 21 this is required in paraconsistent logics that do not treat displaystyle neg and ϕ displaystyle phi to bot as equivalents 1 2 3 von plato 2013 pp 12 13 prawitz 1965 p 20 sfn error no target citerefprawitz1965 help 1 2 3 instead of displaystyle neg neg e one can add reductio ad absurdum as a rule to obtain classical logic 34 35 φ φ displaystyle frac begin array c varphi to bot vdots bot end array varphi raa johansson 1937 1 2 prawitz 1965 p 21 sfn error no target citerefprawitz1965 help 1 2 ayala rincón de moura 2017 pp 17 24 tennant 1990 p 48 1 2 suppes 1999 1 2 lemmon 1978 1 2 lemmon 1978 pp passim especially 39 40 1 2 3 4 5 arthur 2017 1 2 this is not conform the raa rule the rule which is implicitly applied is e on φ φ displaystyle varphi varphi to bot as negation is not defined as an implication this modus ponens is undocumented see also his book prawitz 1965 harvnb error no target citerefprawitz1965 help prawitz 2006 harvnb error no target citerefprawitz2006 help see the article on lambda calculus for more detail about the concept of substitution kleene 2009 pp 440 516 see also kleene 1980 references edit allen colin hand michael 2022 logic primer 3rd ed cambridge massachusetts the mit press isbn 978 0 262 54364 4 arthur richard t w 2017 an introduction to logic using natural deduction real arguments a little history and some humour 2nd ed peterborough ontario broadview press isbn 978 1 55481 332 2 oclc 962129086 ayala rincón maurício de moura flávio l c 2017 applied logic for computer scientists undergraduate topics in computer science springer doi 10 1007 978 3 319 51653 0 isbn 978 3 319 51651 6 barker plummer dave barwise jon etchemendy john 2011 language proof and logic 2nd ed csli publications isbn 978 1575866321 bostock david 1997 intermediate logic oxford new york oxford university press isbn 978 0 19 875141 0 gallier jean 2005 constructive logics part i a tutorial on proof systems and typed λ calculi archived from the original on 5 july 2017 retrieved 12 june 2014 gentzen gerhard karl erich 1935a untersuchungen über das logische schließen i mathematische zeitschrift 39 2 176 210 doi 10 1007 bf01201353 s2cid 121546341 archived from the original on 24 december 2015 1964 1935 investigations into logical deduction american philosophical quarterly 1 4 249 287 gentzen gerhard karl erich 1935b untersuchungen über das logische schließen ii mathematische zeitschrift 39 3 405 431 doi 10 1007 bf01201363 s2cid 186239837 cite journal cs1 maint deprecated archival service link 1965 1935 investigations into logical deduction american philosophical quarterly 2 3 204 218 girard jean yves 1990 proofs and types cambridge tracts in theoretical computer science cambridge university press cambridge england archived from the original on 4 july 2016 retrieved 20 april 2006 translated and with appendices by paul taylor and yves lafont hansson sven ove hendricks vincent f 2018 introduction to formal philosophy springer undergraduate texts in philosophy cham springer isbn 978 3 030 08454 7 jaśkowski stanisław 1934 on the rules of suppositions in formal logic reprinted in polish logic 1920 39 ed storrs mccall johansson ingebrigt 1937 der minimalkalkül ein reduzierter intuitionistischer formalismus compositio mathematica in german 4 119 136 kleene stephen cole 1980 1952 introduction to metamathematics eleventh ed north holland isbn 978 0 7204 2103 3 kleene stephen cole 2009 1952 introduction to metamathematics...
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