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heses are separated from the succedent by means of a turnstile this modification sometimes goes under the name of localised hypotheses the following diagram summarises the change u 1 u 2 u n j 1 j 2 j n j u 1 j 1 u 2 j 2 u n j n j the collection of hypotheses will be written as γ when their exact composition is not relevant to make proofs explicit we move from the proof less judgment a to a judgment π is a proof of a which is written symbolically as π a following the standard approach proofs are specified with their own formation rules for the judgment π proof the simplest possible proof is the use of a labelled hypothesis in this case the evidence is the label itself u v proof f u proof hyp u a u a let us re examine some of the connectives with explicit proofs for conjunction we look at the introduction rule i to discover the form of proofs of conjunction they must be a pair of proofs of the two conjuncts thus π 1 proof π 2 proof pair f π 1 π 2 proof γ π 1 a γ π 2 b i γ π 1 π 2 a b the elimination rules e 1 and e 2 select either the left or the right conjunct thus the proofs are a pair of projections first fst and second snd π proof fst f fst π proof γ π a b e 1 γ fst π a π proof snd f snd π proof γ π a b e 2 γ snd π b for implication the introduction form localises or binds the hypothesis written using a λ this corresponds to the discharged label in the rule γ u a stands for the collection of hypotheses γ together with the additional hypothesis u π proof λ f λu π proof γ u a π b i γ λu π a b π 1 proof π 2 proof app f π 1 π 2 proof γ π 1 a b γ π 2 a e γ π 1 π 2 b with proofs available explicitly one can manipulate and reason about proofs the key operation on proofs is the substitution of one proof for an assumption used in another proof this is commonly known as a substitution theorem and can be proved by induction on the depth or structure of the second judgment 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 γ π 1 a and γ u a π 2 b then γ π 1 u π 2 b so far the judgment γ π a has had a purely logical interpretation in type theory the logical view is exchanged for a more computational view of objects propositions in the logical interpretation are now viewed as types and proofs as programs in the lambda calculus thus the interpretation of π a is the program π has type a the logical connectives are also given a different reading conjunction is viewed as product implication as the function arrow etc the differences are only cosmetic however type theory has a natural deduction presentation in terms of formation introduction and elimination rules in fact the reader can easily reconstruct what is known as simple type theory from the previous sections the difference between logic and type theory is primarily a shift of focus from the types propositions to the programs proofs type theory is chiefly interested in the convertibility or reducibility of programs for every type there are canonical programs of that type which are irreducible these are known as canonical forms or values if every program can be reduced to a canonical form then the type theory is said to be normalising or weakly normalising if the canonical form is unique then the theory is said to be strongly normalising normalisability is a rare feature of most non trivial type theories which is a big departure from the logical world recall that almost every logical derivation has an equivalent normal derivation to sketch the reason in type theories that admit recursive definitions it is possible to write programs that never reduce to a value such looping programs can generally be given any type in particular the looping program has type although there is no logical proof of for this reason the propositions as types proofs as programs paradigm only works in one direction if at all interpreting a type theory as a logic generally gives an inconsistent logic example dependent type theory 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 like logic type theory has many extensions and variants including first order and higher order versions one branch known as dependent type theory is used in a number of computer assisted proof systems dependent type theory allows quantifiers to range over programs themselves these quantified types are written as π and σ instead of and and have the following formation rules γ a type γ x a b type π f γ πx a b type γ a type γ x a b type σ f γ σx a b type these types are generalisations of the arrow and product types respectively as witnessed by their introduction and elimination rules γ x a π b πi γ λx π πx a b γ π 1 πx a b γ π 2 a πe γ π 1 π 2 π 2 x b γ π 1 a γ x a π 2 b σi γ π 1 π 2 σx a b γ π σx a b σe 1 γ fst π a γ π σx a b σe 2 γ snd π fst π x b dependent type theory in full generality is very powerful it is able to express almost any conceivable property of programs directly in the types of the program this generality comes at a steep price either typechecking is undecidable extensional type theory or extensional reasoning is more difficult intensional type theory for this reason some dependent type theories do not allow quantification over arbitrary programs but rather restrict to programs of a given decidable index domain for example integers strings or linear programs since dependent type theories allow types to depend on programs a natural question to ask is whether it is possible for programs to depend on types or any other combination there are many kinds of answers to such questions a popular approach in type theory is to allow programs to be quantified over types also known as parametric polymorphism of this there are two main kinds if types and programs are kept separate then one obtains a somewhat more well behaved system called predicative polymorphism if the distinction between program and type is blurred one obtains the type theoretic analogue of higher order logic also known as impredicative polymorphism various combinations of dependency and polymorphism have been considered in the literature the most famous being the lambda cube of henk barendregt the intersection of logic and type theory is a vast and active research area new logics are usually formalised in a general type theoretic setting known as a logical framework popular modern logical frameworks such as the calculus of constructions and lf are based on higher order dependent type theory with various trade offs in terms of decidability and expressive power these logical frameworks are themselves always specified as natural deduction systems which is a testament to the versatility of the natural deduction approach classical and modal logics 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 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 deducti...
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