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4 i discharging 2 1 6 p p displaystyle p lor p to bot 5 i 1 7 displaystyle bot 1 6 e 8 p p displaystyle p lor p to bot to bot to bot 1 7 i discharging 1 9 p p displaystyle p lor p to bot 8 dn q e d remark valery glivenko proved the following theorem if φ displaystyle varphi is a propositional formula then φ displaystyle varphi is a classical tautology if and only if φ displaystyle neg neg varphi is an intuitionistic tautology this implies that all classical propositional theorems φ displaystyle varphi can be proved like in this example prove φ displaystyle neg neg varphi within intuitionistic logic i e without e displaystyle neg neg _ e apply e displaystyle neg neg _ e to get φ displaystyle varphi from φ displaystyle neg neg varphi consistency completeness and normal forms 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 a theory is said to be consistent if falsehood is not provable from no assumptions and is complete if every theorem or its negation is provable using the inference rules of the logic these are statements about the entire logic and are usually tied to some notion of a model however there are local notions of consistency and completeness that are purely syntactic checks on the inference rules and require no appeals to models the first of these is local consistency also known as local reducibility which says that any derivation containing an introduction of a connective followed immediately by its elimination can be turned into an equivalent derivation without this detour it is a check on the strength of elimination rules they must not be so strong that they include knowledge not already contained in their premises as an example consider conjunctions a u b w a b i a e 1 a u displaystyle begin aligned cfrac cfrac cfrac a u qquad cfrac b w a wedge b wedge _ i a wedge _ e1 end aligned quad rightarrow quad cfrac a u dually local completeness says that the elimination rules are strong enough to decompose a connective into the forms suitable for its introduction rule again for conjunctions a b u a b u a e 1 a b u b e 2 a b i displaystyle cfrac a wedge b u quad rightarrow quad begin aligned cfrac cfrac cfrac a wedge b u a wedge _ e1 qquad cfrac cfrac a wedge b u b wedge _ e2 a wedge b wedge _ i end aligned these notions correspond exactly to β reduction beta reduction and η conversion eta conversion in the lambda calculus using the curry howard isomorphism by local completeness we see that every derivation can be converted to an equivalent derivation where the principal connective is introduced in fact if the entire derivation obeys this ordering of eliminations followed by introductions then it is said to be normal in a normal derivation all eliminations happen above introductions in most logics every derivation has an equivalent normal derivation called a normal form the existence of normal forms is generally hard to prove using natural deduction alone though such accounts do exist in the literature most notably by dag prawitz in 1961 42 it is much easier to show this indirectly by means of a cut free sequent calculus presentation first and higher order extensions 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 summary of first order system the logic of the earlier section is an example of a single sorted logic i e a logic with a single kind of object propositions many extensions of this simple framework have been proposed in this section we will extend it with a second sort of individuals or terms more precisely we will add a new category term denoted t displaystyle mathcal t we shall fix a countable set v displaystyle v of variables another countable set f displaystyle f of function symbols and construct terms with the following formation rules v v v t var f displaystyle frac v in v v mathcal t hbox var _ f and f f t 1 t t 2 t t n t f t 1 t 2 t n t app f displaystyle frac f in f qquad t_ 1 mathcal t qquad t_ 2 mathcal t qquad cdots qquad t_ n mathcal t f t_ 1 t_ 2 cdots t_ n mathcal t hbox app _ f for propositions we consider a third countable set p of predicates and define atomic predicates over terms with the following formation rule ϕ p t 1 t t 2 t t n t ϕ t 1 t 2 t n f pred f displaystyle frac phi in p qquad t_ 1 mathcal t qquad t_ 2 mathcal t qquad cdots qquad t_ n mathcal t phi t_ 1 t_ 2 cdots t_ n mathcal f hbox pred _ f the first two rules of formation provide a definition of a term that is effectively the same as that defined in term algebra and model theory although the focus of those fields of study is quite different from natural deduction the third rule of formation effectively defines an atomic formula as in first order logic and again in model theory to these are added a pair of formation rules defining the notation for quantified propositions one for universal and existential quantification x v a f x a f f x v a f x a f f displaystyle frac x in v qquad a mathcal f forall x a mathcal f forall _ f qquad qquad frac x in v qquad a mathcal f exists x a mathcal f exists _ f the universal quantifier has the introduction and elimination rules a t u a x a x a i u a x a t t t x a e displaystyle cfrac begin array c cfrac a mathcal t text u vdots a x a end array forall x a forall _ i u a qquad qquad frac forall x a qquad t mathcal t t x a forall _ e the existential quantifier has the introduction and elimination rules t x a x a i a t u a x a v x a c c e a u v displaystyle frac t x a exists x a exists _ i qquad qquad cfrac begin array cc underbrace cfrac a mathcal t hbox u quad cfrac a x a hbox v vdots exists x a quad c end array c exists _ e a u v in these rules the notation t x a stands for the substitution of t for every visible instance of x in a avoiding capture 43 as before the superscripts on the name stand for the components that are discharged the term a cannot occur in the conclusion of i such terms are known as eigenvariables or parameters and the hypotheses named u and v in e are localised to the second premise in a hypothetical derivation although the propositional logic of earlier sections was decidable adding the quantifiers makes the logic undecidable so far the quantified extensions are first order they distinguish propositions from the kinds of objects quantified over higher order logic takes a different approach and has only a single sort of propositions the quantifiers have as the domain of quantification the very same sort of propositions as reflected in the formation rules p f u a f p a f f u p f u a f p a f f u displaystyle cfrac begin matrix cfrac p mathcal f hbox u vdots a mathcal f end matrix forall p a mathcal f forall _ f u qquad qquad cfrac begin matrix cfrac p mathcal f hbox u vdots a mathcal f end matrix exists p a mathcal f exists _ f u a discussion of the introduction and elimination forms for higher order logic is beyond the scope of this article it is possible to be in between first order and higher order logics for example second order logic has two kinds of propositions one kind quantifying over terms and the second kind quantifying over propositions of the first kind proofs and 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 the presentation of natural deduction so far has concentrated on the nature of propositions without giving a formal definition of a proof to formalise the notion of proof we alter the presentation of hypothetical derivations slightly we label the antecedents with proof variables from some countable set v of variables and decorate the succedent with the actual proof the antecedents or hypotheses 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 l...
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