Logic Colloquium 73
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Get my own profile Cited by View all All Since Citations h-index 17 12 iindex 31 Verified email at qmul. Articles Cited by Co-authors. Annals of Pure and Applied Logic , , Mathematical structures in computer science 20 2 , , Conference on Computability in Europe, , Mathematical Structures in Computer Science 16 2 , , Notre Dame Journal of Formal Logic 47 2 , , The Law of Excluded Middle does not hold for propositions in intuitionistic type theory.
Foundations and logical aspects of constructivism
In the ordered pair, the type of the second term can depend on the value of the first term. For example, the first term of the pair might be a natural number and the second term's type might be a vector of length equal to the first term. Such a type would be written:. Using set-theory terminology, this is similar to an indexed disjoint unions of sets.
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In the case of usual ordered pairs, the type of the second term does not depend on the value of the first term. An example of a dependently-typed 3-tuple is two integers and a proof that the first integer is smaller than the second integer, described by the type:.
Its type would be:.
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As with typical function types, they consist of an input type and an output type. They are more powerful than typical function types however, in that the return type can depend on the input value. Functions in type theory are different from set theory. In set theory, you look up the argument's value in a set of ordered pairs. In type theory, the argument is substituted into a term and then computation "reduction" is applied to the term. This type could be written more consistently as:.
The type would be. The terms of that new type represent proofs that the pair reduce to the same canonical term. Equality of proofs is an area of active research in proof theory and has led to the development of homotopy type theory and other type theories. Inductive types allow the creation of complex, self-referential types. For example, a linked list of natural numbers is either an empty list or a pair of a natural number and another linked list.
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Inductive types can be used to define unbounded mathematical structures like trees, graphs, etc.. Proofs on inductive types are made possible by induction. Each new inductive type comes with its own inductive rule. Inductive types in intuitionistic type theory are defined in terms of W-types, the type of well-founded trees. Later work in type theory generated coinductive types, induction-recursion, and induction-induction for working on types with more obscure kinds of self-referentiality. Higher inductive types allow equality to be defined between terms.
The universe types allow proofs to be written about all the types created with the other type constructors. Universe types are a tricky feature of type theories. Later research covered topics such as "super universes", "Mahlo universes", and impredicative universes. The formal definition of intuitionistic type theory is written using judgements. This second level of the type theory can be confusing, particularly where it comes to equality. It is a statement that two terms reduce to the same canonical term.
Obviously, terms of this type are generated using the term-equality judgement. An object that depends on an object from another type can be done two ways. If the object is "abstracted", then it is written. The object-depending-on-object can also be declared as a constant as part of a recursive type. An example of a recursive type is:. It is not associated with an abstraction.
So, objects and types and these relations are used to express formulae in the theory. The following styles of judgements are used to create new objects, types and relations from existing ones:. By convention, there is a type that represents all other types. To implement logic, each proposition is given its own type. The objects in those types represent the different possible ways to prove the proposition. About this Item: Springer. Condition: Fair.
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