What is a type type theory?

A “type” in type theory has a role similar to a “type” in a programming language: it dictates the operations that can be performed on a term and, for variables, the possible values it might be replaced with. Some type theories serve as alternatives to set theory as a foundation of mathematics.

What is a constructive type?

Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972.

What is type logic?

Type logic is a logical system based on Russell’s theory of types. Every expression of a type-logical language belongs to a particular type indicating the set-theoretical denotation of that expression. There are two basic types, the type e (from entity) and the type t (from truth value).

What is a type church?

Church’s type theory, aka simple type theory, is a formal logical language which includes classical first-order and propositional logic, but is more expressive in a practical sense. It is used, with some modifications and enhancements, in most modern applications of type theory.

What are the 3 types of theory?

Although there are many different approaches to learning, there are three basic types of learning theory: behaviorist, cognitive constructivist, and social constructivist.

What is Russell’s theory of types?

theory of types, in logic, a theory introduced by the British philosopher Bertrand Russell in his Principia Mathematica (1910–13) to deal with logical paradoxes arising from the unrestricted use of predicate functions as variables.

What is cubical type theory?

Cubical type theory is a version of homotopy type theory in which univalence is not just an axiom but a theorem, hence, since this is constructive, has “computational content”. Cubical type theory models the infinity-groupoid-structure implied by Martin-Löf identity types on constructive cubical sets, whence the name.

Is type theory constructive?

Intuitionistic type theory (also constructive type theory or Martin-Löf type theory) is a formal logical system and philosophical foundation for constructive mathematics.

What are the 4 types of logic?

The four main logic types are:

  • Informal logic.
  • Formal logic.
  • Symbolic logic.
  • Mathematical logic.

What are the 3 types of church?

Churches Militant, Penitent, and Triumphant.

Is Catholicism a form of Christianity?

Christianity is an important world religion that stems from the life, teachings, and death of Jesus. Roman Catholicism is the largest of the three major branches of Christianity. Thus, all Roman Catholics are Christian, but not all Christians are Roman Catholic.

What is the theory of semantics in philosophy?

Within philosophy, proof-theoretic semantics has mostly figured under the heading “theory of meaning”. This terminology follows Dummett, who claimed that the theory of meaning is the basis of theoretical philosophy, a view which he attributed to Frege.

What is proof theoretic semantics?

It is based on the fundamental assumption that the central notion in terms of which meanings are assigned to certain expressions of our language, in particular to logical constants, is that of proof rather than truth. In this sense proof-theoretic semantics is semantics in terms of proof.

What is Martin-Löf’s type theory?

However, Martin-Löf’s type theory has at least two characteristic features which go beyond other approaches in proof-theoretic semantics: The consideration of proof objects and the corresponding distinction between proofs-as-objects and proofs-as-demonstrations.

What is “standard semantics?

“Standard semantics” here not only means standard proof-theoretic semantics, but also classical model-theoretic semantics, where these dogmas are assumed as well.