TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, and
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (inclu ...
, a type theory is the formal presentation of a specific
type system In computer programming, a type system is a logical system comprising a set of rules that assigns a property called a type to every "term" (a word, phrase, or other set of symbols). Usually the terms are various constructs of a computer prog ...
, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
as a foundation of mathematics. Two influential type theories that were proposed as foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's
intuitionistic type theory 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 ...
. Most computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of Inductive Constructions.

# History

Type theory was created to avoid a paradox in a mathematical foundation based on
naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It ...
and
formal logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
.
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains a ...
, which was discovered by
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
, existed because a set could be defined using "all possible sets", which included itself. Between 1902 and 1908, Bertrand Russell proposed various "theories of type" to fix the problem. By 1908 Russell arrived at a "ramified" theory of types together with an " axiom of reducibility" both of which featured prominently in Whitehead and Russell's '' Principia Mathematica'' published between 1910 and 1913. This system avoided Russell's paradox by creating a hierarchy of types and then assigning each concrete mathematical entity to a type. Entities of a given type are built exclusively of subtypes of that type, thus preventing an entity from being defined using itself. Russell's theory of types ruled out the possibility of a set being a member of itself. Types were not always used in logic. There were other techniques to avoid Russell's paradox.''Stanford Encyclopedia of Philosophy'
(rev. Mon Oct 12, 2020) Russell’s Paradox
3. Early Responses to the Paradox
Types did gain a hold when used with one particular logic, Alonzo Church's
lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computatio ...
. The most famous early example is Church's simply typed lambda calculus. Church's theory of types helped the formal system avoid the Kleene–Rosser paradox that afflicted the original untyped lambda calculus. Church demonstrated that it could serve as a foundation of mathematics and it was referred to as a
higher-order logic mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expre ...
. The phrase "type theory" now generally refers to a typed system based around lambda calculus. One influential system is Per Martin-Löf's
intuitionistic type theory 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 ...
, which was proposed as a foundation for constructive mathematics. Another is Thierry Coquand's calculus of constructions, which is used as the foundation by Coq, Lean, and other "proof assistants" (computerized proof writing programs). Type theories are an area of active research, as demonstrated by homotopy type theory.

# Introduction

There are many type theories, which makes it difficult to produce a comprehensive taxonomy; this article is not an exhaustive categorization. What follows is an introduction for those unfamiliar with type theory, covering some of the major approaches.

## Basics

### Terms and types

In type theory, every term has a type. A term and its type are often written together as "''term'' : ''type''". A common type to include in a type theory is the
Natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
, often written as "$\mathbb N$" or "nat". Another is
Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
values. So, some very simple terms with their types are: * 0 : nat * 42 : nat * true : bool Terms can be built out of other terms using function calls. In type theory, a function call is called "function application". Function application takes a term of a given type and results in a term of another given type. Function application is written "''function'' ''argument'' ''argument'' ...", instead of the conventional "''function''(''argument'',''argument'', ...)". For natural numbers, it is possible to define a function called "add" that takes two natural numbers. Thus, some more terms with their types are: * add 0 0 : nat * add 2 3 : nat * add 1 (add 1 (add 1 0)) : nat In the last term, parentheses were added to indicate the order of operations. Technically, most type theories require the parentheses to be present for every operation, but, in practice, they are not written and authors assume readers can use precedence and associativity to know where they are. For similar ease, it is a common notation to write "$x + y$" instead of "add $x$ $y$". So, the above terms might be rewritten as: * 0 + 0 : nat * 2 + 3 : nat * 1 + (1 + (1 + 0)) : nat Terms may also contain variables. Variables always have a type. So, assuming "x" and "y" are variables of type "nat", the following are also valid terms: * x : nat * x + 2 : nat * x + (x + y) : nat There are more types than "nat" and "bool". We have already seen the term "add", which is not a "nat", but a function that, when applied to two "nat"s, computes to a "nat". The type of "add" will be covered later. First, we need to describe "computes to".

### Computation

Type theory has a built-in notation of computation. The following terms are all different: * 1 + 4 : nat * 3 + 2 : nat * 0 + 5 : nat but they all compute to the term "5 : nat". In type theory, we use the words "reduction" and "reduce" to refer to computation. So, we say "0 + 5 : nat" reduces to "5 : nat". It can be written "0 + 5 : nat $\twoheadrightarrow$ 5 : nat". The computation is mechanical, accomplished by rewriting the term's syntax. Terms that contain variables can be reduced too. So the term "x + (1 + 4) : nat" reduces to "x + 5 : nat". (We can reduce any sub-term within a term, thanks to the Church-Rosser theorem.) A term without any variables that cannot be reduced further is a "canonical term". All the terms above reduce to "5 : nat", which is a canonical term. The canonical terms of the natural numbers are: * 0 : nat * 1 : nat * 2 : nat * etc. Obviously, terms that compute to the same term are equal. So, assuming "x : nat", the terms "x + (1 + 4) : nat" and "x + (4 + 1) : nat" are equal because they both reduce to "x + 5 : nat". When two terms are equal, they can be substituted for each other. Equality is a complex topic in type theory and there are many kinds of equality. This kind of equality, where two terms compute to the same term, is called "judgemental equality".

## Functions

In type theory, functions are terms. Functions can either be lambda terms or defined "by rule".

### Lambda terms

A lambda term looks like "(λ ''variablename'' : ''type1'' . ''term'')" and has type "''type1'' $\to$ ''type2''". The type "''type1'' $\to$ ''type2''" indicates that the lambda term is a function that takes a parameter of type "''type1''" and computes to a term of type "''type2''". The term inside the lambda term must be a value of "''type2''", assuming the variable has type "''type1''". An example of a lambda term is this function which doubles its argument: * (λ x : nat . (add x x)) : nat $\to$ nat The variable name is "x" and the variable has type "nat". The term "(add x x)" has type "nat", assuming "x : nat". Thus, the lambda term has type "nat $\to$ nat", which means if it is given a "nat" as an argument, it will compute to a "nat". Reduction (a.k.a. computation) is defined for lambda terms. When the function is applied (a.k.a. called), the argument is
substituted A substitution reaction (also known as single displacement reaction or single substitution reaction) is a chemical reaction during which one functional group in a chemical compound is replaced by another functional group. Substitution reactions ar ...
for the parameter. Earlier, we saw that function application is written by putting the parameter after the function term. So, if we want to call the above function with the parameter "5" of type "nat", we write: * (λ x : nat . (add x x)) 5 : nat The lambda term was type "nat $\to$ nat", which meant that given a "nat" as an argument, it will produce a term of type "nat". Since we have given it the argument "5", the above term has type "nat". Reduction works by substituting the argument "5" for the parameter "x" in the term "(add x x)", so the term computes to: * (add 5 5) : nat which obviously computes to * 10 : nat A lambda term is often called an "anonymous function" because it has no name. Often, to make things easier to read, a name is given to a lambda term. This is merely a notation and has no mathematical meaning. Some authors call it "notational equality". A name might be given to the function above using the notation: * double : nat $\to$ nat ::= (λ x : nat . (add x x)) This is the same function as above, just a different way to write it. So the term * double 5 : nat still computes to * 10 : nat

### Dependent typing

Dependent typing is when the type returned by a function depends on the value of its argument. For example, when a type theory has a rule that defines the type "bool", it also defines the function "if". The function "if" takes 3 arguments and "if true b c" computes to "b" and "if false b c" computes to "c". But what is the type of "if a b c"? If "b" and "c" have the same type, it is obvious: "if a b c" has the same type as "b" and "c". Thus, assuming "a : bool", * if a 2 4 : nat * if a false true : bool But if "b" and "c" have different types, then the type of "if a b c" depends on the value of "a". We use the symbol "Π" to indicate a function that takes an argument and returns a type. Assuming we have some types "B" and C" and "a : bool", "b : B" and "c : C", then * if a b c : (Π a : bool . B $\to$ C $\to$ if a B C) That is, the type of the "if" term is either the type of the second or third argument, depending on the value of the first argument. In actuality, "if a B C" isn't defined using "if", but that gets into details too complicated for this introduction. Because the type can contain computation, dependent typing is amazingly powerful. When mathematicians say "there exists a number $x$ such that $x$ is prime" or "there exists a number $x$ such that property $P\left(x\right)$ holds", it can be expressed as a dependent type. That is, the property is proven for the specific "$x$" and that is visible in the type of the result. There are many details to dependent typing. They are too long and complicated for this introduction. See the article on dependent typing and the lambda cube for more information.

### Universes

Π-terms return a type. So what is the type of their return value? Well, there must be a type that contains types. A type that contains other types is called a "
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univer ...
". It is often written with the symbol $U$. Sometimes there is a hierarchy of universes, with "$U_0$ : $U_1$", "$U_1$ : $U_2$", etc.. If a universe contains itself, it can lead to paradoxes like Girard's Paradox. For example:

## Common "by rule" types and terms

Type theories are defined by their
rules of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
. There are rules for a "functional core", described above, and rules that create types and terms. Below is a non-exhaustive list of common types and their associated terms. The list ends with "inductive types", which is a powerful technique that is able to construct all the other ones in the list. The mathematical foundations used by the proof assistants "Coq" and "Lean" are based on the "Calculus for Inductive Constructions" which is the "Calculus of Constructions" (its "functional core") with inductive types.

### Empty type

The empty type has no terms. The type is usually written "$\bot$" or "$\mathbb 0$". It is used to show that something is uncomputable. If for a type "A", you can create a function of type "A $\to \bot$", you know that "A" has no terms. An example for the type "A" might be "there exists a number $x$ such that both $x$ is even and $x$ is odd". (See "Product Type" below for how the example "A" is constructed.) When a type has no terms, we say it is "uninhabited".

### Unit type

The
unit type In the area of mathematical logic and computer science known as type theory, a unit type is a type that allows only one value (and thus can hold no information). The carrier (underlying set) associated with a unit type can be any singleton set. ...
has exactly 1 canonical term. The type is written "$\top$" or "$\mathbb 1$" and the single canonical term is written "*". The unit type is used to show that something exists or is computable. If for a type "A", you can create a function of type "$\top \to$ A", you know that "A" has one or more terms. When a type has at least 1 term, we say it is "inhabited".

### Boolean type

The Boolean type has exactly 2 canonical terms. The type is usually written "bool" or "$\mathbb B$" or "$\mathbb 2$". The canonical terms are usually "true" and "false". The Boolean type is defined with an eliminator function "if" such that: * if true b c $\twoheadrightarrow$ b * if false b c $\twoheadrightarrow$ c

### Product type

The
product type In programming languages and type theory, a product of ''types'' is another, compounded, type in a structure. The "operands" of the product are types, and the structure of a product type is determined by the fixed order of the operands in the prod ...
has terms that are ordered pairs. For types "A" and "B", the product type is written "A $\times$ B". Canonical terms are created by the constructor function "pair". The terms are "pair a b", where "a" is a term of type "A" and "b" is a term of type "B". The product type is defined with eliminator functions "first" and "second" such that: * first (pair a b) $\twoheadrightarrow$ a * second (pair a b) $\twoheadrightarrow$ b Besides ordered pairs, this type is used for the logical operator "and", because it holds an "A" and a "B". It is also used for
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
, because it holds one of both types. If a type theory has dependent typing, it has dependent pairs. In a dependent pair, the second type depends on the value of the first term. Thus, the type is written "$\Sigma$ a:A . B(a)" where "B" has type "A $\to$ U". It is useful when showing
existence Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being. Etymology The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistenti ...
of an "a" with property "B(a)".

### Sum type

The sum type is a "tagged union". That is, for types "A" and "B", the type "A + B" holds either a term of type "A" or a term of type "B" and it knows which one it holds. The type comes with the constructors "injectionLeft" and "injectionRight". The call "injectionLeft a" takes "a : A" and returns a canonical term of type "A + B". Similarly, injectionRight b" takes "b : B" and returns a canonical term of type "A + B". The type is defined with an eliminator function "match" such that for a type "C" and functions "f : A $\to$ C" and "g : B $\to$ C": * match (injectionLeft a) C f g $\twoheadrightarrow$ (f a) * match (injectionRight b) C f g $\twoheadrightarrow$ (g b) The sum type is used for logical or and for union.

### Natural numbers

The natural numbers are usually implemented in the style of
Peano Arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
. There is a canonical term, "0 : nat" for zero. Canonical values larger than zero use the constructor function "S : nat $\to$ nat". Thus, "S 0" is one. "S (S 0)" is two. "S (S (S 0)))" is three. Etc. The decimal numbers are just notationally equal to those terms. * 1 : nat ::= S 0 * 2 : nat ::= S (S 0) * 3 : nat ::= S (S (S 0)) * ... The natural numbers are defined with an eliminator function "R" that uses recursion to define a function for all nats. It takes a function "P : nat $\to$ U" which is the type of the function to define. It also takes a term "PZ : P 0" which is the value at zero and a function "PS : P n $\to$ P (S n)" which says how to transform the value at "n" into the value at "n + 1". Thus, its computation rules are: * R P PZ PS 0 $\twoheadrightarrow$ PZ * R P PZ PS (S $n$) $\twoheadrightarrow$ PS (R P PZ PS $n$) The function "add", that was used earlier, can be defined using "R". * add : nat$\to$nat$\to$nat ::= R (λ n:nat . nat$\to$nat) (λ n:nat . n) (λ g:nat$\to$nat . (λ m:nat . S (g m)))

### Identity type

The identity type is the third concept of equality in type theory. The first is "notational equality", which is for definitions like "2 : nat ::= (S (S 0))" that have no mathematical meaning but are useful to readers. The second is "judgemental equality", which is when two terms compute to the same term, like "x + (1 + 4)" and "x + (4 + 1)", which both compute to "x + 5". But type theory needs another form of equality, known as the "identity type" or "propositional equality". The reason it needs the identity type is because some equal terms do not compute to the same term. Assuming "x : nat", the terms "x + 1" and "1 + x" do not compute to the same term. Recall that "+" is a notation for the function "add", which is a notation for the function "R". We cannot compute on "R" until the value for "x" is specified and, until it is specified, two different calls to "R" will not compute to the same term. An identity type requires two terms "a" and "b" of the same type and is written "a = b". So, for "x + 1" and "1 + x", the type would be "x+1 = 1+x". Canonical terms are created with the constructor "reflexivity". The call "reflexivity a" takes a term "a" and returns a canonical term of the type "a = a". Computation with the identity type is done with the eliminator function "J". The function "J" lets a term dependent on "a", "b", and a term of type "a = b" to be rewritten so that "b" is replaced by "a". While "J" is one directional, only able to substitute "b" with "a", it can be proven that the identity type is reflexive,
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and transitive. If the canonical terms are always "a=a" and "x+1" does not compute to the same term as "1+x", how do we create a term of "x+1 = 1+x"? We use the "R" function. (See "Natural Numbers" above.) The "R" function's argument "P" is defined to be "(λ x:nat . x+1 = 1+x)". The other arguments act like the parts of an induction proof, where "PZ : P 0" becomes the base case "0+1 = 1+0" and "PS : P n $\to$ P (S n)" becomes the inductive case. Essentially, this says that when "x+1 = 1+x" has "x" replaced with a canonical value, the expression will be the same as "reflexivity (x+1)". This application of the function "R" has type "x : nat $\to$ x+1 = 1+x". We can use it and the function "J" to substituted "1+x" for "x+1" in any term. In this way, the identity type is able to capture equalities that are not possible with judgemental equality. To be clear, it is possible to create the type "0 = 1", but there will not be a way to create terms of that type. Without a term of type "0 = 1", it will not be possible to use the function "J" to substitute "0" for "1" in another term. The complexities of equality in type theory make it an active research area, see homotopy type theory.

### Inductive types

Inductive types is a way to create a large variety of types. In fact, all the types described above and more can be defined using the rules of inductive types. Once the type's constructors are specified, the eliminator functions and computation is determined by structural recursion. There are similar, more powerful ways to create types. These include induction-recursion and induction-induction. There is also a way to create similar types using only lambda terms, called Scott encoding. (NOTE: Type theories do not usually include coinductive types. They represent an infinite data type and most type theories limit themselves to functions that can be proven to halt.)

# Differences from set theory

The traditional foundation for mathematics has been set theory paired with a logic. The most common one cited is
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
, known as "ZF" or, with the
Axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection o ...
, "ZFC". Type theories differ from this foundation in a number of ways. * Set theory has both
rules Rule or ruling may refer to: Education * Royal University of Law and Economics (RULE), a university in Cambodia Human activity * The exercise of political or personal control by someone with authority or power * Business rule, a rule pert ...
and
axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
, while type theories only have rules. Set theories are built on top of logic. Thus, ZFC is defined by both the rules of
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...
and its own axioms. (An
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
is a logical statement accepted as true without a logical derivation.) Type theories, in general, do not have axioms and are defined by their rules of inference. * Set theory and logic have the
law of excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncon ...
. That is, every theorem is true or false. When a type theory defines the concepts of "and" and "or" as types, it leads to
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, system ...
, which does not have the law of excluded middle. However, the law can be proven for some types. * In set theory, an element is not restricted to one set. The element can appear in subsets and unions with other sets. In type theory, terms (generally) belong to only one type. Where a subset would be used, type theory can use a predicate function or use a dependently-typed product type, where each element $x$ is paired with a proof that the subset's property holds for $x$. Where a union would be used, type theory uses the sum type, which contains new canonical terms. * Type theory has a built-in notion of computation. Thus, "1+1" and "2" are different terms in type theory, but they compute to the same value. Moreover, functions are defined computationally as lambda terms. In set theory, "1+1=2" means that "1+1" is just another way to refer the value "2". Type theory's computation does require a complicated concept of equality. * Set theory usually encodes numbers as sets. (0 is the empty set, 1 is a set containing the empty set, etc. See Set-theoretic definition of natural numbers.) Type theory can encode numbers as functions using
Church encoding In mathematics, Church encoding is a means of representing data and operators in the lambda calculus. The Church numerals are a representation of the natural numbers using lambda notation. The method is named for Alonzo Church, who first encoded ...
or more naturally as inductive types. The constructors "0" and "S" created by the inductive type closely resemble Peano's axioms. * Set theory has
set-builder notation In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy. Definin ...
. It can create any set that can be defined. This allows it to create
Uncountable set In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal n ...
s. Type theories are syntactic, which limits them to a
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
terms. Additionally, most type theories require computation to always halt and limit themselves to recursively generable terms. As a result, most type theories do not use the
Real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s but the Computable numbers. * In set theory, the
Axiom of Choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection o ...
is an axiom and is controversial, particularly when applied to uncountable sets. In type theory, the equivalent statement is a theorem (type) and is provable (inhabited by a term). * In type theory, proofs are mathematical objects. The type "x+1 = 1+x" cannot be used unless there is a term of the type. That term represents a proof that "x+1 = 1+x". Thus, type theory opens up proofs to be studied as mathematical objects. Proponents of type theory will also point out its connection to constructive mathematics through the BHK interpretation, its connected to logic by the Curry–Howard isomorphism, and its connections to
Category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
.

# Technical details

A type theory is a
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
. It is a collection of
rules of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
that result in judgements. Most logics have judgements meaning "The term $x$ is true." or "The term $x$ is a
well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can ...
.". A type theory has additional judgements that define types and relate terms to types.

## Terms

A term in logic is recursively defined as a constant symbol, variable, or a function application, where a term is applied to another term. Some constant symbols will be "0" of the natural numbers, "true" of the Booleans, and functions like "S" and "if". Thus some terms are "0", "(S 0)", "(S (S x))", and "if true 0 (S 0)".

## Judgements

Most type theories have 4 judgements: * "$T$ is a type." * "$t$ is a term of type $T$." * "Type $T_1$ is equal to type $T_2$." * "Terms $t_1$ and $t_2$ are both of type $T$ and are equal." The judgements can be made under an assumption. Thus, we might say, "assuming $x$ is a term of type "bool" and $y$ is a term of type "nat" , (if x y y) is a term of type "nat"". The mathematical notation for assumptions is a comma-separate list of "''term'' : ''type''" that precede the turnstile symbol '$\vdash$'. Thus, the example statement is formally written: * x:bool, y:nat $\vdash$ (if x y y) : nat If there are no assumptions, there will be nothing to the left of the turnstile. * $\vdash$ S : nat $\to$ nat The list of assumptions is called the "context". It is very common to see the symbol '$\Gamma$' used to represent some or all of the assumptions. Thus, the formal notation for the 4 different judgements is usually: (NOTE: The judgement of equality of terms is where the phrase "judgemental equality" comes from. ) The judgements enforce that every term has a type. The type will restrict which rules can be applied to a term.

## Rules

A type theory's
rules Rule or ruling may refer to: Education * Royal University of Law and Economics (RULE), a university in Cambodia Human activity * The exercise of political or personal control by someone with authority or power * Business rule, a rule pert ...
say what judgements can be made, based on the existence of other judgements. The rules are expressed using a horizontal line, with the required input judgements above the line and the resulting judgement below the line. The rule for creating a lambda term is: $\begin \Gamma , a:A \vdash b : B \\ \hline \Gamma \vdash ( \lambda a:A . b ) : A \to B \\ \end$ The judgements required to create the lambda term go above the line. In this case, only one judgement is required. It is that there is some term "b" of some type "B", assuming there is some term "a" of some type "A" and some other assumptions "$\Gamma$". (Note: "$\Gamma$" "a", "A", "b", and "B" are all metavariables in the rule.) The resulting judgement goes below the line. This rule's resulting judgement states that the new lambda term has type "A $\to$ B" under the other assumptions $\Gamma$. The rules are syntactic and work by rewriting. Thus, the metavariables like "$\Gamma$", "a", "A", etc. may actually consist of complex terms that contain many function applications, not just single symbols. To generate a particular judgement in type theory, there must be a rule to generate it. Then, there must be rules to generate all of that rule's required inputs. And then rules for all the inputs for those rules. The applied rules form a proof tree. This is usually drawn Gentzen-style, where the target judgement (root) is at the bottom and rules that do not require any inputs (leaves) at the top. (See Natural deduction#Proofs_and_type_theory.) An example of a rule that does not require any inputs is one that states there is a term "0" of type "nat": $\begin \hline \vdash 0 : nat \\ \end$ A type theory usually has a number of rules, including ones to: * create a context * add an assumption to the context ("weakening") * rearrange the assumptions * use an assumption to create a variable * define reflexivity, symmetry and transitivity for judgemental equality * define substitution for application of lambda terms * all the interactions of equality, substitution, etc. * define universes Also, for each "by rule" type, there are 4 different kinds of rules * "type formation" rules say how to create the type * "term introduction" rules define the canonical terms and constructor functions, like "pair" and "S". * "term elimination" rules define the other functions like "first", "second", and "R". * "computation" rules specify how computation is performed with the type-specific functions. Examples of rules:
Rules to Martin-Löf's Intuitionistic Type Theory
* Appendix A.2 o
Homotopy Type Theory
book

## Properties of type theories

Terms usually belong to a single type. However, there are set theories that define "subtyping". Computation takes place by repeated application of rules. Many type theories are strongly normalizing, which means that any order of applying the rules will always end in the same result. However, some are not. In a normalizing type theory, the one-directional computation rules are called "reduction rules" and applying the rules "reduces" the term. If a rule is not one-directional, it is called a "conversion rule". Some combinations of types are equivalent to other combinations of types. When functions are considered "exponentiation", the combinations of types can be written similar to algebraic identities. Thus, $+ A \cong A$, $\times A \cong A$, $+ \cong$, $A^ \cong A^B \times A^C$, $A^ \cong \left(A^B\right)^C$.

## Axioms

Most type theories do not have
axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
. This is because a type theory is defined by its rules of inference. (See "
Rules Rule or ruling may refer to: Education * Royal University of Law and Economics (RULE), a university in Cambodia Human activity * The exercise of political or personal control by someone with authority or power * Business rule, a rule pert ...
" above). This is a source of confusion for people familiar with Set Theory, where a theory is defined by both the rules of inference for a logic (such as
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...
) and axioms about sets. Sometimes, a type theory will add a few axioms. An axiom is a judgement that is accepted without a derivation using the rules of inference. They are often added to ensure properties that cannot be added cleanly through the rules. Axioms can cause problems if they introduce terms without a way to compute on those terms. That is, axioms can interfere with the normalizing property of the type theory. Some commonly encountered axioms are: * "Axiom K" ensures "uniqueness of identity proofs". That is, that every term of an identity type is equal to reflexivity. * "Univalence Axiom" holds that equivalence of types is equality of types. The research into this property led to cubical type theory, where the property holds without needing an axiom. * "Law of Excluded Middle" is often added to satisfy users who want
classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this clas ...
, instead of intuitionistic logic. The
Axiom of Choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection o ...
does not need to be added to type theory, because in most type theories it can be derived from the rules of inference. This is because of the
constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...
nature of type theory, where proving that a value exists requires a method to compute the value. The Axiom of Choice is less powerful in type theory than most set theories, because type theory's functions must be computable and, being syntax-driven, the number of terms in a type must be countable. (See .)

## Decision problems

A type theory is naturally associated with the
decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whet ...
of type inhabitation.

### Type inhabitation

The decision problem of ''type inhabitation'' (abbreviated by $\exists e.\Gamma \vdash e : \tau?$) is: :Given a type environment $\Gamma$ and a type $\tau$, decide whether there exists a term $e$ that can be assigned the type $\tau$ in the type environment $\Gamma$. Girard's paradox shows that type inhabitation is strongly related to the consistency of a type system with Curry–Howard correspondence. To be sound, such a system must have uninhabited types. The opposition of terms and types can also be views as one of ''implementation'' and ''specification''. By program synthesis (the computational counterpart of) type inhabitation (see below) can be used to construct (all or parts of) programs from specification given in form of type information.

### Type inference

Many programs that work with type theory (e.g., interactive theorem provers) also do type inferencing. It lets them select the rules that the user intends, with fewer actions by the user.

## Research areas

Homotopy type theory differs from
intuitionistic type theory 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 ...
mostly by its handling of the equality type. In 2016 cubical type theory was proposed, which is a homotopy type theory with normalization.

# Interpretations

Type theory has connections to other areas of mathematics. Proponents of type theory as a foundation often mention these connections as justification for its use.

## Types are propositions; terms are proofs

When used as a foundation, certain types are interpreted as
propositions In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
(statements that can be proven) and a term of the type is a proof of that proposition. Thus, the type "Π x:nat . x+1=1+x" represents that, for any "x" of type "nat", "x+1" and "1+x" are equal. And a term of that type represents its proof.

## Curry-Howard correspondence

The Curry–Howard correspondence is the observed similarity between logics and programming languages. The implication in logic, "A $\to$ B" resembles a function from type "A" to type "B". For a variety of logics, the rules are a similar to expression in a programming language's types. The similarity goes farther, as applications of the rules resemble programs in the programming languages. Thus, the correspondence is often summarized as "proofs as programs". The logic operators "
for all In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In oth ...
" and " exists" led Per Martin-Löf to invent dependent type theory.

## Intuitionistic logic

When some types are interpreted as propositions, there is a set of common types that can be used to connect them to make a logic out of types. However, that logic is not
classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this clas ...
but
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, system ...
. That is, it does not have the
law of excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncon ...
nor double negation. There is a natural relation of types to logical propositions. If "A" is a type representing a proposition, being able to create a function of type "$\top \to$ A" indicates that A has a proof and being able to create the function "A $\to \bot$" indicates that A does not have a proof. That is, inhabitable types are proven and uninhabitable types are disproven. ''WARNING: This interpretation can lead to a lot of confusion. A type theory may have ''terms'' "true" and "false" of type "bool", which act like a
Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
, and at the same time have ''types'' $\top$ and $\bot$ to represent "true" (provable) and "false" (disproven), as part of a intuitionistic logic for proposition.'' Under this intuitionistic interpretation, there are common types that act as the logical operators: But under this interpretation, there is no law of excluded middle. That is, there is no term of type Π A . A + (A $\to \bot$). Likewise, there is no double negation. There is no term of type Π A . ((A $\to \bot$) $\to \bot$) $\to$ A. (Note: Intuitionistic logic does allow $\lnot \lnot \lnot A \to \lnot A$ and there is a term of type (((A $\to \bot$) $\to \bot$) $\to \bot$) $\to$ (A $\to \bot$).) Thus, the logic-of-types is an intuitionistic logic. Type theory is often cited as an implementation of the
Brouwer–Heyting–Kolmogorov interpretation In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the r ...
. It is possible to include the law of excluded middle and double negation into a type theory, by rule or assumption. However, terms may not compute down to canonical terms and it will interfere with the ability to determine if two terms are judgementally equal to each other.

## Constructive mathematics

Per Martin-Löf proposed his intuitionistic type theory as a foundation for constructive mathematics. Constructive mathematics requires when proving "There exists an $x$ with property P($x$)", there must be a particular $x$ and a proof that it has property "P". In type theory, existence is accomplished using the dependent product type and, its proof, requires a term of that type. For the term $t$, "first $t$" will produce the $x$ and "second $t$" will produce the proof of P($x$). An example of a non-constructive proof is a "proof by contradiction". The first step is assuming that $x$ does not exist and refuting it by contradiction. The conclusion from that step is "it is not the case that $x$ does not exist". The last step is, by double negation, concluding that $x$ exists. To be clear, constructive mathematics still allows "refute by contradiction". It can prove that "it is not the case that $x$ does not exist". But constructive mathematics does not allow the last step of removing the double negation to conclude that $x$ exists. Constructive mathematics has often used intutionistic logic, as evidenced by the
Brouwer–Heyting–Kolmogorov interpretation In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the r ...
. Most of the type theories proposed as foundations are constructive. This includes most of the ones used by proof assistants. It is possible to add non-constructive features to a type theory, by rule or assumption. These include operators on continuations such as call with current continuation. However, these operators tend to break desirable properties such as canonicity and parametricity.

## Category theory

Although the initial motivation for
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
was far removed from foundationalism, the two fields turned out to have deep connections. As John Lane Bell writes: "In fact categories can ''themselves'' be viewed as type theories of a certain kind; this fact alone indicates that type theory is much more closely related to category theory than it is to set theory." In brief, a category can be viewed as a type theory by regarding its objects as types (or sorts), i.e. "Roughly speaking, a category may be thought of as a type theory shorn of its syntax." A number of significant results follow in this way: *
cartesian closed categories In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in ...
correspond to the typed λ-calculus ( Lambek, 1970); * C-monoids (categories with products and exponentials and one non-terminal object) correspond to the untyped λ-calculus (observed independently by Lambek and Dana Scott around 1980); * locally cartesian closed categories correspond to Martin-Löf type theories (Seely, 1984). The interplay, known as
categorical logic __NOTOC__ Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, cate ...
, has been a subject of active research since then; see the monograph of Jacobs (1999) for instance. Homotopy type theory attempts to combine type theory and category theory. It focuses on equalities, especially equalities between types.

# List of type theories

## Major

* Simply typed lambda calculus which is a
higher-order logic mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expre ...
*
intuitionistic type theory 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 ...
* system F * LF is often used to define other type theories * calculus of constructions and its derivatives

## Minor

* Automath * ST type theory * UTT (Luo's Unified Theory of dependent Types) * some forms of
combinatory logic Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of com ...
* others defined in the lambda cube (also known as pure type systems) * others under the name typed lambda calculus

## Active research

* Homotopy type theory explores equality of types * Cubical Type Theory is an implementation of homotopy type theory

# Applications

## Mathematical foundations

The first computer proof assistant, called Automath, used type theory to encode mathematics on a computer. Martin-Löf specifically developed
intuitionistic type theory 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 ...
to encode ''all'' mathematics to serve as a new foundation for mathematics. There is ongoing research into mathematical foundations using homotopy type theory. Mathematicians working in
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
. This led to proposals such as Lawvere's Elementary Theory of the Category of Sets (ETCS). Homotopy type theory continues in this line using type theory. Researchers are exploring connections between dependent types (especially the identity type) and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classi ...
(specifically
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
).

## Proof assistants

Much of the current research into type theory is driven by proof checkers, interactive proof assistants, and automated theorem provers. Most of these systems use a type theory as the mathematical foundation for encoding proofs, which is not surprising, given the close connection between type theory and programming languages: * LF is used by
Twelf Twelf is an implementation of the logical framework LF developed by Frank Pfenning and Carsten Schürmann at Carnegie Mellon University. It is used for logic programming and for the formalization of programming language theory. Introduction At i ...
, often to define other type theories; * many type theories which fall under
higher-order logic mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expre ...
are used by the HOL family of provers and PVS; * computational type theory is used by NuPRL; * calculus of constructions and its derivatives are used by Coq, Matita, and Lean; * UTT (Luo's Unified Theory of dependent Types) is used by Agda which is both a programming language and proof assistant Many type theories are supported by
LEGO Lego ( , ; stylized as LEGO) is a line of plastic construction toys that are manufactured by The Lego Group, a privately held company based in Billund, Denmark. The company's flagship product, Lego, consists of variously colored interlock ...
and
Isabelle Isabel is a female name of Spanish origin. Isabelle is a name that is similar, but it is of French origin. It originates as the medieval Spanish form of '' Elisabeth'' (ultimately Hebrew '' Elisheva''), Arising in the 12th century, it became popu ...
. Isabelle also supports foundations besides type theories, such as ZFC.
Mizar Mizar is a second- magnitude star in the handle of the Big Dipper asterism in the constellation of Ursa Major. It has the Bayer designation ζ Ursae Majoris ( Latinised as Zeta Ursae Majoris). It forms a well-known naked eye ...
is an example of a proof system that only supports set theory.

## Programming languages

Any
static program analysis In computer science, static program analysis (or static analysis) is the analysis of computer programs performed without executing them, in contrast with dynamic program analysis, which is performed on programs during their execution. The term ...
, such as the type checking algorithms in the semantic analysis phase of
compiler In computing, a compiler is a computer program that translates computer code written in one programming language (the ''source'' language) into another language (the ''target'' language). The name "compiler" is primarily used for programs th ...
, has a connection to type theory. A prime example is Agda, a programming language which uses UTT (Luo's Unified Theory of dependent Types) for its type system. The programming language ML was developed for manipulating type theories (see LCF) and its own type system was heavily influenced by them.

## Linguistics

Type theory is also widely used in formal theories of semantics of
natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languag ...
s, especially
Montague grammar __notoc__ Montague grammar is an approach to natural language semantics, named after American logician Richard Montague. The Montague grammar is based on mathematical logic, especially higher-order predicate logic and lambda calculus, and makes ...
Cooper, Robin.
Type theory and semantics in flux
" Handbook of the Philosophy of Science 14 (2012): 271-323.
and its descendants. In particular,
categorial grammar Categorial grammar is a family of formalisms in natural language syntax that share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and seman ...
s and pregroup grammars extensively use type constructors to define the types (''noun'', ''verb'', etc.) of words. The most common construction takes the basic types $e$ and $t$ for individuals and
truth-value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some progr ...
s, respectively, and defines the set of types recursively as follows: * if $a$ and $b$ are types, then so is $\langle a,b\rangle$; * nothing except the basic types, and what can be constructed from them by means of the previous clause are types. A complex type $\langle a,b\rangle$ is the type of functions from entities of type $a$ to entities of type $b$. Thus one has types like $\langle e,t\rangle$ which are interpreted as elements of the set of functions from entities to truth-values, i.e.
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\ ...
s of sets of entities. An expression of type $\langle\langle e,t\rangle,t\rangle$ is a function from sets of entities to truth-values, i.e. a (indicator function of a) set of sets. This latter type is standardly taken to be the type of natural language quantifiers, like '' everybody'' or '' nobody'' ( Montague 1973, Barwise and Cooper 1981).

## Social sciences

Gregory Bateson Gregory Bateson (9 May 1904 – 4 July 1980) was an English anthropologist, social scientist, linguist, visual anthropologist, semiotician, and cyberneticist whose work intersected that of many other fields. His writings include '' Steps ...
introduced a theory of logical types into the social sciences; his notions of
double bind A double bind is a dilemma in communication in which an individual (or group) receives two or more reciprocally conflicting messages. In some scenarios (e.g. within families or romantic relationships) this can be emotionally distressing, creating ...
and logical levels are based on Russell's theory of types.

*
Foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...

* * * Covers type theory in depth, including polymorphic and dependent type extensions. Gives categorical semantics. * * Provides a historical survey of the developments of the theory of types with a focus on the decline of the theory as a foundation of mathematics over the four decades following the publication of the second edition of 'Principia Mathematica'. * Intended as a type theory counterpart of
Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operato ...
's (1960) '' Naïve Set Theory'' * * * A good introduction to simple type theory for computer scientists; the system described is not exactly Church's STT though
Book review
* * *

# References

## Introductory material

Type Theory at nLab
which has articles on many topics.
Intuitionistic Type Theory
article at the Stanford Encyclopedia of Philosophy
Lambda Calculus with Types
book by Henk Barendregt
Calculus of Constructions / Typed Lambda Calculus
textbook style paper by Helmut Brandl
Intuitionistic Type Theory
notes by Per Martin-Löf
Programming in Martin-Löf ’s Type Theory
book
Homotopy Type Theory
book, which proposed homotopy type theory as a mathematical foundation.

*
The TYPES Forum
— moderated e-mail forum focusing on type theory in computer science, operating since 1987. * tp://ftp.cs.cornell.edu/pub/nuprl/doc/book.ps.gz The Nuprl Book
Introduction to Type Theory.

of summer schools 2005–2008 ** Th

has introductory lectures

many lectures and some notes.

includin