Categorial grammar is a term used for a family of formalisms in natural language syntax motivated by the principle of compositionality and organized according to the view that syntactic constituents should generally combine as functions or according to a function-argument relationship. Image File history File links Question_book-3. ... The term natural language is used to distinguish languages spoken and signed (by hand signals and facial expressions) by humans for general-purpose communication from constructs such as writing, computer-programming languages or the languages used in the study of formal logic, especially mathematical logic. ... For other uses, see Syntax (disambiguation). ... The Principle of Compositionality in semantics is the principle that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them. ... If grammar is seen as studying the relations and architectures of language, such relations and structures would then have supports and anchors, forces and motions, features and property equivalencies, and roles galore just as a building or organixm has. ...

Basics of categorial grammar

A categorial grammar shares some features with the simply-typed lambda calculus. Whereas the lambda calculus has only one function type A → B, a categorial grammar typically has more. For example, a simple categorial grammar for English might have two function types A/B and AB, depending on whether the function takes its argument from the left or the right. Such a grammar would have only two rules: left and right function application. Such a grammar might have three basic categories (N,NP, and S), putting count nouns in the category N, adjectives in the category N/N, determiners in the category NP/N, names in the category NP, intransitive verbs in the category SNP, and transitive verbs in the category (SNP)/NP. Categorial grammars of this form (having only function application rules) are equivalent in generative capacity to context-free grammar and are thus often considered inadequate for theories of natural language syntax. Unlike CFGs, categorial grammars are lexicalized, meaning that only a small number of (mostly language-independent) rules are employed, and all other syntactic phenomena derive from the lexical entries of specific words. The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... A count noun is a noun which is itself counted, or the units which are used to count it. ... In grammar, an adjective is a word whose main syntactic role is to modify a noun or pronoun (called the adjectives subject), giving more information about what the noun or pronoun refers to. ... Determiners are words which quantify or identify nouns. ... It has been suggested that Verbal agreement be merged into this article or section. ... In formal language theory, a context-free grammar (CFG) is a grammar in which every production rule is of the form V → w where V is a single nonterminal symbol, and w is a string of terminals and/or nonterminals (possibly empty). ...

Another appealing aspect of categorial grammars is that it is often easy to assign them a compositional semantics, by first assigning interpretation types to all the basic categories, and then associating all the derived categories with appropriate function types. The interpretation of any constituent is then simply the value of a function at an argument. With some modifications to handle intensionality and quantification, this approach can be used to cover a wide variety of semantic phenomena. In cognitive psychology, a basic category is a category at a particular level of the category inclusion hierarchy (i. ... In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). ... If grammar is seen as studying the relations and architectures of language, such relations and structures would then have supports and anchors, forces and motions, features and property equivalencies, and roles galore just as a building or organixm has. ... Intension refers to the meanings or characteristics encompassed by a given word. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...

Historical notes

The basic ideas of categorial grammar date from work by Kazimierz Ajdukiewicz (in 1935) and Yehoshua Bar-Hillel (in 1953). In 1958, Joachim Lambek introduced a syntactic calculus that formalized the function type constructors along with various rules for the combination of functions. This calculus is a forerunner of linear logic in that it is a substructural logic. Montague grammar uses an ad hoc syntactic system for English that is based on the principles of categorial grammar. Although Montague's work is sometimes regarded as syntactically uninteresting, it helped to bolster interest in categorial grammar by associating it with a highly successful formal treatment of natural language semantics. More recent work in categorial grammar has focused on the improvement of syntactic coverage. One formalism which has received considerable attention in recent years is Steedman and Szabolcsi's Combinatory Categorial Grammar which builds on combinatory logic invented by Moses Schonfinkel and Haskell Curry. Kazimierz Ajdukiewicz (born on December 12, 1890 in Tarnopol, Galicia (now Ternopil, Ukraine) - April 12, 1963 in Warsaw, Poland) was a Polish philosopher, mathematician and logician. ... Yehoshua Bar-Hillel (1915-1975) was a philosopher, mathematician and linguist at MIT and the Hebrew University. ... Joachim Lambek (1922-) is the Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD in 1950. ... Noncommutative logic is the name given to a family of substructural logics in which the exchange rule is inadmissible. ... In mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. ... In mathematical logic, in particular in connection with proof theory, a number of substructural logics have been introduced, as systems of propositional calculus that are weaker than the conventional one. ... Montague grammar is an approach to natural language semantics, based on formal logic, especially lambda calculus and set theory. ... Richard Merett Montague (1930–1971) was an American mathematician and philosopher. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Not to be confused with combinational logic, a topic in digital electronics. ... Moses Schönfinkel was a Russian logician. ... Haskell Brooks Curry (September 12, 1900, Millis, Massachusetts - September 1, 1982, State College, Pennsylvania) was an American mathematician and logician. ...

There are a number of related formalisms of this kind in linguistics. Another one is Type logical grammar.

Some definitions

• Derivation

A derivation is a binary tree that encodes a proof.

• Parse tree
• Functor and Argument

In a left (right) function application, the node of the type AB (A/B) is called the functor, and the node of the type A is called an argument.

• Functor-argument structure

Refinements of categorial grammar

A variety of changes to categorial grammar have been proposed to improve syntactic coverage. Some of the most common ones are listed below.

Features and subcategories

Most systems of categorial grammar subdivide categories. The most common way to do this is by tagging them with features, such as person, gender, number, and tense. Sometimes only atomic categories are tagged in this way. In Montague grammar, it is traditional to subdivide function categories using a multiple slash convention, so A/B and A//B would be two distinct categories of left-applying functions, that took the same arguments but could be distinguished between by other functions taking them as arguments. For other uses, see Person (disambiguation). ... Gender in common usage refers to the sexual distinction between male and female. ... For other uses, see Number (disambiguation). ... Grammatical tense is a way languages express the time at which an event described by a sentence occurs. ...

Function composition

Rules of function composition are included in many categorial grammars. An example of such a rule would be one that allowed the concatenation of a constituent of type A/B with one of type B/C to produce a new constituent of type A/C. The semantics of such a rule would simply involve the composition of the functions involved. Function composition is important in categorial accounts of conjunction and extraction, especially as they relate to phenomena like right node raising. The introduction of function composition into a categorial grammar leads to many kinds of derivational ambiguity that are vacuous in the sense that they do not correspond to semantic ambiguities. In chemistry, liquid-liquid extraction is a useful method to separate components (compounds) of a mixture. ...

Conjunction

Many categorial grammars include a typical conjunction rule, of the general form X CONJ X → X, where X is a category. Conjunction can generally be applied to nonstandard constituents resulting from type raising or function composition..

Type raising

Rules of type raising allow one to convert an expression of category X into one of category Y/(YX) or Y(Y/X), for some other category Y. These rules essentially reverse the function-argument relationship.