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. In the Philosophy of language, a natural language (or ordinary language) is a Language that is spoken or written in phonemic-alphabetic or phonemically-related In Linguistics, syntax (from Ancient Greek grc συν- syn-, "together" and grc τάξις táxis, "arrangement" is the In Mathematics, Semantics, and Philosophy of language, the Principle of Compositionality is the principle that the meaning of a complex expression is determined In Linguistics, grammatical functions or ( grammatical relations) refer to syntactic relationships between Parts of speech such as subject
A categorial grammar shares some features with the simply-typed lambda calculus. In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function Whereas the lambda calculus has only one function type A → B, a categorial grammar typically has more. In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function For example, a simple categorial grammar for English might have two function types A/B and A\B, 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 NP\S, and transitive verbs in the category (NP\S)/NP. In Linguistics, a count noun (also countable noun) is a noun which can be modified by a Numeral and occur in both singular and Plural In Grammar, an adjective is a word whose main syntactic role is to modify a Noun or Pronoun, giving more information about the For English usage of verbs see the wiki article English verbs. 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. In Formal language theory, a context-free grammar ( CFG) is a grammar in which every production rule is of the form V &rarr 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.
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. 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 an Abelian category C is a construction of Homological algebra introduced In Linguistics, grammatical functions or ( grammatical relations) refer to syntactic relationships between Parts of speech such as subject 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. Not to be confused with the homophone Intention; or the related concept of Intentionality. Quantification has two distinct meanings In Mathematics and Empirical science, it refers to human acts known as Counting and Measuring
The basic ideas of categorial grammar date from work by Kazimierz Ajdukiewicz (in 1935) and Yehoshua Bar-Hillel (in 1953). Kazimierz Ajdukiewicz ( December 12, 1890, Tarnopol, Galicia – April 12, 1963, Warsaw, Poland) was Yehoshua Bar-Hillel (יהושע בר-הלל born 1915 in Vienna; died 1975 in Jerusalem) was a Philosopher, Mathematician, and linguist In 1958, Joachim Lambek introduced a syntactic calculus that formalized the function type constructors along with various rules for the combination of functions. Joachim Lambek (born 5 Dec 1922 is Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph An extension of linear logic Noncommutative logic is an extension of Linear logic which combines the commutative connectives of linear logic with the noncommutative multiplicative This calculus is a forerunner of linear logic in that it is a substructural logic. 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 Montague grammar uses an ad hoc syntactic system for English that is based on the principles of categorial grammar. Montague grammar is an approach to Natural language Semantics, named after American Logician Richard Montague. 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. Richard Merett Montague (September 20 1930 Stockton California – March 7 1971 Los Angeles) was an American Mathematician and Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from 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 Schönfinkel and Haskell Curry. Mark J Steedman, FBA FRSE (born 18 September 1946) is a computational linguist and cognitive scientist Anna Szabolcsi is a linguist She was born and educated in Hungary Combinatory Categorial grammar (CCG is an efficiently parseable yet linguistically expressive grammar formalism Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for Variables in Mathematical logic Moses Ilyich Schönfinkel, also known as Moisei Isai'evich Sheinfinkel' Шейнфинкель (September 4 1889 Ekaterinoslav (now Dnipropetrovsk, Haskell Brooks Curry ( September 12, 1900 – September 1, 1982) was an American Mathematician and Logician.
There are a number of related formalisms of this kind in linguistics, such as type logical grammar.
A derivation is a binary tree that encodes a proof.
In a left (right) function application, the node of the type A\B (A/B) is called the functor, and the node of the type A is called an argument.
A variety of changes to categorial grammar have been proposed to improve syntactic coverage. Some of the most common ones are listed below.
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. The term person is used in Common sense to mean an individual Human being. Gender comprises a range of differences between men and women extending from the biological to the social A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. 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.
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. In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of 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.
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. .
Rules of type raising allow one to convert an expression of category X into one of category Y/(Y\X) or Y\(Y/X), for some other category Y. These rules essentially reverse the function-argument relationship.