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 quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
In linguistics, syntax is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), agreement, the nature of crosslinguistic variation, and the relationship between form and meaning. There are numerous approaches to syntax which differ in their central assumptions and goals.
In computer science, a tree is a widely used abstract data type that simulates a hierarchical tree structure, with a root value and subtrees of children with a parent node, represented as a set of linked nodes.
Phrase structure rules are a type of rewrite rule used to describe a given language's syntax and are closely associated with the early stages of transformational grammar, proposed by Noam Chomsky in 1957. They are used to break down a natural language sentence into its constituent parts, also known as syntactic categories, including both lexical categories and phrasal categories. A grammar that uses phrase structure rules is a type of phrase structure grammar. Phrase structure rules as they are commonly employed operate according to the constituency relation, and a grammar that employs phrase structure rules is therefore a constituency grammar; as such, it stands in contrast to dependency grammars, which are based on the dependency relation.
In logic and computer science, unification is an algorithmic process of solving equations between symbolic expressions.
In mathematics, the transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets it is the unique minimal transitive superset of R.
Government and binding is a theory of syntax and a phrase structure grammar in the tradition of transformational grammar developed principally by Noam Chomsky in the 1980s. This theory is a radical revision of his earlier theories and was later revised in The Minimalist Program (1995) and several subsequent papers, the latest being Three Factors in Language Design (2005). Although there is a large literature on government and binding theory which is not written by Chomsky, Chomsky's papers have been foundational in setting the research agenda.
In mathematics and computer science, the syntactic monoid of a formal language is the smallest monoid that recognizes the language .
In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinite sequence of elements from contains an increasing pair with .
Categorial grammar is a family of formalisms in natural language syntax which share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and semantic composition, since it typically treats syntactic categories as corresponding to semantic types. Categorial grammars were developed in the 1930s by Kazimierz Ajdukiewicz, Yehoshua Bar-Hillel, and Joachim Lambek. It saw a surge of interest in the 1970s following the work of Richard Montague, whose Montague grammar assumed a similar view of syntax. It continues to be a major paradigm, particularly within formal semantics.
Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke.
The term phrase structure grammar was originally introduced by Noam Chomsky as the term for grammar studied previously by Emil Post and Axel Thue. Some authors, however, reserve the term for more restricted grammars in the Chomsky hierarchy: context-sensitive grammars or context-free grammars. In a broader sense, phrase structure grammars are also known as constituency grammars. The defining trait of phrase structure grammars is thus their adherence to the constituency relation, as opposed to the dependency relation of dependency grammars.
In linguistics, focus is a grammatical category that conveys which part of the sentence contributes new, non-derivable, or contrastive information. In the English sentence "Mary only insulted BILL", focus is expressed prosodically by a pitch accent on "Bill" which identifies him as the only person Mary insulted. By contrast, in the sentence "Mary only INSULTED Bill", the verb "insult" is focused and thus expresses that Mary performed no other actions towards Bill. Focus is a cross-linguistic phenomenon and a major topic in linguistics. Research on focus spans numerous subfields including phonetics, syntax, semantics, pragmatics, and sociolinguistics.
In generative grammar and related frameworks, a node in a parse tree c-commands its sister node and all of its sister's descendants. In these frameworks, c-command plays a central role in defining and constraining operations such as syntactic movement, binding, and scope. Tanya Reinhart introduced c-command in 1976 as a key component of her theory of anaphora. The term is short for "constituent command".
Answer set programming (ASP) is a form of declarative programming oriented towards difficult search problems. It is based on the stable model semantics of logic programming. In ASP, search problems are reduced to computing stable models, and answer set solvers—programs for generating stable models—are used to perform search. The computational process employed in the design of many answer set solvers is an enhancement of the DPLL algorithm and, in principle, it always terminates.
In linguistics, antisymmetry is a theory of syntactic linearization presented in Richard S. Kayne's 1994 monograph The Antisymmetry of Syntax. It asserts that hierarchical structure in natural language maps universally onto a particular surface linearization, namely specifier-head-complement branching order. The theory derives a version of X-bar theory. Kayne hypothesizes that all phrases whose surface order is not specifier-head-complement have undergone syntactic movements that disrupt this underlying order. Subsequently, others have attempted to derive specifier-complement-head as the basic word order.
In linguistics, locality refers to the proximity of elements in a linguistic structure. Constraints on locality limit the span over which rules can apply to a particular structure. Theories of transformational grammar use syntactic locality constraints to explain restrictions on argument selection, syntactic binding, and syntactic movement.
In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact.
In artificial intelligence and related fields, an argumentation framework is a way to deal with contentious information and draw conclusions from it using formalized arguments.
Dynamic Syntax (DS) is a grammar formalism and linguistic theory whose overall aim is to explain the real-time twin processes of language understanding and production. Under the DS approach, syntactic knowledge is understood as the ability to incrementally analyse the structure and content of spoken and written language in context and in real-time. While it posits representations similar to those used in Combinatory Categorial Grammars (CCG), it builds those representations left-to-right going word-by-word. Thus it differs from other syntactic models which generally abstract way from features of everyday conversation such as interruption, backtracking, and self-correction. Moreover, it differs from other approaches in that it does not postulate an independent level of syntactic structure over words.
