Formal semantics is the study of

grammatical In linguistics, grammaticality is determined by the conformity to language usage as derived by the grammar of a particular variety (linguistics), speech variety. The notion of grammaticality rose alongside the theory of generative grammar, the go ...

meaning in

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 languages ...

s using formal tools from

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

theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the ...

. It is an interdisciplinary field, sometimes regarded as a subfield of both

linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Ling ...

and

philosophy of language In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of Meaning (philosophy of language), meanin ...

. It provides accounts of what linguistic expressions mean and how their meanings are composed from the meanings of their parts. The enterprise of formal semantics can be thought of as that of reverse-engineering the semantic components of natural languages' grammars.

Overview

Formal semantics studies the

denotation In linguistics and philosophy, the denotation of an expression is its literal meaning. For instance, the English word "warm" denotes the property of being warm. Denotation is contrasted with other aspects of meaning including connotation. For insta ...

s of natural language expressions. High-level concerns include

compositionality In semantics, mathematical logic and related disciplines, the principle of compositionality 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. ...

reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a '' name'' ...

, and the nature of meaning. Key topic areas include

scope Scope or scopes may refer to: People with the surname * Jamie Scope (born 1986), English footballer * John T. Scopes (1900–1970), central figure in the Scopes Trial regarding the teaching of evolution Arts, media, and entertainment * Cinema ...

modality Modality may refer to: Humanities * Modality (theology), the organization and structure of the church, as distinct from sodality or parachurch organizations * Modality (music), in music, the subject concerning certain diatonic scales * Modaliti ...

, binding, tense, and aspect. Semantics is distinct from

pragmatics In linguistics and related fields, pragmatics is the study of how context contributes to meaning. The field of study evaluates how human language is utilized in social interactions, as well as the relationship between the interpreter and the int ...

, which encompasses aspects of meaning which arise from interaction and communicative intent. Formal semantics is an interdisciplinary field, often viewed as a subfield of both

and

philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...

, while also incorporating work from

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 Applied science, practical discipli ...

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 forma ...

, and

cognitive psychology Cognitive psychology is the scientific study of mental processes such as attention, language use, memory, perception, problem solving, creativity, and reasoning. Cognitive psychology originated in the 1960s in a break from behaviorism, which ...

. Within philosophy, formal semanticists typically adopt a Platonistic ontology and an

externalist Internalism and externalism are two opposite ways of integration of explaining various subjects in several areas of philosophy. These include human motivation, knowledge, justification, meaning, and truth. The distinction arises in many areas of de ...

view of meaning. Within linguistics, it is more common to view formal semantics as part of the study of linguistic

cognition Cognition refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses all aspects of intellectual functions and processes such as: perception, attention, though ...

. As a result, philosophers put more of an emphasis on conceptual issues while linguists are more likely to focus on the syntax–semantics interface and crosslinguistic variation.

Central concepts

Truth conditions

The fundamental question of formal semantics is what you know when you know how to interpret expressions of a language. A common assumption is that knowing the meaning of a sentence requires knowing its truth conditions, or in other words knowing what the world would have to be like for the sentence to be true. For instance, to know the meaning of the English sentence "Nancy smokes" one has to know that it is true when the person Nancy performs the action of smoking. However, many current approaches to formal semantics posit that there is more to meaning than truth-conditions. In the formal semantic framework of inquisitive semantics, knowing the meaning of a sentence also requires knowing what issues (i.e. questions) it raises. For instance "Nancy smokes, but does she drink?" conveys the same truth-conditional information as the previous example but also raises an issue of whether Nancy drinks. Other approaches generalize the concept of truth conditionality or treat it as epiphenomenal. For instance in

dynamic semantics Dynamic semantics is a framework in logic and natural language semantics that treats the meaning of a sentence as its potential to update a context. In static semantics, knowing the meaning of a sentence amounts to knowing when it is true; in dynam ...

, knowing the meaning of a sentence amounts to knowing how it updates a context. Pietroski treats meanings as instructions to build concepts.

Compositionality

The Principle of Compositionality is the fundamental assumption in formal semantics. This principle states that the

of a complex expression is determined by the denotations of its parts along with their mode of composition. For instance, the denotation of the

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...

sentence "Nancy smokes" is determined by the meaning of "Nancy", the denotation of "smokes", and whatever semantic operations combine the meanings of subjects with the meanings of predicates. In a simplified semantic analysis, this idea would be formalized by positing that "Nancy" denotes Nancy herself, while "smokes" denotes a function which takes some individual ''x'' as an argument and returns the truth value "true" if ''x'' indeed smokes. Assuming that the words "Nancy" and "smokes" are semantically composed via

function application In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abst ...

, this analysis would predict that the sentence as a whole is true if Nancy indeed smokes.

Phenomena

Scope

Scope can be thought of as the semantic order of operations. For instance, in the sentence "''Paulina doesn't drink beer but she does drink wine''," the

proposition 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 ...

that Paulina drinks beer occurs within the scope of

negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...

, but the proposition that Paulina drinks wine does not. One of the major concerns of research in formal semantics is the relationship between operators' syntactic positions and their semantic scope. This relationship is not transparent, since the scope of an operator need not directly correspond to its surface position and a single surface form can be semantically ambiguous between different scope construals. Some theories of scope posit a level of syntactic structure called

logical form In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguou ...

, in which an item's syntactic position corresponds to its semantic scope. Others theories compute scope relations in the semantics itself, using formal tools such as type shifters, monads, and continuations.

Binding

Binding is the phenomenon in which anaphoric elements such as

pronoun In linguistics and grammar, a pronoun ( abbreviated ) is a word or a group of words that one may substitute for a noun or noun phrase. Pronouns have traditionally been regarded as one of the parts of speech, but some modern theorists would not ...

s are grammatically associated with their antecedents. For instance in the English sentence "Mary saw herself", the anaphor "herself" is bound by its antecedent "Mary". Binding can be licensed or blocked in certain contexts or syntactic configurations, e.g. the pronoun "her" cannot be bound by "Mary" in the English sentence "Mary saw her". While all languages have binding, restrictions on it vary even among closely related languages. Binding was a major for the

government and binding theory A government is the system or group of people governing an organized community, generally a state. In the case of its broad associative definition, government normally consists of legislature, executive, and judiciary. Government is a ...

paradigm.

Modality

Modality is the phenomenon whereby language is used to discuss potentially non-actual scenarios. For instance, while a non-modal sentence such as "Nancy smoked" makes a claim about the actual world, modalized sentences such as "Nancy might have smoked" or "If Nancy smoked, I'll be sad" make claims about alternative scenarios. The most intensely studied expressions include modal auxiliaries such as "could", "should", or "must"; modal adverbs such as "possibly" or "necessarily"; and modal adjectives such as "conceivable" and "probable". However, modal components have been identified in the meanings of countless natural language expressions including

counterfactuals Counterfactual conditionals (also ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactua ...

propositional attitudes A propositional attitude is a mental state held by an agent toward a proposition. Linguistically, propositional attitudes are denoted by a verb (e.g. "believed") governing an embedded "that" clause, for example, 'Sally believed that she had won ...

, evidentials, habituals and generics. The standard treatment of linguistic modality was proposed by

Angelika Kratzer Angelika Kratzer is a professor emerita of linguistics in the department of linguistics at the University of Massachusetts Amherst. Biography She was born in Germany, and received her PhD from the University of Konstanz in 1979, with a dissertat ...

in the 1970s, building on an earlier tradition of work in

modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...

.Kaufmann, S.; Condoravdi, C. & Harizanov, V. (2006
Formal approaches to modality
Formal approaches to modality. In: Frawley, W. (Ed.). The Expression of Modality. Berlin, New York: Mouton de Gruyter

History

The

al analysis of the meaning of

declarative sentence In linguistics and grammar, a sentence is a linguistic expression, such as the English example " The quick brown fox jumps over the lazy dog." In traditional grammar, it is typically defined as a string of words that expresses a complete thoug ...

s began with

Aristotelian logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, ...

. However, it took until the early 1970s (with the pioneering work of the philosopher and logician

Richard Montague Richard Merritt Montague (September 20, 1930 – March 7, 1971) was an American mathematician and philosopher who made contributions to mathematical logic and the philosophy of language. He is known for proposing Montague grammar to formaliz ...

) for formal semantics to emerge as a major area of research. Montague proposed a formal system now known as

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 ...

which consisted of a novel syntactic formalism for English, a logical system called

Intensional Logic Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe (''extensions''), by additional quantifiers that range over terms that may have such individuals ...

, and a set of homomorphic translation rules linking the two. In retrospect, Montague Grammar has been compared to a

Rube Goldberg machine A Rube Goldberg machine, named after American cartoonist Rube Goldberg, is a chain reaction-type machine or contraption intentionally designed to perform a simple task in an indirect and (impractically) overly complicated way. Usually, these mach ...

, but it was regarded as earth-shattering when first proposed, and many of its fundamental insights survive in the various semantic models which have superseded it. Barbara_partee

Montague Grammar was a major advance because it showed that natural languages could be treated as interpreted

formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sym ...

s. Before Montague, many linguists had doubted that this was possible, and logicians of that era tended to view logic as a replacement for natural language rather than a tool for analyzing it. Montague's work was published during the

Linguistics Wars The linguistics wars were a protracted academic dispute inside American theoretical linguistics which took place mostly in the 1960s and 1970s, stemming from an intellectual falling-out between Noam Chomsky and some of his early colleagues and doct ...

, and many linguists were initially puzzled by it. While linguists wanted a restrictive theory that could only model phenomena that occur in human languages, Montague sought a flexible framework that characterized the concept of meaning at its most general. At one conference, Montague told

Barbara Partee Barbara Hall Partee (born June 23, 1940) is a Distinguished University Professor Emerita of Linguistics and Philosophy at the University of Massachusetts Amherst (UMass). Biography Born in Englewood, New Jersey, Partee grew up in the Baltimore ...

that she was "the only linguist who it is not the case that I can't talk to". Formal semantics grew into a major subfield of linguistics in the late 1970s and early 1980s, due to the seminal work of Barbara Partee. Partee developed a linguistically plausible system which incorporated the key insights of both Montague Grammar and

Transformational grammar In linguistics, transformational grammar (TG) or transformational-generative grammar (TGG) is part of the theory of generative grammar, especially of natural languages. It considers grammar to be a system of rules that generate exactly those combi ...

. Early research in linguistic formal semantics used Partee's system to achieve a wealth of empirical and conceptual results. Later work by Irene Heim,

Tanya Reinhart Tanya Reinhart ( he, טניה ריינהרט; July 1943 – March 17, 2007) was an Israeli linguist who wrote frequently on the Israeli-Palestinian conflict. She contributed columns to the Israeli newspaper '' Yediot Aharonot'' and longer articles ...

, Robert May and others built on Partee's work to further reconcile it with the

generative Generative may refer to: * Generative actor, a person who instigates social change * Generative art, art that has been created using an autonomous system that is frequently, but not necessarily, implemented using a computer * Generative music, mus ...

approach to syntax. The resulting framework is known as the ''Heim and Kratzer'' system, after the authors of the textbook ''Semantics in Generative Grammar'' which first codified and popularized it. The Heim and Kratzer system differs from earlier approaches in that it incorporates a level of syntactic representation called

which undergoes semantic interpretation. Thus, this system often includes syntactic representations and operations which were introduced by translation rules in Montague's system. However, work by others such as

Gerald Gazdar Gerald James Michael Gazdar, FBA (born 24 February 1950) is a British linguist and computer scientist. Education He was educated at Heath Mount School, Bradfield College, the University of East Anglia (BA, 1970) and the University of Reading ( ...

proposed models of the syntax-semantics interface which stayed closer to Montague's, providing a system of interpretation in which denotations could be computed on the basis of surface structures. These approaches live on in frameworks such as categorial grammar and

combinatory categorial grammar Combinatory categorial grammar (CCG) is an efficiently parsable, yet linguistically expressive grammar formalism. It has a transparent interface between surface syntax and underlying semantic representation, including predicate–argument structur ...

Cognitive semantics Cognitive semantics is part of the cognitive linguistics movement. Semantics is the study of linguistic meaning. Cognitive semantics holds that language is part of a more general human cognitive ability, and can therefore only describe the world a ...

emerged as a reaction against formal semantics, but there have been recently several attempts at reconciling both positions.