Formal semantics is the scientific study of linguistic meaning through formal tools from logic and mathematics. It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of language. Formal semanticists rely on diverse methods to analyze natural language. Many examine the meaning of a sentence by studying the circumstances in which it would be true. They describe these circumstances using abstract mathematical models to represent entities and their features. The principle of compositionality helps them link the meaning of expressions to abstract objects in these models. This principle asserts that the meaning of a compound expression is determined by the meanings of its parts.
Propositional and predicate logic are formal systems used to analyze the semantic structure of sentences. They introduce concepts like singular terms, predicates, quantifiers, and logical connectives to represent the logical form of natural language expressions. Type theory is another approach utilized to describe sentences as nested functions with precisely defined input and output types. Various theoretical frameworks build on these systems. Possible world semantics and situation semantics evaluate truth across different hypothetical scenarios. Dynamic semantics analyzes the meaning of a sentence as the information contribution it makes.
Using these and similar theoretical tools, formal semanticists investigate a wide range of linguistic phenomena. They study quantificational expressions, which indicate the quantity of something, like the sentence "all ravens are black". An influential proposal analyzes them as relations between two setsthe set of ravens and the set of black things in this example. Quantifiers are also used to examine the meaning of definite and indefinite descriptions, which denote specific entities, like the expression "the president of Kenya". Formal semanticists are also interested in tense and aspect, which provide temporal information about events and circumstances. In addition to studying statements about what is true, semantics also investigates other sentences types such as questions and imperatives. Other investigated linguistic phenomena include intensionality, modality, negation, plural expressions, and the influence of contextual factors.
Formal semantics is relevant to various fields. In logic and computer science, formal semantics refers to the analysis of meaning in artificially constructed logical and programming languages. In cognitive science, some researchers rely on the insights of formal semantics to study the nature of the mind. Formal semantics has its roots in the development of modern logic starting in the late 19th century. Richard Montague's work in the late 1960s and early 1970s was pivotal in applying these logical principles to natural language, inspiring many scholars to refine his insights and apply them to diverse linguistic phenomena.
Definition Formal semantics is a branch of linguistics and philosophy that studies linguistic meaning using formal methods. To analyze language in a precise and systematic manner, it incorporates ideas from logic, mathematics, and philosophy of language, like the concepts of truth conditions, model theory, and compositionality. Because of the prominence of these concepts, formal semantics is also referred to as truth-conditional semantics and model-theoretic semantics.
The primary focus of formal semantics is the analysis of natural languages such as English, Sanskrit, Quechua, and American Sign Language. They investigate diverse linguistic phenomena, including reference, quantifiers, plurality, tense, aspect, vagueness, modality, scope, binding, conditionals, questions, and imperatives.
Understood in a wide sense, formal semantics also includes the study of artificial or constructed languages. This covers the formal languages used in the logical analysis of arguments, such as the language of first-order logic, and programming languages in computer science, such as C++, JavaScript, and Python. Formal semantics is related to formal pragmatics since both are subfields of formal linguistics. One key difference is that formal pragmatics centers on how language is used in communication rather than the problem of meaning in general.
Methodology Formal semanticists rely on diverse methods, conceptual tools, and background assumptions, which distinguish the field from other branches of semantics. Most of these principles originate in logic, mathematics, and the philosophy of language. One key principle is that an adequate theory of meaning needs to accurately predict sentences' truth conditions. A sentence's truth conditions are the circumstances under which it would be true. For example, the sentence "Tina is tall and happy" is true if Tina has the property of being tall and also the property of being happy; these are its truth conditions. This principle reflects the idea that understanding a sentence requires knowing how it relates to reality and under which circumstances it would be appropriate to use it.
Formal semanticists test their theories using the linguistic intuitions of competent speakers. A key source of data comes from judgments concerning entailment, a relation between sentencescalled premises and conclusionsin which truth is preserved. For instance, the sentence "Tina is tall and happy" entails the sentence "Tina is tall" because the truth of the first sentence guarantees the truth of the second. One aspect of understanding the meaning of a sentence is comprehending what it does and does not entail. The study of entailment relations helps formal semanticists build new theories and verify whether existing theories accurately reflect the intuitions of native speakers.
One major goal of research in formal semantics is to uncover and explain the range of variation across human languages. As part of this typological work, researchers have sought to uncover principles that apply to all languages known as semantic universals. A proposed universal is that languages only allow determiners which have the property of being conservative, indicating that the human language faculty cannot lexicalize many quantifiers that are possible in principle. Crosslinguistic research has also been used to inform theoretical issues within semantics. For example, early research on the semantics of English sparked debate over whether tense markers are quantificational or referential. Subsequent work on Javanese, Atayal, St’átʼimcets, and Tlingit has been taken as evidence that natural languages actually make both interpretations available. Similarly, evidence from sign languages has been used to support certain analyses of coreference which are hard to test in spoken languages because the crucial elements are covert. In order to collect data on languages that they themselves do not speak, formal semanticists carry out fieldwork.
Formal semanticists typically analyze language by proposing a system of model theoretic interpretation. In such a system, a sentence is evaluated relative to a mathematical structure called a model that maps each expression to an abstract object. Models rely on set theory and introduce abstract objects for all entities in this situation. For example, a model of a situation where Tina is tall and happy may include an abstract object representing Tina and two sets of objectsone for all tall entities and another for all happy entities. Different models introduce different objects and group them into different sets while interpretation functions link natural language expressions to these objects and sets. Using this approach, semanticists can precisely calculate the truth values of sentences. For example, the sentence "Tina is tall" is true in models where the abstract object corresponding to the name "Tina" is in the set corresponding to the predicate "tall". In this way, model theoretic interpretation makes it possible to formally capture truth conditions and entailments.
The principle of compositionality is another key methodological assumption for analyzing the meaning of natural language sentences and linking them to abstract models. It states that the meaning of a compound expression is determined by the meanings of its parts and the way they are combined. According to this principle, if a person knows the meanings of the name "Tina", the verb "is", and the adjective "happy", they can understand the sentence "Tina is happy" even if they have never heard this specific combination of words before. The principle of compositionality explains how language users can comprehend an infinite number of sentences based on their knowledge of a finite number of words and rules.
Semantic formalisms differ in whether they adopt the method of direct interpretation or indirect interpretation. In direct interpretation, the semantic interpretation rules map natural language expressions directly to their denotations, whereas indirect interpretation maps them to expressions of some formal language. For example, in direct interpretation the name "Barack Obama" would be mapped to the man himself, while a system of indirect interpretation would map it to a logical constant "o" which would then be mapped to the man himself. In this way, direct interpretation can be seen as providing a model theory for natural languages themselves, while indirect interpretation provides a system of logic translation and then relies on the model theory of the formal language. These two methods are in principle equivalent, but are often chosen based on practical or expository considerations. Formal semanticists sometimes limit their studies to particular fragments of whatever language they are studying to isolate core phenomena and simplify analysis.
Formal systems and theories Propositional and predicate logic
Formal semanticists often represent truth conditions using propositional or predicate logic. Propositional logic employs letters like and to represent simple statements. These can be combined to form complex formulas using logical connectives such as (and), (or), and (if...then). For example, using the method of indirect interpretation, the sentence "Alice is happy and Bob is rich" can be translated into propositional logic with the formula , where stands for "Alice is happy" and stands for "Bob is rich". Of key interest to semantic analysis is that the truth value of these compound statements is directly determined by the truth values of the simple statements. For instance, the formula is only true if both and are true; otherwise, it is false.
Predicate logic extends propositional logic using symbols for singular terms, predicates, and quantifiers. Singular terms refer to specific entities, whereas predicates represent properties of and relations between entities. For instance, the sentence "Alice is happy" can be translated as the formula , where is a logical constant and is a predicate. Quantifiers express that a certain condition applies to some or all entities. For example, the sentence "Someone is happy" can be represented with the formula , where the existential quantifier indicates that happiness applies at least to one person. Similarly, the idea that everyone is happy can be expressed through the formula , where is the universal quantifier.