Formal semantics (natural language)

Formal semantics is the scientific study of grammatical meaning in natural languages using formal concepts from logic, mathematics and theoretical computer science. It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of language. 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.

Definition

Formal semantics is an approach to the study of linguistic meaning that uses ideas from logic and philosophy of language to characterize the relationships between expressions and their denotations. These tools include the concepts of truth conditions, model theory, and compositionality.

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. Formal semanticists examine a wide range of linguistic phenomena, including reference, quantifiers, plurality, tense, aspect, vagueness, modality, scope, binding, conditionals, questions, and imperatives.

Formal semantics is an interdisciplinary field, often viewed as a subfield of both linguistics and philosophy, while also incorporating work from computer science, mathematical logic, and cognitive psychology. Formal semanticists typically adopt an externalist view of meaning that interprets meaning as the entities to which expressions refer. This focus on the connection between language and the external world sets formal semantics apart from semantic theories that concentrate on the cognitive processes and mental representations involved in understanding language.

The primary focus of formal semantics is the analysis of natural language such as English, Spanish, and Japanese. This enterprise faces challenges due to the complexity and context-dependence of natural language. As a result, theorists sometimes limit their studies to specific fragments or subsets of these languages to avoid these complexities. 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.

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 involves the analysis of the meaning of a sentence by studying its truth conditions. A truth condition is a specific situation or set of circumstances under which a sentence would be true. For example, a truth condition of the sentence "It is raining" is that raindrops are falling outside. 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.

A closely related methodological consideration is the problem of entailment. Entailment is a relation between sentences—called premises and conclusions—in which truth is preserved. For instance, the sentence "Tina is tall and thin" 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.

To analyze truth conditions and entailment relations in a precise manner, formal semanticists typically employ model theory. In this context, a model is an abstract representation of a hypothetical situation. 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 thin may include an abstract object representing Tina and two sets of objects—one for all tall entities and another for all thin entities. Using this approach, it is possible to define truth conditions and mimic linguistic phenomena through mathematical relations between abstract objects, such as the relation of set membership between the object corresponding to Tina and the set of tall objects.

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 thin, they can understand the sentence "Tina is thin" 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. Following this principle, formal semanticists connect natural language sentences to abstract models through a form of translation, for instance, by defining an interpretation function that maps the name "Tina" to an abstract object and the adjective "thin" to a set of objects. This makes it possible to precisely calculate the truth values of sentences relative to abstract models.

Within formal semantics, there are diverse ways how to construct models and relate linguistic expressions to them. Some rely on the contrast between grammatical and logical forms. The grammatical form of an expression is the arrangement of words and phrases on its surface, following rules of syntax. The logical form of an expression abstracts away from linguistic conventions to reveal the underlying logical relations at the semantic level. The rule-to-rule hypothesis, proposed by Richard Montague, seeks to bridge the gap between syntax and semantics. It states that for every syntactic rule, governing how a sentence may be formed, there is a corresponding semantic rule, governing how this procedure influences the meaning of the sentence.

To test the adequacy of their theories, formal semanticists typically rely on the linguistic intuitions of competent speakers as a form of empirical validation. For instance, intuitions can be used to assess whether a theory accurately predicts entailment relations between specific sentences.

Formal systems and theories

Propositional and predicate logic

Main articles: Propositional logic and Predicate logic

Formal semanticists often rely on propositional and predicate logic to analyze the semantic structure of sentences. Propositional logic can be used to examine compound sentences made up of several independent clauses. It employs letters like $A$ and $B$ to represent simple statements. Compound statements are created by combining simple statements with logical connectives, such as $\land$ (and), $\lor$ (or), and $\to$ (if...then), which express the relationships between the statements. For example, the sentence "Alice is happy and Bob is rich" can be translated into propositional logic with the formula $A\land B$, where $A$ stands for "Alice is happy" and $B$ 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 $A\land B$ is only true if both $A$ and $B$ are true; otherwise, it is false.

Predicate logic extends propositional logic by articulating the internal structure of non-compound sentences through concepts like singular term, predicate, and quantifier. Singular terms refer to specific entities, whereas predicates describe characteristics of and relations between entities. For instance, the sentence "Alice is happy" can be represented with the formula $Happy(alice)$, where $alice$ is a singular term and $Happy$ 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 $\exists xHappy(x)$, where the existential quantifier $\exists$ indicates that happiness applies at least to one person. Similarly, the idea that everyone is happy can be expressed through the formula $\forall xHappy(x)$, where $\forall$ is the universal quantifier.

There are different ways how natural language sentences can be translated into predicate logic. A common approach interprets verbs as predicates. Intransitive verbs, like sleeps and dances, have a subject but no objects and are interpreted as one-place predicates. Transitive verbs, like loves and gives, have one or more objects and are interpreted as predicates with two or more places. For example, the sentence "Bob loves Alice" can be formalized as $Loves(bob,alice)$, using the two-place predicate $Loves$. Typically, not every word in natural language sentences has a direct counterpart symbol in the logic translation, and in some cases, the pattern of the logical formula differs significantly from the surface structure of the natural language sentence. For example, sentences like "all cats are animals" are usually translated as $\forall x(Cat(x)\to Animal(x))$ (for all entities, if the entity is a cat then the entity is an animal) even though the expression "if...then" ($\to$) is not present in the original sentence.

Logic translations face challenges as a result of attempting to associate vague and ambiguous ordinary language expressions with precise logical formulas. This process frequently requires case-by-case interpretation without a generally accepted algorithm to cover all cases. Many early approaches to formal semantics, such as the works of Gottlob Frege, Rudolf Carnap, and Donald Davidson, relied primarily on predicate logic.

Type theory

Main article: Type theory

Type theory is another approach to formal semantics that was popularized by Montague. Its core idea is that expressions belong to different types, which describe how the expressions can be used and combined with other expressions. Type theory, typically in the form of typed lambda calculus, provides a formalism for this endeavor. It begins by defining a small number of basic types, which can be fused to create new types.

According to a common approach, there are only two basic types: entities ($e$) and truth values ($t$). Entities are the denotations of names and similar noun phrases, while truth values are the denotations of declarative sentences. All other types are constructed from these two types as functions that have entities, truth values, or other functions as inputs and outputs. This way, a sentence is analyzed as a complex function made up of several internal functions. When all functions are evaluated, the output is a truth value. Simple intransitive verbs without objects are functions that take an entity as input and produce a truth value as output. The type of this function is written as $\langle e,t\rangle$, where the first letter indicates the input type and the second letter the output type. According to this approach, the sentence "Alice sleeps" is analyzed as a function that takes the entity Alice as input to produce a truth value. Transitive verbs with one object, such as the verb likes, are complex or nested functions. They take an entity as input and output a second function, which itself requires an entity as input to produce a truth value, formalized as $\langle e,\langle e,t\rangle \rangle$. This way, the sentence "Alice likes Bob" corresponds to a nested function to which two entities are applied. Similar types of analyses are provided for all relevant expressions, including logical connectives and quantifiers.

Others

Possible worlds are another central concept used in the analysis of linguistic meaning. A possible world is a complete and consistent version of how everything could have been, similar to a hypothetical alternative universe. For instance, the dinosaurs were wiped out in the actual world but there are possible worlds where they survived. Possible worlds have various applications in formal semantics, usually to study expressions or aspects of meaning that are difficult to explain when referring only to entities of the actual world. They include modal statements about what is possible or necessary and descriptions of the contents of mental states, such as what people believe and desire. Possible worlds are also used to explain how two expressions can have different meanings even though they refer to the same entity, such as the expressions "the morning star" and "the evening star", which both refer to the planet Venus. One way to include possible worlds in the model-theoretic formalism is to define a set of all possible worlds as one additional component of a model. The interpretation of the meanings of different expressions is then modified to account for this change. For example, to explain that a sentence may be true in one possible world and false in another, one can interpret its meaning not directly as a truth value but as a function from a possible world to a truth value.

Situation semantics is a theory closely related to possible world semantics. Situations, like possible worlds, present possible circumstances. However, unlike possible worlds, they do not encompass a whole universe but only capture specific parts or fragments of possible worlds. This modification reflects the observation that many statements are context-dependent and aim to describe the speaker's specific circumstances rather than the world at large. For example, the sentence "every student sings" is false when interpreted as an assertion about the universe as a whole. However, speakers may use this sentence in the context of a limited situation, such as a specific high school musical, in which it can be true.

Dynamic semantics interprets language usage as a dynamic process in which information is continually updated against the background of an existing context. It rejects static approaches that associate a given expression with a fixed meaning. Instead, this theory argues that meaning depends on the information that is already present in the context, understanding the meaning of a sentence as the change in information it produces. This view reflects the idea that sentences are usually not interpreted in isolation but form part of a larger discourse, to which they contribute in some way. For example, update semantics—one form of dynamic semantics—defines an information state as the set of all possible worlds compatible with the current information, reflecting the idea that the information is incomplete and cannot determine which of these worlds is the right one. Sentences introducing new information update the information state by excluding some possible worlds, thereby decreasing uncertainty.

Phenomena

Scope

Further information: Scope (formal semantics)

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 that Paulina drinks beer occurs within the scope of negation, 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 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

Further information: Binding (linguistics)

Binding is the phenomenon in which anaphoric elements such as pronouns 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 component to the government and binding theory paradigm.

Modality

Further information: Linguistic 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, propositional attitudes, evidentials, habituals and generics. The standard treatment of linguistic modality was proposed by Angelika Kratzer in the 1970s, building on an earlier tradition of work in modal logic.

History

Formal semantics emerged as a major area of research in the early 1970s, with the pioneering work of the philosopher and logician Richard Montague. Montague proposed a formal system now known as Montague grammar which consisted of a novel syntactic formalism for English, a logical system called Intensional Logic, and a set of homomorphic translation rules linking the two. In retrospect, Montague Grammar has been compared to a Rube Goldberg machine, 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 is one of the founders and major contributors to the field.

Montague Grammar was a major advance because it showed that natural languages could be treated as interpreted formal languages. 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, 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 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. Early research in linguistic formal semantics used Partee's system to achieve a wealth of empirical and conceptual results. Later work by Irene Heim, Angelika Kratzer, Tanya Reinhart, Robert May and others built on Partee's work to further reconcile it with the generative 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 logical form 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 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.

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

See also

• Alternative semantics
• Compositionality
• Computational semantics
• Discourse representation theory
• Dynamic semantics
• Frame semantics (linguistics)
• Inquisitive semantics
• Philosophy of language
• Pragmatics
• Traditional grammar
• Syntax–semantics interface

References

Sources

• Akman, V. (2009). "Situation Semantics". In Allan, Keith (ed.). Concise Encyclopedia of Semantics. Elsevier. pp. 890–893. ISBN 978-0-08-095969-6.
• Barba, Juan (2007). "Formal semantics in the age of pragmatics". Linguistics and Philosophy. 30 (6): 637–668. doi:10.1007/s10988-008-9031-4.
• Baumgartner, Michael; Lampert, Timm (September 2008). "Adequate formalization". Synthese. 164 (1): 93–115. doi:10.1007/s11229-007-9218-1. S2CID 15396554.
• Bezuidenhout, A. (2009). "Semantics–Pragmatics Boundary". In Allan, Keith (ed.). Concise Encyclopedia of Semantics. Elsevier. ISBN 978-0-080-95969-6. Retrieved 2024-02-04.
• Cann, Ronnie (1993). Formal Semantics: An Introduction. Cambridge University Press.
• Fox, Chris (2014). "The Meaning of Formal Semantics". In Stalmaszczyk, Piotr (ed.). Semantics and Beyond: Philosophical and Linguistic Inquiries. Walter de Gruyter GmbH & Co KG. pp. 85–107. ISBN 978-3-11-039114-5.
• Griffiths, Patrick; Cummins, Chris (2023). An Introduction to English Semantics and Pragmatics (3rd ed.). Edinburgh University Press. ISBN 978-1-399-50460-7.
• Groenendijk, J.; Stokhof, M. (2009). "Dynamic Semantics". In Allan, Keith (ed.). Concise Encyclopedia of Semantics. Elsevier. ISBN 978-0-080-95969-6.
• Hurley, Patrick J. (2018). A concise introduction to logic (13th ed.). Boston, MA: Cengage Learning. ISBN 978-1-305-95809-8.
• Janssen, Theo M. V.; Zimmermann, Thomas Ede (2025). "Montague Semantics". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 28 May 2025.
• Kearns, Kate (2011). Semantics (2nd ed.). Bloomsbury Academic. ISBN 978-0-230-23229-7.
• King, Jeffrey C. (2009). "Formal Semantics". In Lepore, Ernie; Smith, Barry C. (eds.). The Oxford Handbook of Philosophy of Language (1 ed.). Oxford University Press. pp. 557–573. ISBN 978-0-19-955223-8.
• Klement, Kevin C. "Propositional Logic". Internet Encyclopedia of Philosophy. Retrieved 24 March 2025.
• Lappin, Shalom (2003). "An Introduction to Formal Semantics". In Aronoff, Mark; Ress-Miller, Janie (eds.). The Handbook of Linguistics (1st ed.). Wiley. pp. 369–393. doi:10.1002/9780470756409.ch15. ISBN 978-0-631-20497-8.
• Matthews, P. H. (2007). "Rule to Rule Hypothesis". The Concise Oxford Dictionary of Linguistics. Oxford University Press. ISBN 978-0-19-920272-0.
• Moeschler, Jacques (2007). "Introduction to Semantics". In Rajman, Martin (ed.). Speech and Language Engineering. EPFL Press. ISBN 978-0-824-72219-7.
• Nouwen, Rick; Brasoveanu, Adrian; van Eijck, Jan; Visser, Albert (2022). "Dynamic Semantics". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Archived from the original on 25 February 2024. Retrieved 13 February 2024.
• Partee, Barbara H. (2016). "1. Formal Semantics". In Aloni, Maria; Dekker, Paul (eds.). The Cambridge Handbook of Formal Semantics. Cambridge University Press. pp. 3–32. ISBN 978-1-316-55273-5.
• Portner, Paul H. (2005). What is Meaning?: Fundamentals of Formal Semantics. John Wiley & Sons. ISBN 978-1-4051-0918-5.
• Portner, Paul H.; Partee, Barbara H. (2002). Formal Semantics: The Essential Readings. Blackwell Publishers. ISBN 0-631-21541-7.
• Saeed, John I. (2009). Semantics (3rd ed.). Wiley-Blackwell. ISBN 978-1-405-15639-4.
• Shapiro, Stewart; Kouri Kissel, Teresa (2024). "Classical Logic". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 31 May 2025.
• Stokhof, Martin (2007). "Hand or Hammer? On Formal and Natural Languages in Semantics". Journal of Indian Philosophy. 35 (5–6): 597–626. doi:10.1007/s10781-007-9023-7.
• Stokhof, Martin (2013). "Formal Semantics and Wittgenstein: An Alternative?". Monist. 96 (2): 205–231. doi:10.5840/monist20139629.
• von Fintel, Kai (2023). "What is semantics?" (Document). Unpublished MIT course notes.
• Winter, Yoad (2016). Elements of Formal Semantics: An Introduction to the Mathematical Theory of Meaning in Natural Language. Edinburgh University Press. ISBN 978-0-7486-7777-1.
• Yalcin, Seth (2014). "Semantics and metasemantics in the context of generative grammar". In Alexis Burgess; Brett Sherman (eds.). Metasemantics: new essays on the foundations of meaning. Oxford University Press. ISBN 9780199669592.

Further reading

• Max Cresswell (2006). "Formal semantics". In Michael Devitt, Richard Hanley (ed.). The Blackwell guide to the philosophy of language. Wiley-Blackwell. ISBN 978-0-631-23142-4. A very accessible overview of the main ideas in the field.
• John I. Saeed (2008). Semantics. Introducing linguistics (3rd ed.). Wiley-Blackwell. ISBN 978-1-4051-5639-4. Chapter 10, Formal semantics, contains the best chapter-level coverage of the main technical directions
• Johan van Benthem; Alice Ter Meulen (2010). Handbook of Logic and Language (2nd ed.). Elsevier. ISBN 978-0-444-53726-3. The most comprehensive reference in the area.
• Emmon W. Bach (1989). Informal lectures on formal semantics. SUNY Press. ISBN 978-0-88706-772-3. One of the first textbooks. Accessible to undergraduates.
• Ronnie Cann (1993). Formal semantics: an introduction. Cambridge University Press. ISBN 978-0-521-37610-5.
• Irene Heim; Angelika Kratzer (1998). Semantics in generative grammar. Wiley-Blackwell. ISBN 978-0-631-19713-3.
• Gennaro Chierchia; Sally McConnell-Ginet (2000). Meaning and grammar: an introduction to semantics (2nd ed.). MIT Press. ISBN 978-0-262-53164-1.
• Sean A. Fulop (2004). On the Logic and Learning of Language. Trafford Publishing. ISBN 978-1-4120-2381-8.
• Glyn V. Morrill (1994). Type logical grammar: categorial logic of signs. Springer. ISBN 978-0-7923-3095-0.
• Reinhard Muskens. Type-logical Semantics. Routledge Encyclopedia of Philosophy Online.
• Bob Carpenter (1998). Type-logical semantics. MIT Press. ISBN 978-0-262-53149-8.
• Johan van Benthem (1995). Language in action: categories, lambdas, and dynamic logic. MIT Press. ISBN 978-0-262-72024-3.
• Barbara H. Partee. Reflections of a formal semanticist as of Feb 2005. Ample historical information. (An extended version of the introductory essay in Barbara H. Partee: Compositionality in Formal Semantics: Selected Papers of Barbara Partee. Blackwell Publishers, Oxford, 2004.)