Formal Semantics and Pragmatics — Deep
Formal semantics is the branch of linguistics that uses mathematical and logical tools to model the truth-conditional content of natural-language expressions.
Formal pragmatics extends those tools to phenomena that depend on context, speaker intention, and inferential reasoning.
The field emerged in its modern form in the late 1960s and early 1970s with Richard Montague’s program of treating English as a formal language with model-theoretic semantics, and with Paul Grice’s work on conversational implicature.
By 2026 the field has integrated experimental psycholinguistics, computational modeling, neuroscience, and large-language-model probing alongside its analytical core.
Montague Grammar and the foundational program
Richard Montague’s three short papers English as a Formal Language (1970), Universal Grammar (1970), and The Proper Treatment of Quantification in Ordinary English (PTQ, 1973) launched formal semantics as a distinct subfield.
Montague’s claim, against the prevailing view, was that there is no important theoretical difference between natural and formal languages and that both can be given a single mathematical treatment.
Montague’s semantics uses a typed lambda calculus over a domain that includes entities (type e), truth values (type t), possible worlds, and times.
Common types include e for entities, t for truth values, <e,t> for one-place predicates, <<e,t>,t> for generalized quantifiers, and <e,<e,t>> for two-place relations.
Compositionality is implemented by functional application: the meaning of a complex expression is determined by the meanings of its parts and the way they are combined.
Intensional logic introduces possible worlds and times to handle non-extensional contexts like attitude reports and modal operators.
Lambda abstraction implements variable binding and is essential for capturing quantifier scope.
Barbara Partee’s Possible Worlds in Semantics (1973) and her Lectures on Semantics at Linguistic Society of America summer institutes through the 1970s and 1980s brought Montague’s program into mainstream linguistics.
Irene Heim and Angelika Kratzer’s textbook Semantics in Generative Grammar (1998) is the field’s standard graduate introduction, formalizing the integration of Montagovian semantics with generative syntax through type-driven interpretation, predicate modification, and the principle of full interpretation.
Generalized quantifiers
Generalized quantifier theory generalizes the classical first-order quantifiers (universal and existential) to a uniform treatment of natural-language determiners.
Jon Barwise and Robin Cooper’s Generalized Quantifiers and Natural Language (1981) laid mathematical foundations.
Determiners like every, some, most, exactly three, and no denote functions from sets to sets of sets, of type <<e,t>,<<e,t>,t>>.
A determiner D applied to a noun-phrase restrictor A yields a generalized quantifier D(A), which is true of a predicate B just in case the appropriate set-theoretic condition holds.
Conservativity is the universal that, for any determiner D, D(A)(B) is equivalent to D(A)(A and B); the determiner only “looks at” the intersection of restrictor and scope.
Monotonicity properties partition determiners into upward-entailing (UE) and downward-entailing (DE) classes, with mixed-monotonicity determiners like most.
William Ladusaw’s Polarity Sensitivity as Inherent Scope Relations (1979 dissertation) connected downward entailment to the licensing of negative polarity items.
Irene Heim’s A Note on Negative Polarity and Downward Entailingness (1984) refined the licensing condition to Strawson downward entailment.
Anastasia Giannakidou’s Polarity Sensitivity as (Non)veridical Dependency (1998) recast NPI licensing in terms of nonveridicality, accommodating a broader range of cross-linguistic data.
Edward Keenan and Jonathan Stavi’s A Semantic Characterization of Natural Language Determiners (1986) explored the mathematical universe of possible determiners.
Tense, aspect, and event semantics
Hans Reichenbach’s Elements of Symbolic Logic (1947) introduced the three-time analysis of tense in terms of speech time S, reference time R, and event time E.
Wolfgang Klein’s Time in Language (1994) reformulated the system as a relation between topic time and time of situation, more tractable for cross-linguistic application.
Zeno Vendler’s Verbs and Times (1957) introduced the four-way classification of lexical aspect (Aktionsart): states, activities, accomplishments, and achievements.
David Dowty’s Word Meaning and Montague Grammar (1979) provided a formal treatment of Vendlerian classes and addressed the imperfective paradox, the question of why John was crossing the street does not entail John crossed the street.
Donald Davidson’s The Logical Form of Action Sentences (1967) introduced event arguments into first-order logic, allowing adverbial modifiers to be analyzed as predicates of events.
Terence Parsons’s Events in the Semantics of English (1990) developed the neo-Davidsonian variant in which thematic roles (agent, patient, instrument, theme) are introduced as separate predicates of the event.
Recent treatments of aspect include Susan Rothstein’s Structuring Events (2004) and Hana Filip’s work on Slavic aspect and quantization.
The interaction of tense, aspect, and modality is treated in Sabine Iatridou’s work on counterfactuals (The Grammatical Ingredients of Counterfactuality, 2000) and in Cleo Condoravdi’s Temporal Interpretation of Modals (2002).
Plurality and mass
Godehard Link’s The Logical Analysis of Plurals and Mass Terms (1983) introduced an algebraic semantics in which the domain of entities is partially ordered by a part-of relation.
Pluralities are sums of individuals, and mass terms denote material parts of substances.
Cumulative readings (The boys lifted the boxes) and distributive readings are derived from independent operators over the same logical structure.
Collective predicates (The students gathered) and reciprocal predicates (The boys hit each other) impose plurality requirements.
Partitives (some of the boys) and proportional quantifiers (most of the wine) draw on the part-of structure.
Gennaro Chierchia’s Reference to Kinds Across Languages (1998) proposed the Nominal Mapping Parameter to capture cross-linguistic variation in mass-count grammar, predicting differences in determiner systems and bare-noun argumenthood.
Lucas Champollion’s Parts of a Whole (2017) develops an event-based theory of plurality that unifies nominal and verbal pluractionality.
Definiteness
Bertrand Russell’s On Denoting (1905) analyzed definite descriptions as quantificational expressions with a uniqueness presupposition.
Irene Heim’s The Semantics of Definite and Indefinite Noun Phrases (1982 dissertation) introduced file-change semantics, in which indefinites introduce new discourse referents and definites refer to existing ones.
Hans Kamp’s A Theory of Truth and Semantic Representation (1981) introduced Discourse Representation Theory (DRT) on similar principles, with discourse referents represented in nested boxes.
The two frameworks converged on a dynamic view of meaning as context-change potential.
Florian Schwarz’s Two Types of Definites in Natural Language (2009 dissertation) argued, drawing on German der versus contracted preposition-article forms, for a distinction between uniqueness definites and familiarity (or anaphoric) definites, a distinction now widely accepted with crosslinguistic evidence from Akan, Mandarin, Spanish, and many other languages.
Bridging definites (I entered the room. The chandelier was dusty.) require accommodation of a definite without prior explicit introduction.
Modality
Angelika Kratzer’s series of papers (The Notional Category of Modality 1981, Modality 1991) provides the standard framework for modal expressions in terms of two contextual parameters: a modal base and an ordering source.
The modal base supplies a set of accessible worlds; the ordering source ranks them by some standard.
Epistemic modality (It must be raining) relies on an epistemic modal base.
Deontic modality (You must leave) relies on a circumstantial modal base and a deontic ordering source.
Other flavors include teleological, bouletic, and stereotypical.
David Lewis’s Counterfactuals (1973) developed possible-world semantics for counterfactual conditionals using a similarity ordering.
Kratzer’s Conditionals (1986) introduced the restrictor view, on which if-clauses restrict overt or covert modal operators.
Subjunctive and indicative conditionals are distinguished by their interaction with the modal base.
The interaction of modality and tense is the locus of much recent work, including Sabine Iatridou (2000), Magdalena Kaufmann’s Interpreting Imperatives (2012), and Valentine Hacquard’s Aspects of Modality (2006 dissertation, MIT).
Anaphora and binding
Tanya Reinhart’s Anaphora and Semantic Interpretation (1983) formalized syntactic binding theory and its semantic correlates.
The integration of binding into Heim-Kratzer’s framework treats binding as a relation between a binder index introduced syntactically and a variable in its scope.
Donkey anaphora (Every farmer who owns a donkey beats it), studied by Peter Geach (Reference and Generality, 1962) and named by him, motivated the development of dynamic semantics.
The E-type approach (Gareth Evans 1977, Stephen Neale 1990) treats it as a definite description recoverable from the antecedent.
The dynamic approach (Heim 1982, Kamp 1981, Jeroen Groenendijk and Martin Stokhof’s Dynamic Predicate Logic 1991) extends binding across sentential boundaries.
Sloppy and strict identity in ellipsis (John loves his mother and Bill does too) is treated by Daniel Büring’s Binding Theory (2005) and by Pauline Jacobson’s variable-free semantics.
Focus and information structure
Mats Rooth’s dissertation Association with Focus (1985) and his 1992 paper introduced alternative semantics, in which focused expressions denote sets of contextually relevant alternatives.
The squiggle operator marks the set anaphorically related to the focus.
Focus particles only, even, and also are analyzed as quantifiers over the focus alternatives.
Manfred Krifka’s work on topic-focus articulation (Basic Notions of Information Structure, 2008) integrates these tools with broader information-structural categories.
Nomi Erteschik-Shir’s Information Structure: The Syntax-Discourse Interface (2007) develops a syntactic theory of focus and topic.
Craige Roberts’s Information Structure in Discourse (1996) develops a question-under-discussion framework, on which the discourse is organized around a hierarchy of explicit and implicit questions.
Question-answer congruence requires that an assertion address the current question with appropriate focus marking.
Conversational implicature
H. Paul Grice’s William James Lectures, published as Logic and Conversation (1975, in Syntax and Semantics vol. 3), introduced the Cooperative Principle and the four maxims of conversation: Quantity, Quality, Relation, and Manner.
Conversational implicatures are inferences that arise when an utterance respects the Cooperative Principle but appears to violate a maxim at the level of what is said.
Generalized conversational implicatures arise by default, particularized only in specific contexts.
Scalar implicature derives the not-all reading of some from the Quantity maxim: a cooperative speaker who knew all to hold would have said all, so some implicates not all.
Laurence Horn’s On the Semantic Properties of Logical Operators in English (1972) introduced Horn scales relating quantifiers and modal expressions by logical strength.
Uli Sauerland’s Scalar Implicatures in Complex Sentences (2004) introduced primary and secondary implicatures.
Gennaro Chierchia’s Scalar Implicatures, Polarity Phenomena, and the Syntax/Pragmatics Interface (2004) argued for a grammatical view of scalar implicature, computed locally with each scope-bearing operator.
Stephen Levinson’s Presumptive Meanings (2000) treats generalized implicature as a level of default inference between literal meaning and particularized context.
Presupposition
Gottlob Frege’s Über Sinn und Bedeutung (1892) implicitly distinguished assertion from presupposition.
Peter Strawson’s On Referring (1950) challenged Russell’s analysis of definites with the observation that The king of France is bald does not seem false but rather lacks a truth value.
Robert Stalnaker’s Presuppositions (1973) and subsequent work treats presupposition as a constraint on the common ground of conversation.
Irene Heim’s On the Projection Problem for Presuppositions (1983) developed satisfaction theory, on which presuppositions project to the global context unless locally satisfied.
The projection problem asks how presuppositions of embedded clauses project to the matrix sentence; the standard inheritance patterns for connectives, conditionals, and quantifiers have been a major topic.
David Beaver’s Presupposition and Assertion in Dynamic Semantics (2001) integrates presupposition into a dynamic framework.
Accommodation, introduced by David Lewis’s Scorekeeping in a Language Game (1979), describes the process by which a hearer adds an unstated presupposition to the common ground to allow an utterance to be interpreted.
Speech acts and dynamic pragmatics
J. L. Austin’s How to Do Things with Words (1962, William James Lectures 1955) distinguished locutionary, illocutionary, and perlocutionary acts and introduced the notion of performative utterances.
John Searle’s Speech Acts (1969) and Expression and Meaning (1979) systematized the typology of illocutionary force into assertives, directives, commissives, expressives, and declaratives.
Kent Bach and Robert Harnish’s Linguistic Communication and Speech Acts (1979) integrates speech-act theory with Gricean pragmatics.
Robert Stalnaker’s Assertion (1978) treats assertion as a context-update operation that adds the proposition expressed to the common ground.
Craige Roberts’s Information Structure in Discourse (1996, revised 2012) develops discourse structure as a stack of questions under discussion.
Jonathan Ginzburg’s The Interactive Stance (2012) elaborates dialogue as a record of question-answer interaction.
The semantics of questions begins with Charles Hamblin’s Questions in Montague English (1973), which treats questions as sets of possible answers.
Lauri Karttunen’s Syntax and Semantics of Questions (1977) takes the partition of logical space induced by question to be the question’s denotation.
Jeroen Groenendijk and Martin Stokhof’s Studies on the Semantics of Questions and the Pragmatics of Answers (1984 dissertation) develops the partition semantics in depth.
Experimental and computational approaches
Experimental semantics and pragmatics emerged in the 1990s and 2000s as a methodological complement to armchair theorizing.
Julie Sedivy, Michael Tanenhaus, Sarah Brown-Schmidt, and colleagues developed the visual world paradigm, in which participants’ eye movements during spoken-language comprehension reveal moment-by-moment processing of contextual cues, definites, scalar terms, and other phenomena.
Lewis Bott and Ira Noveck’s Some Utterances Are Underinformative (2004) demonstrated that scalar implicature is cognitively costly to compute, supporting context-sensitive rather than default models.
Yi Ting Huang and Jesse Snedeker’s Online Interpretation of Scalar Quantifiers (2009) used the visual world paradigm to show that scalar implicatures emerge gradually in incremental comprehension.
The Rational Speech Acts (RSA) framework, introduced by Michael Frank and Noah Goodman in Predicting Pragmatic Reasoning in Language Games (Science 2012), models pragmatic inference as recursive Bayesian reasoning between speakers and listeners.
Subsequent work by Goodman and Frank, Pragmatic Language Interpretation as Probabilistic Inference (Trends in Cognitive Sciences 2016), surveys the rapidly growing RSA literature, which has been applied to scalar implicature, hyperbole, irony, gradable adjectives, and quantity-quality trade-offs.
Probabilistic pragmatics models continue to develop in dialogue with linguistic theory, computational psycholinguistics, and increasingly machine learning.
Formal semantics and large language models
The intersection of formal semantics with large language models has been a productive frontier since 2020.
Emily Bender and Alexander Koller’s Climbing Towards NLU (ACL 2020), the so-called octopus paper, argued that a model trained only on form cannot acquire meaning in the sense relevant to natural language understanding.
The argument turns on a thought experiment about an intelligent octopus eavesdropping on a transatlantic telegraph cable.
Sayash Kapoor, Ryan Cotterell, and others have pushed back with arguments about what counts as grounding.
Kyle Mahowald, Anna Ivanova, and colleagues’ Dissociating Language and Thought in Large Language Models (Trends in Cognitive Sciences 2024) argues for distinguishing formal linguistic competence (which LLMs largely possess) from functional linguistic competence (which they do not reliably exhibit).
Albert Webson and Ellie Pavlick’s Do Prompt-Based Models Really Understand the Meaning of Their Prompts? (NAACL 2022) showed that prompted LLMs are robust to systematically wrong prompt content.
Gašper Beguš and colleagues’ work probes whether models internalize formal semantic generalizations.
LLMs are increasingly evaluated against benchmarks designed for formal semantic phenomena (BIG-Bench, BLiMP for syntax, BIG-bench-hard for compositional reasoning) with mixed results on quantifier scope, modal reasoning, presupposition projection, and counterfactuals.
Degree semantics and gradability
Gradable adjectives such as tall, expensive, and smart pose a distinctive set of analytical problems.
Christopher Kennedy’s Projecting the Adjective (1997 dissertation, published 1999) developed a degree-based semantics in which gradable adjectives denote relations between individuals and degrees on a scale.
Kennedy and Louise McNally’s Scale Structure, Degree Modification, and the Semantics of Gradable Predicates (Language 2005) classified gradable predicates by scale structure into relative (open-scale), maximum-standard, minimum-standard, and totally-closed types, with predictions about modifier compatibility.
Comparatives (taller than Mary, more intelligent than expected) are treated as quantification over the degree argument.
Equatives (as tall as Mary) similarly involve degree comparison.
Probabilistic models of vagueness, including RSA-based accounts (Lassiter and Goodman 2017), treat the gradable threshold as inferred contextually.
Heather Burnett’s Gradability in Natural Language (2017) integrates degree semantics with sociolinguistic variation.
Expressive content and conventional implicature
Conventional implicatures, distinguished from conversational implicatures by Grice but largely neglected for decades, were systematically reanalyzed by Christopher Potts.
Potts’s The Logic of Conventional Implicatures (2005) provided a multidimensional semantics in which expressive content (appositive relative clauses, supplements, honorifics, expressives like damn and slurs) is computed on a separate dimension from at-issue content.
The framework has been extended to pejoratives and slurs in work by Robin Jeshion, Geoffrey Nunberg, and Elin McCready.
Conventional implicature is now recognized as a third major dimension of meaning alongside truth-conditional and presuppositional content.
Iconicity and gesture
Formal semantics has expanded to engage non-verbal modalities.
Philippe Schlenker’s What Is Super Semantics? (2018, Annual Review of Linguistics) sets out a program for extending formal semantics to gesture, signs, music, and visual narrative.
Sign language semantics, including work on ASL by Schlenker and on LSF by Carlo Geraci, integrates manual modality into the same framework as spoken language.
Iconic dimensions of meaning, where the form of a sign or gesture resembles its referent, force extensions of the standard model-theoretic apparatus.
Gesture-speech integration has been studied by Cornelia Ebert and others using formal semantic tools.
Neurolinguistics of composition
Liina Pylkkänen’s group at NYU has used magnetoencephalography (MEG) to identify neural signatures of compositional operations.
The left anterior temporal lobe (LATL) shows reliable activation increases for semantic composition relative to lexical retrieval (Bemis and Pylkkänen, Journal of Neuroscience 2011, and subsequent work).
The ventromedial prefrontal cortex contributes additional combinatorial processing for complex compositions.
This neurolinguistic evidence connects formal semantic theory to brain mechanisms, addressing the long-standing question of whether the theoretical apparatus corresponds to identifiable cognitive operations.
Stanislas Dehaene, Christophe Pallier, and colleagues have used fMRI to identify hierarchical structure-building regions in the language network.
Computational sociolinguistics and meaning change
The integration of formal semantics with computational and sociolinguistic methods has produced new tools for studying meaning change.
Word-embedding methods (Word2Vec, Mikolov et al. 2013; GloVe, Pennington et al. 2014; and subsequent contextual embeddings from BERT and successors) provide a way to measure semantic similarity at scale.
Diachronic word embeddings (Hamilton, Leskovec, Jurafsky Diachronic Word Embeddings Reveal Statistical Laws of Semantic Change, ACL 2016) have documented quantitative regularities in lexical-semantic drift across English over two centuries.
Robert Henderson’s Variable Force in Maya (Linguistics and Philosophy 2019) integrates fieldwork with formal modal semantics.
The rapid growth of corpus-based pragmatics has supplemented introspective judgment with naturalistic data.
Frontier topics, 2024 to 2026
Experimental work on vagueness and gradable adjectives, especially in the framework of degree semantics, continues with probabilistic models of threshold-setting.
Sub-sentential pragmatics, including pejoratives, slurs, and expressive content, connects formal pragmatics to social meaning.
Neurolinguistic work on the brain bases of compositionality probes which formal-semantic operations correspond to identifiable neural signatures.
The integration of formal semantics with computational sociolinguistics, including the study of how meaning conventions shift across communities, continues to expand the discipline’s scope.
LLM probing studies have proliferated as a methodology for testing whether models internalize formal semantic distinctions, with results varying by phenomenon and model scale.
Hossep Dolatian and Jonathan Rawski’s work on subregular phonology has parallels in subregular semantic computation.
Theories of generic sentences (Carlson and Pelletier’s The Generic Book, 1995, and ongoing work by Sarah-Jane Leslie, Bernhard Nickel, and Ariel Cohen) continue to develop in dialogue with social-cognition research on stereotype formation.
The pragmatics of polite indirection, conventionalized implicature in formulaic language, and cross-linguistic variation in implicature patterns are active areas at the intersection of formal pragmatics and sociopragmatics.
Methodological pluralism
The contemporary field is methodologically pluralistic to a degree unusual in linguistic theory.
Armchair judgment-based theorizing continues alongside experimental psycholinguistics, computational modeling, corpus analysis, neuroscience, and LLM evaluation.
The mainstream is broadly converged on a model-theoretic, compositional, dynamic-pragmatic architecture, with controversies localized to specific phenomena rather than foundational frameworks.
Maria Aloni’s work on free-choice inferences, Anna Szabolcsi on quantifier scope, Mandy Simons on presupposition, Stefano Predelli on context, Jeffrey King on propositional content, Daniel Lassiter on probabilistic semantics, and many others continue to extend the program.