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.^[1]^[a]

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.^[3] Formal semanticists examine a wide range of linguistic phenomena, including reference, quantifiers, plurality, tense, aspect, vagueness, modality, scope, binding, conditionals, questions, and imperatives.^[4]

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.^[5]

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.^[6]

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.^[7] 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.^[8]^[b]

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.^[13]^[c]

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.^[15]

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.^[16] Following this principle, formal semanticists connect natural language sentences to abstract models^[d] through a form of translation, for instance, by defining an interpretation function that maps the name "Tina" to an abstract object^[e] and the adjective "thin" to a set of objects.^[19]^[f] This makes it possible to precisely calculate the truth values of sentences relative to abstract models.^[21]

Within formal semantics, there are diverse ways how to construct models and relate linguistic expressions to them.^[22] 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.^[23] 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.^[24]

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.^[25]

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 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.^[26]^[27]^[28]^[29]

Binding

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

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.^[30]^[31]^[32]

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.^[33]^[34]^[35]

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.^[35] 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".^[35]

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.^[35] 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.^[36]^[35] 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.^[37]^[35]

Cognitive semantics emerged as a reaction against formal semantics, but there have been recently several attempts at reconciling both positions.^[38]

References

Notes

^ Because of the prominence of these tools, the terms truth-conditional semantics and model-theoretic semantics are sometimes used to refer to formal semantics even though they do not have exactly the same meaning.^[2]
^ However, many current approaches to formal semantics posit that there is more to meaning than truth-conditions.^[9] 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.^[10] Other approaches generalize the concept of truth conditionality or treat it as epiphenomenal. For instance in dynamic semantics, knowing the meaning of a sentence amounts to knowing how it updates a context.^[11] Pietroski treats meanings as instructions to build concepts.^[12]
^ Typically, entailment relations only go in one direction. However, they can also go in both directions if two sentences entail each other, like the sentences "Tina is tall and thin" and "Tina is thin and tall". In such cases, the two sentences are said to be equivalent.^[14]
^ This general method also reflects the externalist theory of meaning common in formal semantics: the meaning of an expression is interpreted as the entities it denotes in an abstract model, without focusing on cognitive processes internal to language users.^[17]
^ This can be expressed symbolically through the use of double brackets. For example, the formula $[[{\text{Tina}}]]^{M}$ refers to the object denoted by the name "Tina" in the model "M".^[18]
^ In this context, semanticists typically speak of denotation: the denotation of an expression is the entities to which it refers.^[20]

Citations

^
- Portner & Partee 2002, pp. 1–2
- Partee 2016, pp. 3–4
- Winter 2016, pp. 3–4
^
- Portner 2005, pp. 14
- Moeschler 2007, p. 32
- Saeed 2009, p. 305
^
- Moeschler 2007, pp. 31–32
- Griffiths & Cummins 2023, p. 1
- Bezuidenhout 2009, p. 875
^
- Cann 1993, pp. ix–xii, 1–3
- Winter 2016, pp. 5–6
^
- Portner & Partee 2002, pp. 1–2
- Partee 2016, pp. 3–4
- Lappin 2003, pp. 370–371
- Yalcin 2014, pp. 17–54
- von Fintel 2023
^
- King 2009, pp. 557–558
- Fox 2014, pp. 85–87
- Portner 2005, pp. 214–216
- Barba 2007, pp. 637–639
^
- Portner & Partee 2002, pp. 1–2
- Partee 2016, pp. 3–4
- Winter 2016, pp. 3–4
- Kearns 2011, p. 24
^
- Lappin 2003, pp. 375–376
- Kearns 2011, pp. 6, 8–11
- Winter 2016, pp. 17–21
^ Stefano Predelli (2013). Meaning without truth. Oxford Scholarship. ISBN 9780199695638.
^ Ciardelli, Ivano; Groenendijk, Jeroen; Roelofsen, Floris (2019). Inquisitive Semantics (PDF). Oxford University Press.
^ Veltman, Frank (1996). "Defaults in Update Semantics" (PDF). Journal of Philosophical Logic. 25 (3). doi:10.1007/BF00248150. S2CID 19377671.
^ Paul Pietroski (2018). Conjoining meanings. Oxford University Press. ISBN 9780198812722.
^
- Winter 2016, pp. 12–13
- Kearns 2011, pp. 11–12
^ Winter 2016, p. 16
^
- Winter 2016, pp. 17–18
- Fox 2014, pp. 86, 90–92
^
- Winter 2016, pp. 28–29
- Lappin 2003, pp. 370, 374
- Stokhof 2013, pp. 208–209
- Fox 2014, pp. 90–91
^
- Winter 2016, pp. 18, 240–241
- Portner & Partee 2002, pp. 1–2
- Partee 2016, pp. 3–4
- Lappin 2003, pp. 370–371
^ Winter 2016, p. 18
^
- Fox 2014, pp. 86, 90, 93
- Winter 2016, pp. 18, 240–241
- Saeed 2009, pp. 309–310
^
- Winter 2016, p. 18
- Saeed 2009, pp. 309–310
^ Winter 2016, pp. 24–27
^
- Lappin 2003, pp. 370–371
- Moeschler 2007, pp. 32
- Fox 2014, p. 92
^
- Stokhof 2007, pp. 609–611
- Stokhof 2013, pp. 209–210
^
- Janssen & Zimmermann 2025, § 2.3 Logic and Translation
- Matthews 2007
^
- Stokhof 2013, pp. 210–213
- Winter 2016, pp. 12–16
^ Heim, Irene; Kratzer, Angelika (1998). Semantics in Generative Grammar. Oxford: Wiley Blackwell. pp. 194–198.
^ Ruys, Eddy; Winter, Yoad (2011). "Quantifier scope in formal linguistics." (PDF). In Gabbay, Dov; Guenthner, Franz (eds.). Handbook of Philosophical Logic (2 ed.). Dordrecht: Springer. pp. 159–225. doi:10.1007/978-94-007-0479-4_3. ISBN 978-94-007-0478-7.
^ Barker, Chris (2015). "Scope" (PDF). In Lappin, Shalom; Fox, Chris (eds.). Handbook of Contemporary Semantics (2 ed.). Wiley Blackwell. Section 4.3. doi:10.1002/9781118882139.ch2. ISBN 9781118882139.
^ Szabolcsi, Anna (2010). Quantification. Cambridge University Press. p. 92.
^ Portner, Paul (2009). Modality. Oxford: Oxford University Press. ISBN 978-0-19-929242-4.
^ 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
^ Starr, Will (2019). "Supplement to "Counterfactuals": Indicative and Subjunctive Conditionals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.
^ Barwise, Jon; Cooper, Robin (1981). "Generalized quantifiers and natural language". In Kulas, J; Fetzer, J.H.; Rankin, T.L. (eds.). Philosophy, Language, and Artificial Intelligence. Studies in Cognitive Systems. Vol. 2. Springer. pp. 241–301. doi:10.1007/978-94-009-2727-8_10. ISBN 978-94-010-7726-2. S2CID 62189594.
^ For a very readable and succinct overview of how formal semantics found its way into linguistics, see The formal approach to meaning: Formal semantics and its recent developments by Barbara Abbott. In: Journal of Foreign Languages (Shanghai), 119:1 (January 1999), 2–20.
^ ^a ^b ^c ^d ^e ^f Partee, Barbara (2011). "Formal semantics: Origins, issues, early impact". The Baltic International Yearbook of Cognition, Logic and Communication. 6. CiteSeerX 10.1.1.826.5720.
^ Crnič, Luka; Pesetsky, David; Sauerland, Uli (2014). "Introduction: Biographical Notes" (PDF). In Crnič, Luka; Sauerland, Uli (eds.). The art and craft of semantics: A Festschrift for Irene Heim.
^ Michael Moortgat (1988). Categorial investigations: logical and linguistic aspects of the Lambek calculus. Walter de Gruyter. ISBN 978-90-6765-387-9. Retrieved 5 April 2011.
^ Hamm, Fritz; Kamp, Hans; Lambalgen, Michiel van (2006-09-01). "There is no opposition between Formal and Cognitive Semantics". Theoretical Linguistics. 32 (1): 1–40. CiteSeerX 10.1.1.80.6574. doi:10.1515/tl.2006.001. ISSN 1613-4060. S2CID 17691054.

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