Formal Semantics and Applied Mathematics: An Inferential Account


In this paper, I utilise the growing literature on scientific modelling to investigate the nature of formal semantics from the perspective of the philosophy of science. Specifically, I incorporate the inferential framework proposed by Bueno and Colyvan (Nous 45(2): 345–374, 2011) in the philosophy of applied mathematics to offer an account of how formal semantics explains and models its data. This view produces a picture of formal semantic models as involving an embedded process of inference and representation applying indirectly to linguistic phenomena. The final aim of the paper is directed at proposing a novel account of the syntax–semantics interface while shedding light on empty categories, semantically null forms, underspecified content and compositionality as a whole.

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Fig. 1


  1. 1.

    Yalcin (2018) deems it a mistake to separate theory from modelling as they are mutually reinforcing enterprises. This is indeed often the case as models are sometimes designed to test theories and theories are often explained in terms of their models. However, there is also a wide range of cases in which this ideal is practically unfeasible or even unnecessary. For instance, some formal models are designed with practical efficiency or applicability in mind and not as explanations of natural phenomena (they might even be false with relation to this target). See Weisberg (2013) for examples of “modelling for modelling sake” in contemporary biological research.

  2. 2.

    Hodges view can be taken to argue that model-theory is not scientific modelling. I will argue that this is not the case in the use of model-theory in formal semantics nor necessarily even in mathematical logic.

  3. 3.

    This approach is not merely a convenience. I follow Giere (1988) here in starting from an analysis of textbooks as opposed to active research in a field (I do not maintain his focus on them throughout though). “If we wish to learn what a theory is from the standpoint of scientists who use that theory, one way to proceed is by examining the textbooks from which they learned most of what they know about that theory” (Giere 1988: 63).

  4. 4.

    These languages are learnable but cannot be acquired. When pidgins do get adopted by linguistic communities, they tend to undergo what is referred to as “creolisation” which involves the establishment of systematic syntactic and semantic rules. It should be stated that this latter point is not uncontroversial.

  5. 5.

    However, even the term “compositional” might not be as theory neutral as assumed here. There are a number of arguments against compositionality in the literature. Although I think most of these arguments are pitched at the level of the target system and semantics concerns a theoretical distortion of that system. Non-compositional semantics abounds as a viable alternative but it reflects an alternative modelling practice. I thank an anonymous reviewer for stressing the need to mention this point.

  6. 6.

    Schiffer (2015) claims that the kind of formal semantics inspired by Montague is antagonistic to the so-called Generative Grammar Hypothesis or the view that the task of semantics is to investigate the semantic component of generative grammar or a speaker’s tacit knowledge of her language. The view urged here disrupts this alleged distinction.

  7. 7.

    Interestingly, the role of intuitions in linguistics has led to major criticism of the field in its aspiration as an empirical science (see Hintikka 1999; Stokhof 2011). Semantics viewed as modelling mitigates some of these concerns. Modellers are often driven by intuitions in their model construction and their status as empirical scientists is not usually called into question.

  8. 8.

    This is an oversimplification. Derivational morphology can change both the meaning and grammatical category of words in a compositional fashion. Consider the differences between employment, employer, unemployable or more clearly compose, decompose, decomposable, decomposability.

  9. 9.

    Johnson (2007) makes a similar point in defence of semantic mininalism against the onslaught of radical contextualism.

  10. 10.

    A construal is composed of an assignment, intended scope and two fidelity criteria for Weisberg, where the latter are “the standards that theorists use to evaluate a model’s ability to represent real phenomena” (2013: 76).

  11. 11.

    Larson and Segal (1995) and Heim and Kratzer (1998) would also fall within this tradition.

  12. 12.

    It should be noted that Brandom (2008) alters this claim somewhat in his holistic non-compositional or “incompatibility” semantics. There the view is that the learnability and productivity properties essential for understanding natural language need not require compositionality, which fails in particular cases.

  13. 13.

    The landscape is changing, however. For a survey of various methods being explored in semantic analysis, see Krifka (2011).

  14. 14.

    For Cann (1993: 2) “this theory is a formal theory of semantics and is distinguished from general linguistic semantics by its greater use of mathematical techniques and reliance on logical precision”.

  15. 15.

    The mathematics involved in the intensional variant is very much the same as the extensional theory with the exception of the possible world type s and functions involving it.

  16. 16.

    Most current accounts in syntax use DPs or determiner phrases instead of NPs even when there is a null determiner. I ignore these details for the sake of simplicity. Similarly for I(nflectional) P(hrasal) and C(omplementiser) P(hrasal) categories.

  17. 17.

    The two logical forms of the sentences are:

    1. (1)

      \(\forall x(Boy(x)\rightarrow \exists y(Toy(y)\wedge likes(y)(x)))\)

    2. (2)

      \(\exists y(Toy(y)\wedge \forall x(Boy(x)\rightarrow likes(y)(x)))\).

  18. 18.

    See Bunt and Muskens (1999) for a proof of this based on the ambiguity in an average Dutch sentence.

  19. 19.

    There are some specifics involved here, namely three steps which include immersion or the initial mapping, the derivation step which is when theorists draw mathematical conclusions in the model and finally the interpretation step in which the previous conclusions are interpreted in the empirical or assumed structure. For more details, see Bueno and Colyvan (2011).

  20. 20.

    This framework is starkly different from the assumed modelling process in linguistics which looks more like the figure below (not to be confused with the postulated architecture of the language faculty complete with generative elements, Logical Form and phonetic spellout). In the figure below, the syntactician models the empirical phenomenon independently of the semanticist who focuses on different idealised aspects of the target system. The dashed line indicates the usual interface assumptions such as compositionality etc.

  21. 21.

    It should be noted that Hughes is not arguing for an inferential account in the quote above but rather what he calls DDI or Denotation, Demonstration and Interpretation. However, the two accounts do have a lot in common. Nevertheless, it is not my intention in the present work to take any specific stance on the philosophy of applied mathematics.

  22. 22.

    There is another (sub)puzzle which has an analogue in semantics as well, namely the problem of prediction. Unlike the original statement of the puzzle or “unreasonable effectiveness of mathematics” courtesy of Wigner (1960), Steiner (1998) argues that the wonder of applied mathematics involves not one puzzle but a family of puzzles. Colyvan (2001) considers the puzzle of prediction to be particularly intriguing, “where the mathematics seems to be playing an active role in the discovery of the correct theory” (267). A canonical case of this in the natural sciences is how Maxwell’s equations predict electromagnetic radiation (this is Colyvan’s test case).

    The NPI licensing literature displays the predictive powers of semantic models similarly. There has been a wealth of linguistic literature on the topic of Negative Polarity Items (NPIs) in the past few decades. Part of the reason that these particles have been so widely studied is that they exhibit strange hybrid characteristics. When they are unlicensed in a particular context (cannot be appropriately used or generally not found) we are left with an infelicity akin to violations of syntactic rules . However, the linguistic situations in which these items are licensed are generally explained purely in terms of semantics. This is interesting for a number of reasons. For one thing, the NPI phenomena were never considered to be semantic data before certain entailment relations were noticed with relation to it. Thus, we saw a shift in what counts as data for semantic theory as discussed in Sect. 2.1. Another reason is that the model theory does predictive and descriptive work (see Rothschild 2009 for an account).

  23. 23.

    For a vigorous debate on where exactly to draw the line between semantics and pragmatics, see Cappelen and Lepore (2005) and Recanati (2004).

  24. 24.

    I thank an anonymous reviewer for pressing me on this point.

  25. 25.

    As further evidence that this present work captures the status quo of semantic modelling consider an approach which explicitly aims to reject this type of modelling and instead propose a direct semantic framework. Szabó (2015) places semantic values at the forefront of the target system directly which stands in contrast to the view of semantics as applied mathematics presented above. He addresses the alleged folly of assigning type-theoretic semantic values to syntactically defined parts of speech and the idea that grammatical notions intervene in such a way as to prevent grammatical categories receiving purely semantic treatment. The orthodoxy in linguistics aims to define parts of speech such as noun, verb, adjective, adverb etc. in terms of syntactic distribution as opposed to the more intuitive semantic descriptions (such as noun \(=\) object, verb \(=\) event etc.). In my view, the original division of labour stems from the strong claim of the autonomy of syntax within early generative grammar. Nevertheless, formal semantics then provides a logical translation for these categories in terms of type theory, lambda calculus and so on, as I have argued above. Thus, there appears to be a disconnect between grammar and logic, at least at the lexical level. Szabó, however, thinks that this detour to the semantic value of lexical categories is an unnecessary (and problematic) one. His view can be seen as a species of Abstract Direct Representation (ADR) discussed by Weisberg (2007) in the philosophy of science.

  26. 26.

    This is another point at which Yalcin’s very recent analysis of formal semantics as model-based science might diverge from mine. Yalcin takes semantic analysis to be an indirect representation of an aspect of “human linguistic competence - informally, knowledge of meaning” (2018: 353). What makes a model a good one for this task is then a matter of how it explains human speech and understanding. I think FSMs are more circumscribed in their workings than this. For the most part they derive model-theoretic structures from other models with the goal of explaining the semantics facts. This process, on my view, is partially autonomous from the explanation of the target system such as linguistic competence. There are, however, many points on which we are in complete agreement.

  27. 27.

    Even very different semantic formalisms such as Discourse Representation Theory (DRT), which are intended to be interpreted cognitively, presupposes a complete syntactic model. This is brought out by one of the chief problems with DRT, namely that it has been argued not to capture the incremental nature of semantic parsing (hence the move to frameworks such as Segmented Discourse Representation Theory). I thank Hans Kamp for this observation.

  28. 28.

    On the other side, Stanley and Szabó (2000) argue from semantic theory to hidden elements in the syntax in order to explain quantifier domain restriction. On this view, NPs have covert argument places for domain restriction in contexts. This is an excellent example of semantic modelling being interpreted back into the syntactic formalism as discussed in the previous section.

  29. 29.

    Of course, linguists can insist that syntax (and semantics) are scientific theories and the issues I mention are theoretical disputes. But this option carries the burden of explaining how mathematical objects such as functions, operations like recursion (or merge), infinite generalisations etc. can be found directly in nature. This burden is taken up by Chomsky et al. (2002) to inimical effect. See Nefdt (2019) for a discussion of these considerations.


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I am indebted to Josh Dever, Ephraim Glick, Patrick Greenough, Hans Kamp, Kate Stanton, Zoltán Szabó and audiences at the Arché Research Centre in St Andrews and Yale Semantics Seminar for helpful comments and invaluable guidance on the content of this article. I also thank two anonymous reviewers of this journal for their necessary and insightful criticisms which only served to make the work better. Any remaining problems are entirely of my own making.

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Nefdt, R.M. Formal Semantics and Applied Mathematics: An Inferential Account. J of Log Lang and Inf 29, 221–253 (2020).

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  • Formal semantics
  • Applied mathematics
  • Scientific modelling
  • Philosophy of linguistics