Abstract
This chapter explores the history of model-theoretic semantics, arguing that such theories can provide objective truth conditions for sentences involving “racist” and “woman.” It covers the contributions of Frege, Tarski, and Montague to contemporary formal semantics, at each stage raising considerations about how these theories may interact with subjugated knowledge and power. The concluding section of this chapter argues for the Informed Speaker Constraint, which says that a speaker’s judgments only provide data about the semantics of an expression of a natural language if they can be considered knowledgeable about the elements in the domain of discourse that the expression semantically functions to represent. I also argue that the Informed Speaker Constraint, when applied to politically contested terminology, entails that a science of semantics cannot be politically neutral.
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Notes
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Locating Frege as the most important source of modern logic is contentious. Another major contender for this role, who I will not discuss in any detail, is Charles Sanders Peirce, who independently arrived at some of the crucial insights of Frege at roughly the same time. Both Frege and Peirce strongly influenced logicians of the next generation. Ramsey, Russell, Whitehead, and more or less every logician at the turn of the twentieth century was acquainted with both Frege’s and Peirce’s theorizing about logic. Sousing out the lines and degrees of influence is beyond the scope of this chapter. I will proceed without further argument with Partee’s narrative that Frege was the foremost founder. Many thanks to David Yaden for bringing this issue to my attention.
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The third important contribution, which I am leaving out of the discussion here, is Frege’s (1879) theory of quantification. My aim in this chapter is to articulate and defend the possibility of giving an objectively true semantics at all, and for this purpose it would only complicate the argument to worry about the details of quantification.
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This was not the only purpose with which Frege was concerned. Much of what he says in defense of his semantic framework draws on our knowledge of natural language, and he also clearly intended his framework to be applicable to languages in general and so to natural languages as well as mathematical ones. Thanks to Josh Dever for emphasizing this point and also for contributing several other insights to this chapter.
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It’s unclear whether Frege himself held this view of natural language. Conservatively, we might suppose that Frege himself was neutral about whether natural languages should be interpreted in accord with a more perfect ideal version of that language, although the footnote in “Sense and Reference” suggests he may have been sympathetic to the ideal-language view I will defend in Chap. 10.
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Note: it is not necessary that every compositional semantic theory assign {true, false} as the compositional value of every sentence. Other kinds of compositional semantics are possible that are not centered on the concept of truth. And even in contemporary truth-functional formal semantics, it is an oversimplification to suppose that every sentence will have true or false as its compositional semantic value, and this supposition is not entailed by compositionality itself. Nevertheless, it is a helpful oversimplification for understanding the basic concepts of a truth-functional compositional semantics.
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This insight is often attributed to Davidson (1967) who identified the task of a theory of meaning for a language with the task of giving a finite recursive theory for the truth conditions of the sentences of that language.
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In the Grundgesetze (Frege, 1893), Frege stipulates that The True is the ‘course of values’ of the identity function, in other words, the sequence of everything which is identical with itself, or in other words, everything; and The False is its compliment, which is the set of things which are not self-identical, which is nothing at all. While this is a definition of sorts, it is not a substantive theory of truth. It’s a stipulation identifying The True and The False with everything and nothing, respectively. While this may present a certain degree of metaphysical satisfaction, depending on your aesthetic preferences, it’s not really any different than stipulating that The True is the number 42 and The False is the number 45. There’s nothing substantive or illuminating about such stipulations. They can’t be used for any further reasoning or justifications. Later Frege (1918) endorses a redundancy theory of truth, asserting that “P” and “P is true” express the same thought, writing that “nothing is added to the thought by my ascribing to it the property of truth.” Thanks again to Josh Dever for encouraging me to get these details down.
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Frege was opposed to the tradition from which model theory is descended. As we will see, model theory is all about stipulating the domain of discourse, the model under consideration. For Frege, logic is not concerned primarily with models; it is concerned with the entire universe. For discussion, see Heijenoort (1967) and Pedriali (2017).
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For an extremely thorough and excellent overview of the development of non-Euclidean geometry and its impact on formal logic, see Nagel (1939). The fact that Tarski himself was following the example of non-Euclidean geometers is evident in his earliest application of his model-theoretic approach (Tarski, 1931). See also Simmons (2009).
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Although whether this is Tarski’s own preferred way of understanding, his theory is a matter of debate—see Mancosu, 2008. Specifically, it is not clear that Tarski thought of himself as theorizing about a correspondence relation between language and reality beyond the words. But it does seem that even if this was not his preferred interpretation, Tarski was happy enough to let others interpret his work on semantics this way.
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In the context of this book it may be important to note that while Frege is known to have endorsed anti-Semitic views in his private diaries, including speculating that Jews should be deprived of their political rights and expelled from Germany, Tarski on the other hand was Jewish and lost many relatives and colleagues to the holocaust, surviving the war by haphazardly finding himself in the US at its outbreak (Eastaugh, 2017). Insofar as Frege’s anti-Semitism might be seen as relevant for assessing the logical tradition he initiated (Nye, 1990), it must be equally important to note that the logical tradition was transformed into its modern guise by a displaced Jewish logician who personally sustained dreadful losses due to racism and anti-Semitism.
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Note that in Tarski (1933) where he first develops this technique of copying the object language into the metalanguage, he did not explicitly describe the metalanguage construction technique in model-theoretic terms; he defines a truth predicate in the metalanguage for the object language without mentioning models or providing tools for interpreting names and predicates differently using different models and interpretation functions. Nevertheless, we can see his procedure as a model-construction technique, one that paved the way for a fuller and more expansive model theory by providing one systematic way of describing a relationship between an object language and a domain of entities and properties, that is, those referred to by expressions of the object language.
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Frege (1879) does give a recursive definition of truth for sentences constructed using the logical operators (and, not, if…then…, etc.), which is part of what makes him arguably a stronger candidate than Peirce to hold the title of foremost founder of modern logic. But Tarski goes further by providing the basis for defining truth more generally for sentences, not just those constructed using logical operators.
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Tarski generalizes the definition further to include all predicates of any arbitrary length: n-place predicates. An n-place predicate has n arguments and so can require up to n distinct objects in the model to satisfy it. Since the predicates of a formal language can have n-place predicates for any number n, a full definition requires defining satisfaction relations involving indefinitely many objects in the model. For this reason, Tarski defines satisfaction using infinitely long sequences of objects. By defining satisfaction in terms of infinite sequences, he guaranteed that the definition would cover every case; it would cover any predicate having any arbitrary number of argument places. Of course, the natural languages we speak don’t use predicates with more than four or five places, so infinite sequences are a bit overkill.
Tarski reasoned that an adequate definition of truth must satisfy what he called condition T. Here’s how condition T works. For every sentence S in the object language, take the name for sentence S—which we write using quotes: “S”—and then use the sentence to specify its own truth condition. So we get a list of truth-condition sentences (T-sentences), one for every sentence S in the language. Then for every sentence S in the language, we have: “S” is true if and only if S. This set includes “John is happy” is true if and only if John is happy. It also includes “Trump’s tweets were racist” is true if and only if Trump’s tweets were racist. This list of T-sentences specifies the goal of a definition of truth, a condition of adequacy. Any adequate definition of truth must entail, for every sentence in the object language, the corresponding T-sentence. Tarski (1933) shows that his method for defining truth in terms of satisfaction by infinite sequences meets condition T for certain regimented formal languages, such as the calculus of classes introduced by Whitehead and Russell (1910). He thinks it can be extended to some fragments of natural language as well, writing “if we translate into colloquial language any definition of a true sentence which has been constructed for some formalized language, we obtain a fragmentary definition of truth” for that colloquial, that is, natural, language (Tarski, 1933, pp. 165, footnote 2), although he believes truth cannot be defined in general for the whole of any natural language (ibid., section 1).
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See the email excerpt from Hans Kamp, Montague’s student, presented in Partee (2011, pp. 23-24) which gives this account of Montague’s impetus for deploying model theory to the task of characterizing the semantics of natural language.
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Although see Etchemendy (1994) for an extensive discussion of Tarski’s concept of logical consequence and reasons to be skeptical that it really is the ultimate theory of real logical consequence.
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Anderson, D.E. (2021). Model-Theoretic Semantics for Politically Contested Terminology. In: Metasemantics and Intersectionality in the Misinformation Age. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-73339-1_5
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