Abstract
Formal semantic theories are generally thought to make contact with pre-theoretic semantic notions of aboutness and reference. The nature of that contact is, however, not always straightforward. This paper addresses two debates where that issue assumes a significant role. I begin with Simchen’s recent argument that Lewisian Interpretationism succumbs to referential indeterminacy. I develop a proposal about the relationship between the theoretical notion of a term’s semantic value and the pre-theoretic notion of reference, and argue that the indeterminacy Simchen identifies does not constitute a form of referential indeterminacy. In the second part, I apply these resources to the debate between singularist and pluralist approaches to the semantics of plural terms, arguing that a certain form of singularism that emerges from the dialectic ends up agreeing with the pluralist at the level of reference, if not semantic value. The paper concludes with some remarks on what a properly referential style of singularism might look like.
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Notes
Let me flag that the question at issue here—concerning what a term refers to, and what sentences containing it are about—is more restricted than the one involved in other discussions of aboutness, such as the more expansive notion of a sentence’s overall subject matter discussed by e.g. Yablo (2014).
Similarly, Link (1984) quips that “if my kids turn the living room into a mess I find it hard to believe that a set has been at work, and my reaction is not likely to be that of a singleton set.” Link’s second remark, about singleton sets, anticipates the treatment of singular terms discussed below.
The metasemantic views of the historical Lewis were of course more complex, but I’ll follow Simchen in working with this simplified version.
For example, Sider (2011) suggests that naturalness be extended “beyond the predicate” and understood as having to do with worldly structure in general, allowing us to speak of more or less natural meanings for connectives, quantifiers, and so on. One might go further and apply the notion not only to the meanings of words, but also to the operations via which they are composed, arguing that standard composition better reflects some aspect of worldly structure (perhaps a predicational structure) than the scrambled alternative. Or one might take a cue from Williams (2007) and maintain that general standards of theoretical simplicity (analyzed in terms of naturalness) will rule out scrambled semantics.
Putnam did also offer indeterminacy arguments involving domain permutations that leave truth conditions unchanged, most explicitly in the Appendix of Putnam (1981). As Taylor (2006) notes, however, it is Putnam’s Argument from Completeness rather than the Argument from Permutation that poses a direct challenge to realism, and Lewis (1983, 1984) seems primarily concerned to answer Putnam’s argument against realism. At any rate, the point I wish to make is just that if we’re going to worry about indeterminacy below the level of truth conditions we ought to be explicit about the reasons for that worry and make sure those reasons apply in the case at hand.
Other semantic proposals that could be agued to exhibit a distinction between reference and semantic value include views on which terms denote second order properties, as in Montague (1973), individual concepts, as in e.g. Aloni (2005), or sets, as in e.g. Evans (1982) or (I shall suggest) the singularist semantics below. Indeed, Evans (1982, p. 32) explicitly notes that his proposal involving sets as term denotations illustrates that “the equation between semantic value and referent is by no means mandatory.” Thanks to an anonymous referee for reminding me of Evans’ discussion. I have previously addressed this kind of distinction in Rieppel (2018).
I hasten to note that Simchen (2017a) is alive to issues in this vicinity, concerning the contrast between the technical notions of semantics and pre-theoretic semantic concepts. He focuses on the semantic values of verbs and quantifier phrases; what I am suggesting is that similar complications arise even in the case of singular terms. A closely related distinction at the sentential level—between compositional semantic value and assertoric content, or “what is said”—has been a recent focus of attention in e.g. Ninan (2010), Rabern (2012), and Yli-Vakkuri (2013), drawing on Dummett (1973) and Lewis (1980).
Compare Dummett’s (1973) notion of specifying an expression’s Bedeutung in a way that “shows” its sense, and the application of that idea to predicate semantic values in Heim and Kratzer (1998, Ch. 2). Similarly, the idea here is that predicate semantic values can be specified in a way that explicitly appeals to the natural property the Lewisian Interpretationist takes the predicate to express. Let me flag that this way of conceiving of the role of natural properties in eligible semantic theories differs from the proposal in Williams (2007). Williams suggests the role of naturalness is to provide an analysis of the general notion of theoretical simplicity in non-subjective terms, and that an eligible semantics is then one that scores high against its competitors on the scale of theoretical simplicity. This approach could be developed into an alternative reply to Simchen, as mentioned in footnote 5 above. Thanks to an anonymous referee for flagging this.
As Simchen (2020) points out, one can construct not just a scrambled semantics, but also a jumbled semantics, one that uses a jumbler function to permute the properties denoted by predicates in the language. On such a semantics, the truth of \(\varphi (t)\) doesn’t require that \(\llbracket t\rrbracket \) instantiate \(\llbracket \varphi \rrbracket \), but rather that it instantiate \(\rho (\llbracket \varphi \rrbracket )\), the image of \(\varphi \)’s denotation under the jumbler \(\rho \). On such a semantics, however, the property expressed by the predicate, in the sense explained above, would then be \(\rho (\llbracket \varphi \rrbracket )\) rather than \(\llbracket \varphi \rrbracket \), since it is the instantiation of \(\rho (\llbracket \varphi \rrbracket )\) rather than \(\llbracket \varphi \rrbracket \) that is required for the truth of atomic sentences containing \(\varphi \).
Pluralist proposals include Boolos (1984), Yi (2005, 2006), McKay (2006), and Oliver and Smiley (2016a). Set-based approaches have been proposed Scha (1981) and Schwarzschild (1996), and broadly property-based approaches include Higginbotham and Schein (1989) and Florio (2014). Another variety of singularism, which I shall not consider here, takes ‘Alice and Béla’ to denote the sum of Alice and Béla, an entity that has each of them as its parts. This view, associated with Link (1983), faces difficulties if sums are construed in standard mereological fashion: pairs like ‘the cards in the two decks’ and ‘the two decks of cards’ will denote the same thing, since the two decks, and the multitude of cards, compose the same mereological whole. To avoid this, the sums in question need to be given a more set-like structure. For a recent discussion and defense, see Florio and Nicolas (2021).
Though set-based singularists sometimes also present their proposals in type-theoretic terms. Scha (1981), for example, explains that he is working within a type system where “every semantic type has a set of entities as its domain” and “the denotation of an expression is necessarily an element of the domain of its type” (Scha, 1981, p. 485). Thus he begins with a type individual, and then introduces a derived type S(individual), whose domain is to consist of sets of individuals, to serve as the denotations of plural terms (as well as singular terms, for the reasons discussed below).
This operation would be an example of what Oliver and Smiley (2016a, Ch. 9) call a multi-valued function: one that, applied to a given argument (a set in this case) returns potentially many values. For the property-based singularist, the suggestion would be that the predicate ‘wrote a book’ denotes the second-order property \((\hat{p}. u(p) \text { wrote a book})\), which holds of a property p just in case the things that instantiate p wrote a book.
Although one could perhaps follow Evans (1982) and include the empty set as the semantic value assigned to empty (i.e. non-referring) terms. Cf. footnote 8.
Allowing plural terms to denote single things (singleton sets in the present context, or singleton properties on the property-based variant) is again quite standard. As Yi (2005) points out, for example, ‘Cicero and Tully’, though syntactically plural, denotes just one thing (or, here, a singleton). And Boolos (1984) proposed that the plurally quantified ‘there are some Fs’ should not be understood to require the existence of more than one F. As Schwarzschild (1996, p. 5) notes, there is good reason to go this route, since we want ‘no students passed’ to come out false (and therefore ‘some students passed’ true) even if just one student passed.
The general picture is not peculiar to set-based singularists. Singularists that appeal to sums generally make use of a structurally isomorphic domain populated by sums instead of sets, see e.g. Link (1998, Ch. 2) and Nouwen (2016). Property-based singularism will generate a similar structure, since the instantiation relation \(\sqsubset \) generates a partial order \(\sqsubseteq \) similar to the way membership does for sets: \(p_1 \sqsubseteq p_2 \equiv _{df} \forall x(x \sqsubset p_1 \rightarrow x \sqsubset p_2)\). Atomicity for properties could then be defined as above. One disanalogy is that if properties are construed non-extensionally, there may not be an analogue of the union operation (since there can be more than one property \(p_3\) instantiated by all and only those things that instantiate both \(p_1\) and \(p_2\)), so the treatment of term-conjunction may have to differ.
There is, as mentioned, a significant worry, going back to Boolos (1984), that singularism cannot allow for unrestricted first-order quantification. For an overview of this debate, see Rayo and Uzquiano (2006). Florio (2014) offers a version of typed property-based singularism designed to address this.
The response to Yi’s argument might be different for the property-based singularist. She might reply that doubly-instantiated properties like \(\hat{x}(x = \text {Russell} \vee x = \text {Whitehead})\) can only be denoted by plural terms. In response to the objection that one should be able to introduce a singular term for any object one likes, the property-based singularist has at least two options. One move, paralleling the set-based singularist’s response, would be to grant that we can introduce a term ‘Genie’ to refer to the property in question, but that what this term will denote is the singly-instantiated property \(\hat{y}(y = \ \hat{x}(x = \text {Russell} \vee x = \text {Whitehead}))\). Alternatively, the singularist could go type-theoretic and insist, in a Fregean vein, that \(\hat{x}(x = \text {Russell} \vee x = \text {Whitehead})\) is not an object but an entity of a genuinely higher logical type, and that singular terms not only don’t denote such entities, but also cannot refer to them, since only objects can ever be referred to. (Though care will need to be taken in articulating this second reply, since the previous sentence on the face of it involves reference to the relevant property in the course of denying that very possibility. I have discussed this kind of difficulty in Rieppel (2018).)
This style of set theory was proposed by Quine (1969). By identifying urelements with their singletons, Quine aimed to allow for an unrestricted formulation of the axiom of extensionality \(\forall x(x \in y \equiv x \in z) \rightarrow y = z\) on which it holds quite generally, for sets and urelements alike. Socrates and Plato won’t be ruled identical by the axiom, because Socrates qua {Socrates} has Socrates as a member, but Plato does not.
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For helpful comments and feedback at various stages of this paper’s development I’d like to thank Thiago de Melo, John Keller, Arc Kocurek, Harvey Lederman, Tom McKay, Ori Simchen, two anonymous referees for this journal, and audience members at the 2020 Syracuse Philosophy Graduate Conference.
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Rieppel, M. Reference by proxy. Synthese 200, 180 (2022). https://doi.org/10.1007/s11229-022-03681-3
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DOI: https://doi.org/10.1007/s11229-022-03681-3