Abstract
This paper investigates the question of how to correctly capture the scope of singular thinking. The first part of the paper identifies a scope problem for the dominant view of singular thought maintaining that, in order for a thinker to have a singular thought about an object o, the thinker has to bear a special epistemic relation to o. The scope problem has it is that this view cannot make sense of the singularity of our thoughts about objects to which we do not or cannot bear any special epistemic relation. The paper focuses on a specific instance of the scope problem by addressing the case of thoughts about the natural numbers. Various possible solutions to the scope problem within the dominant framework are assessed and rejected. The second part of the paper develops a new theory of singular thought which hinges on the contention that the constraints that need to be met in order to think singularly vary depending on the kind of object we are thinking about. This idea is developed in detail by discussing the difference between the somewhat standard case of thoughts about spatio-temporal medium-sized inanimate objects and the case of thoughts about the natural numbers. It is contended that this new Pluralist theory of singular thought can successfully solve the scope problem.
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Notes
An interesting question is whether, and if so why, one of the two constraints has some priority over the other. This question deserves an extensive examination which is better postponed to another separate investigation.
Russell maintains that sense-data are non-mental external particulars, but it is unclear what his views on the relation between sense-data and ordinary objects are. I thank an anonymous referee for urging me to acknowledge this point.
I am using ‘special’ instead of ‘direct’ since I take testimonial chains of communication to be mediated ways of being epistemically related to objects.
These authors challenge (SEV) in different ways. Some of them think that the semantic constraint is both necessary and sufficient to think singular thoughts (so-called semantic instrumentalists such as Hawthorne and Manley 2012). Others replace the epistemic constraint with different constraints (e.g. Crane 2011 and Jeshion 2010). More on this below.
Vlad the Impaler is a character of David Foster Wallace’s The Broom of the System.
A couple of clarifications are in order. First, the number of specific instances of the scope problem might vary depending on one’s views about the nature of objects. For instance, one might hold that fictional objects are abstract objects. Secondly, in order for the scope problem to get off the ground, it is sufficient to single out one kind of object to which we do not bear the special epistemic relation imposed by the epistemic constraint. In the following, I identify and focus on an instance of the scope problem which enables us to steer clear of complications arising from metaphysics.
A clarification is in order. Some authors draw a distinction between epistemically special causal relations and epistemically special contextual relations. Recanati (2012) subsumes them under the label of ‘epistemically rewarding relations’. Examples of contextual relations are the relations we bear to our egocentric spatial and temporal locations, and indexical relations of self-identity. However, authors availing themselves of this distinction do not explain how a contextual relation manages to be epistemically rewarding without being causal. Moreover, it strikes me as implausible to maintain that we bear a sort of indexical or egocentric relation to numbers (nor do authors accepting the causal/contextual distinction think so). For these reasons, I’ll frame my discussion here in terms of causal acquaintance relations only.
For this reason, I won’t here engage with Giaquinto’s (2001) idea that we can be acquainted with cardinal numbers, which are properties of sets, by experiencing instances of them. Giaquinto appeals here to what Dehaene (1997) dubs the number sense, that is, the innate, pre-conceptual faculty whereby we can perceptually estimate the cardinal size of a given plurality. According to Dehaene, our number sense is an analog system that represents approximately the cardinal size of a plurality of perceived objects. It does so by accumulating a continuous variable, e.g. physical magnitude. This amounts to saying that the number sense will never get us a discrete representation of five as opposed to the six; rather, it will give us an approximate representation which fails to be correlated with a discrete number.
In order to have an Idea of an object, T should satisfy the Generality Constraint (1982: p. 104): “If a subject can be credited with the thought that a is F, then he must have the conceptual resources for entertaining the thought that a is G, for every property of being G of which he has a conception”.
I am focusing here on how Evans interprets the epistemic constraint imposed by (SEV). I grant that his view satisfies the semantic constraint.
From this result, Benacerraf draws two conclusions: there is no non-arbitrary reason to choose one reduction over the other and, for this reason, numbers do not exist. The point I want to make is independent of both conclusions, though.
A critic could argue that I haven’t spelled out fundamental Ideas of numbers correctly. For instance, one could reject the claim that numbers must be identified on the basis of their arithmetical properties only and avoid the Benacerraf Problem. One way to do so would be to accept the neo-logicist claim that the essence of numbers is given by what Frege called Hume’s Principle (see Wright 1983). Hume’s Principle says that the cardinal number belonging to the concept F is identical to the cardinal number belonging to the concept G iff the entities falling under F are in 1-1 correspondence with the entities falling under G. Neo-logicists maintain that this principle works as an abstraction operator which produces numbers as its values. By adding Hume’s Principle to second-order logic, we can derive the Dedekind–Peano axioms, thereby accounting for arithmetic to in purely logical terms. I cannot address this option in depth here, but let me notice that controversies about the status of Hume’s Principle are such that, if our account of the possibility of having singular thoughts about numbers had to rely on such controversial principle, it would certainly lack the required generality to make such an account theoretically attractive.
The acquaintance norm is exploited just in case, even if T is not acquainted with the object, T presupposes the acquaintance norm and makes “as if [T] were using the file normally (i.e. according to the norm) to refer to an object of acquaintance” (Recanati 2013: p. 209).
Things get more complicated when we focus on thoughts about animate objects, such as persons. Murez and Smortchkova (2014) argue that, in this case, conceptual and empirical reasons support the introduction of a distinct type of file, what they call a person file. I need not go into these details here.
I’m using here “concept” as synonymous with mental representation to conform to Carey’s terminology.
Supporters of (SEV) might acknowledge this failure yet maintain that, all things considered, this is not enough to give up (SEV). They might insist that (SEV) has further advantages which trump its inability to solve the scope problem. I don’t find this reply incredibly convincing: the scope problem is the problem of capturing the correct extension of singular thinking, and I find it hard to downplay the importance of such an issue. At all events, I think that supporters of (SEV) should grant that, ceteris paribus, a theory of singular thought that solves the scope problem is preferable to one that does not.
Hawthorne and Manley might actually offer a slightly different argument, in that they accept a second principle they call Harmony (2012: p. 38): “Any belief report whose complement clause contains either a singular term or a variable bound from outside by an existential quantifier requires for its truth that the subject believe a singular proposition”. Since, my dissatisfaction with semantic instrumentalism will be largely independent of the specific argument they might use to reject (5), these details won’t matter.
An anonymous referee suggests that Evans’ point might be resisted by saying that any linguistic stipulation would be in some important sense related to some cognitive mechanism. Then, semantic instrumentalism would become the view that our cognitive systems generate singular representations via reference-fixing descriptive representations alone. I cannot examine this hypothesis in detail, but let me notice that those philosophers accepting semantic instrumentalism in virtue of their endorsement of the externalist idea that reference depends on social and causal factors that are external to our mind will not light-heartedly accept that linguistic stipulation is somehow parasitic upon mechanisms internal to the thinker’s mind.
To forestall misunderstandings: there is an intuitive sense in which any version of (SEV)—as well as any version of semantic instrumentalism—countenancing the possibility of singular thoughts about at least two different kinds of object, e.g. concrete and abstract, is pluralist. While I do not deny this, let me stress that I am using “monism” and “pluralism” in the somewhat technical sense of there to be just one (monism) or more than one (pluralism) constraint(s) to be met in order to have singular thoughts about different kinds of object.
As the reader might have foreseen, I am distinguishing different kinds of singular thought on the basis of the kinds of object thoughts are about. I acknowledge that this is far from uncontroversial, especially in light of heated disputes in metaphysics about the nature of certain objects (e.g. fictional objects). Yet, I believe that since the main aim of this paper is to outline an as yet unexplored approach to singular thought, and since this aim is pursued by relying on the generally accepted distinction between concrete and abstract objects (“generally accepted” in the sense that many accept the existence of the distinction while, at the same time, debating on how to trace it), we can postpone a full discussion of the identification method of different kinds of singular thought to a separate investigation.
To cite a few authors making this claim: Bach (1987), Goodman (2016), Murez and Smortchkova (2014), Recanati (2012). Something very similar is proposed by Azzouni (2011: p. 59) and Evans (1982: pp. 35–36). Hawthorne and Manley (2012: pp. 17–19) have reservations about the idea that what is opposed to satisfactional thinking is relational thinking, where the latter is understood in broadly epistemic-causal terms (as is done by supporters of SEV). However, I am not subscribing here to the satisfactional/relational dichotomy, for the pluralist view precisely denies the identification of singular thinking with relational (i.e. epistemic-causal) thinking. Rather, I am starting to characterise the category of singular thought in a negative way, as it were, by saying that it is constituted by all those thoughts which do not involve thinking of objects qua satisfiers of properties. As far as I can see, this claim is not touched by Hawthorne and Manley’s concerns.
I am grateful to an anonymous referee for suggesting this intuitive formulation.
I am borrowing this expression from Murez and Smortchkova (2014: p. 643).
An anonymous referee worries that the appeal to functionalism might not carry the required explanatory weight, in that functionalism could be apt to explain notions on which we have a pre-theoretical take whereas singular thought is a theoretical construct. It is off the agenda of the present discussion to put forward a sustained defence of the functionalist framework. So, I shall confine myself to the following two observations. First, various philosophers have developed functionalist-like accounts of (to some extent) theoretically loaded notions, such as truth (Lynch), theoretical terms (Lewis), and properties (Shoemaker). So, I’m in good company in thinking that the functionalist approach can be extended to more theoretically loaded notions. Secondly, while I agree that we have less pre-theoretical purchase on singular thought than on, say, feeling pain, the phenomenon of singular thinking is not a mere theoretical construct—as opposed, for instance, to the related notion of a singular proposition. It seems to me clear that we have an intuitive take on what thoughts—and other representational vehicles such as utterances—are about, and in explaining the object-directed aim function and the cognitive impenetrability function I have also invoked some intuitive considerations. In light of this, I think that my appeal to a broadly functionalist framework to develop a pluralist approach to singular thought is not misplaced.
Obviously, whether Enoch’s indispensability argument succeeds is not at issue here.
I will be interchangeably using expressions such as ‘using numerals in thought’, ‘mental tokening of numeral-like representations’, ‘tokening the mental analogous of numerals’. I use them as follows: Numerals occur in sentences which express thoughts. If the sentence contains a numeral which aims to refer to a number n, in order for a thinker to entertain the relevant thought expressed by the sentence she has to deploy a mental representation which also aims to refer to n. So, the mental representation in question is a syntactic constituent of the thought just like the numeral is a syntactic constituent of the sentence.
A wealth of research in cognitive science supports the hypothesis that humans represent the same numbers in different forms. The prominent triple-code model (see Dehaene 1997; Dehaene et al. 1999; Spelke and Tsivkin 2001) has it that we represent numbers in an analog magnitude code which is used for approximate calculation, in an Arabic visual number code which is used for parity judgements and multi-digit operations, and in a verbal code which is used for pure computational operations. Clearly, the analog magnitude system is not the one relevant here, in that it produces a merely approximate representation of numbers. So, the cognitive project of mental computation involves representations of numbers belonging to the Arabic visual number code and the verbal code.
The registered advantages hold for other place-value representational systems.
The same applies, perhaps to a greater extent, to more computationally complex mathematical formulas and non-arithmetical descriptions such as ‘The number of gin and tonics my friend Hichem has had tonight’.
An anonymous referee has pointed out that there is an analog of simple pure computations that could be carried out by taking numbers to behave as properties. I do not want to dispute this claim, for the point I am arguing for is compatible with it. To wit, my point is that when we focus on the cognitive project of mentally performing simple computations in which numbers behave as objects, it is instrumentally indispensable to token numeral-like representations of numbers. This is what enables us to regard the object-directed thoughts involved in such a project as singular, as opposed to descriptive.
The problem, in a nutshell, is that Hume’s Principle tells us whether two objects described in mathematical terms are identical or not but it does not tell us which objects can be described in mathematical terms. Thus, Hume’s Principle does not provide us with the resources to establish whether Julius Caesar is a number or not. Neo-logicists offer a solution to the Julius Caesar problem, but their solution is contested by some non-logicist platonists and nominalists. For a nice overview of the debate, see section 6 of MacBride (2003).
Let me emphasise that what I said so far is compatible with the claim that we have acquired numerical concepts through the bootstrapping process described by Carey. Surely, this process requires a certain cognitive effort, and it might seem to involve the acquisition of some descriptive knowledge about numbers. Yet, this does not mean that while deploying these concepts in the computing project thinkers have to access and deploy that descriptive knowledge. This consideration bears close resemblance to a point made in relation to descriptive names by authors defending different theories of singular thought, such as Azzouni (2011), Hawthorne and Manley (2012) and Recanati (2012). They all acknowledge that we can think of Jack the Ripper singularly even if all we have is descriptive information about him.
Two clarifications are in order. First, one might wonder how the instrumental indispensability constraint interacts with the semantic constraint I have presented at the beginning of this paper. I want to remain neutral on the question whether the purely cognitive characterisation of the singular thought-role and the semantic constraint are both necessary to capture the singularity of our thoughts. I wish to stress that if one thinks that they both are (or must be), the instrumental indispensability constraint is compatible with different ways of showing that thoughts about numbers playing the singular thought-role also have singular truth-conditions. To illustrate. If one is a platonist, a standard truth-theoretic semantics will guarantee the semantic singularity of these thoughts (as well as of the correspondent sentences). By contrast, if one is a nominalist, one could show that the relevant representations satisfy the semantic constraint via a pretence-theoretic semantics. Secondly, let me stress the mental representations tokened in the computation project are not object-dependent, in that the existence of numbers is not required to carry out the computing project (see also Azzouni 2009 on this), nor can they be identified with mental files if we stick to Recanati’s acquaintance-based type-individuation of mental files.
The view I am articulating might be taken to bear some similarities to Robin Jeshion’s cognitivist theory of singular thought. Jeshion’s idea, in a nutshell, is that a thinker entertains a singular thought about an object just in case the thinker opens a mental file for the object; to open a mental file for the object, the object must be significant to the thinker’s plans, projects, affective states, motivations, (see Jeshion 2010). Even though a comparison between my view and Jeshion’s should be postponed to another separate investigation, let me stress two important differences. First, Jeshion takes the cognitive significance constraint to govern all kinds of singular thoughts, whereas my pluralist view holds that thoughts about numbers are subject to the instrumental indispensability constraint while thoughts about spatiotemporal inanimate medium-sized objects are subject to an epistemic constraint (more on this below). Secondly, instrumental indispensability for a cognitive project and significance for an individual’s plans, motivations and so on appear to be two rather different properties.
Thanks to an anonymous referee for suggesting this line of reasoning.
The research for this paper has been generously supported by the I + D + I, Spanish Government, research project #FFI2016–80588-R, by the Juan de la Cierva postdoctoral fellowship programme, and by a postdoctoral fellowship from the Centre for Research on Ethics of the University of Montreal and McGill University. I wish to thank audiences in Barcelona, Montreal, and Oslo, in particular Dominic Alford-Duguid, David Davies, Ian Gold, Max Kölbel, Daniel Morgan and Peter Pagin, for helpful feedback on various parts of this material. I am particularly grateful to Anton Alexandrov, Manuel García-Carpintero, J.P. Grodniewicz, Michael Murez, François Recanati, and Carlota Serrahima for providing detailed comments on previous drafts of this paper. I also wish to thank three anonymous referees for this journal, whose extensive and valuable comments improved the paper dramatically. Last but not least, I wish to thank Hichem Naar for a thought-inspiring and particularly nice conversation on the topics of this paper while taking a stroll in Jarry Park in Montreal.
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Palmira, M. Towards a pluralist theory of singular thought. Synthese 195, 3947–3974 (2018). https://doi.org/10.1007/s11229-017-1401-4
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DOI: https://doi.org/10.1007/s11229-017-1401-4