Abstract
In this paper, I argue that Scott Soames’ theory of naturalized cognitive propositions (hereafter, ‘NCP’) faces a serious objection: there are true propositions for which NCP cannot account. More carefully, NCP cannot account for certain truths of mathematics unless it is possible for there to be an infinite intellect. For those who reject the possibility of an infinite intellect, this constitutes a reductio of NCP.
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Notes
It is important to note, right from the outset, that ‘naturalism’, as it occurs in ‘propositional naturalism’, is a term of art. For example, propositional naturalists need not be wedded to the research program of naturalizing the mind, and not all of them are. (The success of King’s theory of naturalized propositions is clearly tied to the success of this research program, but Soames claims that his theory is independent of it (pc).) If these theories are not all committed to the broader naturalistic program of naturalizing the mind, in what sense are they naturalistic, and why do they use this term? I think the basic idea that they all share is that human minds are part of the natural world, so grounding the representational properties of propositions in the representational powers of human minds allows for a more naturalistic conception of propositions in the sense that it is not an utter mystery how they perform this essential function. Note, however, that on all of these theories, propositions are abstract.
Throughout, single quotes will function as mention quotes and corner quotes; double quotes will be used for direct quotation and scare quotes.
Wayne Davis develops a view of propositions similar to that of Soames (and Hanks) in Davis (2002). He also considers and responds to the Scarcity Objection (see pp. 315–316). However, Davis’ view differs in an important respect from Soames’: Davis is not a propositional naturalist, i.e., it is no part of his view that propositional representation is inherited from—let alone just is—the representation of cognitive acts. More importantly, since Davis is a Platonist about types as well as a traditionalist about propositional representation, he can avail himself of the plenitude of abstract thought-types that exist and have their truth-conditions necessarily and eternally—independent of thinkers. Thus, his view escapes the Scarcity Objection. This is not an option for propositional naturalists like Soames.
Jeff Speaks disagrees—he argues that propositions have truth-conditions, but do not represent (see Soames et al. 2014; Richard 2013 and Stalnaker 2012 agree with Speaks on this point). To avoid unnecessary distraction, I’m simply going to narrow my focus in this paper to those (admittedly, the majority) who think that propositions represent.
Soames is using ‘entertaining’ and ‘predicating’ in a technical sense; e.g. one need not be conscious of engaging in the act of predication in Soames’s sense.
Note that only the positive integers are being dealt with here, though the semantics can easily be extended to the negative integers. It is not at all obvious, however, that it can be extended to the reals—a simple Cantor-style diagonalization argument shows that there are infinitely many reals for which there is no linguistic representation in a language of countable symbols.
Isn’t being linguistically inaccessible sufficient for being un-cognizable? Not if, as seems plausible, young children and some non-human animals can cognize many objects without mastery of language. However, it is very plausible that certain sorts of objects—in particular, the sorts of mathematical objects under discussion in this paper—are only cognizable via language. Soames agrees (see Soames 1989).
This is why the Axiom of Choice is objectionable to mathematical constructivists, who offer choice-free alternative set theories such as IZF and CZF, or offer alternative, constructivist-friendly choice-principles such as countable choice and dependent choice (see Crosilla 2014; see also Bell 2011 for discussion of the non-constructive character of Choice and section 3 of Feferman 2000 for discussion of some of the paradoxical results of this axiom). But why isn’t constructivism an option for Soames? It is, of course, an option, as is the denial of the existence of sets, but it would be a pretty bad result if one’s theory of propositions forced one’s hand about mathematics in this way. In what follows, I assume the dominant view among set theorists, a form of realism that is sometimes called ‘Quasi-combinatorialism’ or ‘Combinatorialism’ [for discussion, see Bernays (1935), Ferreirós (2011), Gödel (1944), and Maddy (1997)].
Thanks to John Keller for suggesting the use of AC as an example.
Thanks to Chris Menzel for suggesting this definition of ‘choice-only’.
The notion of a choice-only set is much like what José Ferreirós calls an “arbitrary set”, which depends on the “indispensable use” of AC (Ferreirós 2011: 386). As Ferreirós explains, “Indispensable uses of AC occur precisely in cases where infinitely many sets are assumed to be given, with possibly arbitrary subsets among them, and a set of corresponding elements is needed. Russell’s famous exemplification of the axiom with infinite sets of socks, versus shoes, is aimed precisely at underscoring that AC is unnecessary when a formal predicate can be specified that “does the choosing””(ibid.). Thanks to Laura Crosilla for drawing my attention to Ferreirós’ paper.
One may take issue with my claim about the inaccessibility of choice-only sets if she adopts the Axiom of Constructibility, V = L (where ‘V’ stands for the von Neumann universe of sets and ‘L’ stands for Gödel’s constructible universe), which yields an extension of ZFC according to which, roughly, all sets are definable. A fuller discussion of the implications of this axiom is beyond the scope of this paper, but let me note that V = L is controversial: most set theorists reject V = L because it conflicts with the maximum iterative conception of a set (see Arrigoni 2011: 337–342; see also Maddy 1997, II.4)—I follow their lead in this paper.
The reason for talking about being in a position to cognize a set rather than simply cognizing it is to leave open the possibility that one may, e.g., cognize all of the members of a set S without thereby cognizing S. Suppose there are impure sets: then it’s plausible that one may cognize all of the members of my singleton (i.e., me) without thereby cognizing my singleton. It doesn’t seem plausible that one automatically cognizes a set simply in virtue of cognizing an individual.
Since I am making the negative point that finite agents cannot cognize choice-only sets, I only need to be concerned with necessary conditions for cognizing them: so long as finite agents fail to satisfy the disjunction of necessary conditions, they fail to cognize the relevant sets.
Note that all three of these Conditions are linguistic means for designating sets. In the discussion and arguments to follow, I will simply use the expression ‘cognize’ to cover both our ability to (directly, as it were) cognize a set and our ability to designate a set linguistically. Since the ability to cognize the sets I will talk about would plausibly depend on their linguistic accessibility anyway, this simplification is harmless. It is important to remember, however, that I’m arguing that choice-only sets are uncognizable and linguistically inaccessible—unlike, say, very large numbers.
Another way of making this point: there are no formal predicates in the language of set theory by which such sets are definable. Of course, definability is relative to a language—so the fact that there are no such predicates in any formal language that we currently use doesn’t mean that there aren’t any such predicates in any possible formal language. However, note that any finitary or recursive language will fall short of specifying all of the sets there are—this is simply a result of Cantor’s Theorem: e.g. given a finitary or recursive formal language L, there are only countably many definable subsets of \( {\mathbb{N}} \) in L, but by Cantor’s Theorem, there are uncountably many subsets of \( {\mathbb{N}} \) (cf. Ferreirós 2011: 364–5). And presumably, only such languages can be mastered by finite cognitive beings. I discuss objections from linguistic relativity in more detail in Sect. 4 below.
By ‘propositionalism’, I just mean the view shared by the parties to the debate under discussion: that there are such things as propositions, which sentences express in contexts of utterance, are the contents of the attitudes, etc.
Recall that Soames’ view does not require that a proposition be possibly entertained in order to exist and have truth-conditions—rather, it’s sufficient that the constituents of a proposition are cognized. However, I have argued that choice propositions are not entertained by any possible finite agent by way of arguing that they have constituents that are not cognized by any possible finite agent. So this condition is covered by my argument. For simplicity of presentation, though, I state the argument Against NCP in terms of entertainment.
Though, as the saying goes, “One person’s modus ponens is another person’s modus tollens” (though we’d have to reverse the order in this case). See Keller Forthcoming for a discussion of the modus ponens version of the argument (very roughly: if propositions are thoughts, then there’s an infinite thinker; propositions are thoughts; so, there’s an infinite thinker).
Indrek Reiland raised this objection.
Of course, if there are choice propositions, then not all propositions are possible objects of the attitudes (barring an infinite intellect). Is this a problem? I don’t think so—note that this does not involve denying that propositions play the role of possible contents of the attitudes. This only involves denying the claim that, for all x, if x is a proposition, then x is a possible attitude-content. But one can still hold that, for all x, if x is a possible attitude content, then x is a proposition. Also, note that holding that some propositions are not possible contents of the attitudes (of finite beings) does not lead to a problem of scarcity—there are still propositions enough to play the roles we think they should play—rather, there are more than enough propositions to play the belief-content role, since they are needed to play the primary-truth-bearer role.
Paul Nedelisky pressed this objection.
See, e.g., Soames et al. (2014): 34–35, 239.
Some version of this objection was raised by Paddy Blanchette, David Braun, and Ben Caplan.
This strategy only works on the (uncontroversial) assumption that there are (at least) continuum many worlds.
Versions of this objection were raised by several different people: the version presented here most closely resembles the objection raised by Lewis Michael Powell.
Spacetime may be continuous in the actual world, but it may not if, e.g., the theory of loop quantum gravity is correct. See Smolin (2014).
Though philosophers disagree widely over the sorts of constraints singular thought should meet, it’s generally accepted that thinking of an object under a description is not sufficient. Even Hawthorne and Manley (2012) seem to agree with this point despite defending a quite liberal conception of singular thought.
A strategy of this sort was suggested by Lewis Michael Powell. ‘Dthat’ is an operator invented by David Kaplan to convert singular terms such as definite descriptions into directly referential terms (see Kaplan 1978). For any expression of the form ‘dthat[\( \alpha \)]’, its content is whatever ‘ \( \alpha \) ’ denotes.
Tim Crane, invoking Robin Jeshion (2010), makes a related point about the insufficiency of singular linguistic devices for securing singular thought: “It is not plausible that someone who uses Kaplan’s name ‘Newman I’ to denote the first person born in the next century has any one person in mind or is aiming at any one person in thought. Their thought happens to alight on someone, if there is someone satisfying this condition—and this is the mark of a general thought” (Crane 2011).
See Kripke (1980: 297). I’m using Kripke’s example to support a different, but related, conclusion than the one for which Kripke argues (viz., roughly, that one can competently use the name to refer to Feynman without knowledge of individuating conditions): that the ignorant name-user is capable of singular thought about Feynman.
Thanks to Jeff Speaks for helpful discussion of this point.
Note that ‘V’ is not functioning as name, but as a bound variable.
This was proven in Vitali (1905).
Solovay showed this by constructing, in ZFC plus the axiom that there is an inaccessible cardinal, a model of ZF in which L is true (see Solovay 1970). Thanks to Chris Menzel for directing me to Solovay’s theorem.
References
Arrigoni, T. (2011). V = L and intuitive plausibility in set theory: A case study. Bulletin of Symbolic Logic, 17(3), 337–360.
Bell, J. L. (2011). The axiom of choice in the foundations of mathematics. In G. Sommaruga (Ed.), Foundational theories of classical and constructive mathematics. Berlin: Springer.
Benacerraf, P., & Putnam, H. (1983). Philosophy of mathematics: Selected readings (2nd ed.). Cambridge: Cambridge UP.
Bernays, P. (1935). Sur le platonisme dans les mathématiques. L’Enseignement Mathématique, 34, 52–69.
Crane, T. (2011). The singularity of singular thought. In Proceedings of the aristotelian society supplementary volume (Vol. LXXXV).
Crosilla, L. (2014). Set theory: Constructive and intuitionistic ZF. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy. http://plato.stanford.edu/archives/sum2015/entries/set-theory-constructive/. (Summer 2015 Edition)
Davis, W. (2002). Meaning, expression, and thought. Cambridge: Cambridge University Press.
Feferman, S. (2000). Mathematical intuition versus mathematical monsters. Synthese, 125, 317–332.
Ferreirós, J. (2011). On arbitrary sets and ZFC. The Bulletin of Symbolic Logic, 17(3), 361–393.
Gödel, K. (1944). Russell’s mathematical logic. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics: Selected readings, 1983.
Hanks, P. (2007). The content-force distinction. Philosophical Studies, 134(2), 141–164.
Hanks, P. (2011). Structured propositions as types. Mind, 120, 477.
Hawthorne, J., & Manley, D. (2012). The reference book. Oxford: Oxford University Press.
Jeshion, R. (2010). New essays on singular thought. Oxford: Oxford University Press.
Kaplan, D. (1978). Dthat. In P. Cole (Ed.), Syntax and semantics (Vol. 9, pp. 221–243). New York: Academic Press.
Keller, L. J. Forthcoming. Propositions supernaturalized. In T. Dougherty & J. Walls (eds.), Two dozen (or so) arguments for God. Oxford University Press.
King, J. (2007). The nature and structure of content. Oxford: Oxford University Press.
King, J. (2013). Propositional unity: What’s the problem, who has it, and who solves it? Philosophical Studies, 165(1), 71–93.
Kripke, S. (1980). Naming and necessity. Cambridge: Harvard University Press.
Lewis, D. (1986). A comment on armstrong and forrest. Australasian Journal of Philosophy, 64(1986), 92–93.
Maddy, P. (1997). Naturalism in mathematics. Oxford: Clarendon Press.
Plantinga, A. (1961). A valid ontological argument? Philosophical Review, 70, 93–101.
Richard, M. (2013). What are propositions? Canadian Journal of Philosophy, 43, 702–719.
Russell, B. (1903). Principles of mathematics. Cambridge: Cambridge University Press.
Smolin, L. (2014). Atoms of space and time. Scientific American, 23, 94–103.
Soames, S. (1989). Semantics and semantic competence. Philosophical Perspectives, 3, 575–596.
Soames, S. (1995). Beyond singular propositions? Canadian Journal of Philosophy, 25(4), 514–549.
Soames, S. (2010). What is meaning?. Princeton: Princeton University Press.
Soames, S. (2013). Cognitive propositions. Philosophical Perspectives: Philosophy of Language, 27, 1–23.
Soames, S. (2015). Rethinking language, mind, and meaning. Princeton: Princeton University Press.
Soames, S., King, J., & Speaks, J. (2014). New thinking about propositions. Oxford: OUP.
Solovay, R. (1970). A model of set theory in which every set of reals is lebesgue measurable. Annals of Mathematics, Second Series, 92(1), 1–56.
Stalnaker, R. (2012). Mere possibilities: Metaphysical foundations of modal semantics. Princeton: Princeton University Press.
van Inwagen, P. (2012). Three versions of the ontological argument. In M. Szatkowki (Ed.), Ontological proofs today. Berlin: DeGruyter.
Vitali, G. (1905). Sul problema della misura dei gruppi di punti di una retta. Bologna: Tip. Gamberini e Parmeggiani.
Acknowledgments
I’m indebted to many philosophers for helpful feedback on ancestors and relatives of this paper—I’ve tried to acknowledge their contributions in the relevant parts of the paper. In particular, I’d like to thank those who read and commented on earlier drafts or proper parts thereof: Paddy Blanchette, Laura Crosilla, Chris Menzel, Paul Nedelisky, Josh Rasmussen, and Jeff Speaks. Thanks also to participants at the Propositions Workshop at the University of Leeds (Spring 2014), and to those who attended my talks at the University at Buffalo and the 2015 Central APA for helpful comments on earlier drafts of this paper. Special thanks to John Keller for extremely helpful and detailed comments at all stages of progress.
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Appendix
Appendix
1.1 Construction of a Vitali Set
Letting ‘\( {\mathbb{R}} \)’ denote the set of real numbers and ‘\( {\mathbb{Q}} \)’ the set of rational numbers, we can define the following equivalence relation on \( {\mathbb{R}} \): x~y iff x–y \( \in {\mathbb{Q}} \). The relation ~ partitions \( {\mathbb{R}} \) into uncountably many disjoint classes. Each of these equivalence classes has a non-empty intersection with the interval [0, 1]. By AC, there is a set V with exactly one element from the intersection of each such equivalence class with the interval [0, 1]. For each real number r, there is exactly one v ∊ V such that v–r \( \in {\mathbb{Q}} \). Now we have a Vitali set V ⊆ [0, 1].Footnote 35
1.2 Proof that Vitali Sets are Choice-Only
Vitali sets are not Lebesgue-measurable.Footnote 36 But by Solovay’s theorem, ZF is consistent with the following claim, L: every set of reals is Lebesgue-measurable.Footnote 37 The proof for the existence of Vitali sets relies on AC, but by Solovay’s theorem, the claim that there is a set of reals that is non-Lebesgue-measurable (which contradicts L) is independent of ZF. So Vitali sets are choice-only, relative to ZF (i.e., their existence can be proven in ZFC, but not in ZF).
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Keller, L.J. Against Naturalized Cognitive Propositions. Erkenn 82, 929–946 (2017). https://doi.org/10.1007/s10670-016-9851-5
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DOI: https://doi.org/10.1007/s10670-016-9851-5