Abstract
In Writing the Book of the World, Ted Sider argues that David Lewis’s distinction between those predicates which are ‘perfectly natural’ and those which are not can be extended so that it applies to words of all semantic types. Just as there are perfectly natural predicates, there may be perfectly natural connectives, operators, singular terms and so on. According to Sider, one of our goals as metaphysicians should be to identify the perfectly natural words. Sider claims that there is a perfectly natural first-order quantifier. I argue that this claim is not justified. Quine has shown that we can dispense with first-order quantifiers, by using a family of ‘predicate functors’ instead. I argue that we have no reason to think that it is the first-order quantifiers, rather than Quine’s predicate functors, which are perfectly natural. The discussion of quantification is used to provide some motivation for a general scepticism about Sider’s project. Shamik Dasgupta’s ‘generalism’ and Jason Turner’s critique of ‘ontological nihilism’ are also discussed.
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Notes
We subdivide the columbidae into doves and pigeons, but the division is not in good zoological standing. Roughly, the birds of prettier species are called ‘doves’ while the birds of the uglier species are called ‘pigeons’.
See Burgess (2005) for a brief history of the idea.
Sider (2011); throughout this paper, all references to Sider’s work are to this book.
For example, Sider writes (p. 7): ‘Realism about predicate structure is fairly widely accepted. Many—especially those influenced by David lewis—think that some predicates (like ‘green’) do a better job than others (like ‘grue’) at carving nature at the joints. But this realism should be extended, beyond predicates, to expressions of other grammatical categories, including logical expressions.’ Sider then goes on to use the existential quantifier as an example.
Throughout this paper, ‘justified’ means propositionally justified, not doxastically justified. It might be better to replace ‘if and only if’ with ‘if and only if and to the extent that’, but we need not worry about these subtleties.
Notice also that ‘the’ most virtuous theory need not be ‘our’ most virtuous theory. Suppose that at some particular time, the most virtuous total theory that has been formulated is known to be seriously defective in some respect. In this case, it might well be that researchers would not be justified in believing that the terms in the theory are elite. I would like to thank an anonymous reviewer at Philosophical Studies for this point.
Sider explains his account in Sect. 2.3. ‘[R]egard the ideology of your best theory as carving at the joints,’ he writes. He uses the term ‘indispensable’ in the section: ‘[R]egard as joint-carving the ideology that is indispensable in your best theory. This [way of putting it] is fine, provided “indispensable” is properly understood, as meaning “cannot be jettisoned without sacrificing theoretical virtue.”’
This is from Sect. 9.6.4 of Sider (2011). It should be noted that, according to Sider, the English quantifier ‘there exists’ is not an elite expression. Sider thinks that ‘There exists rocks’ is true in English, but he denies that the elite first-order quantifiers range over compound things such as rocks. I won’t dwell on this issue; I am concerned with the question of whether any quantifier is elite.
Burgess (2005).
Quine (1960).
Rather than presenting Quine’s original system, from Quine (1960), I’ll incorporate some of Quine’s later modifications, from Quine (1971, 1981). I do this partly to make my discussion easier to compare with that in Turner (2010), and partly so as to make the longer formulas more manageable. For proof-systems, see Bacon (1985) and Kuhn (1983).
I’m omitting corner-quotes and adding brackets in an ad hoc way, to make these definitions easier to read.
Perhaps I should explain this result slightly more carefully. It’s easy to see (based on the definitions of the predicate functors given above) how to extend the familiar definition of truth-at-a-model for standard first-order predicate logic to a definition of truth-at-a-model for the language that results from extending first-order predicate logic by adding Quine’s predicate functors. Quine proved that for every name-free sentence \(S\) of first-order predicate logic, there is a nought-place predicate \(P\) composed of just simple predicates and the predicate functors, such that \(S\) and \(P\) are true at precisely the same models.
It’s worth noting too that we can extend Quine’s system so as to dispense with proper names—we need only add an appropriate sentence functor for each name. For example, ‘Rabbit(Peter)’ will become ‘PeterRabbit’, which is to be understood as meaning something like the property rabbit is realised Peterly. More generally, if \(F\) is an \(n\)-place predicate, ‘\(PeterF\)’ is an \((n-1)\)-place predicate such that \([PeterFx_{1} \ldots x_{n-1}]\) is equivalent to \([Fx_{1} \ldots x_{n-1} Peter]\).
Turner (2010).
I’ll assume that ‘\(\exists _{p}\)’ can also be prefixed to simple predicates, and that the resulting formula are interpreted in the obvious way. For example, ‘\(\exists _{p} F\)’ is equivalent to ‘\(\exists x F x\)’.
Turner spends some time in his paper clarifying and defending this principle—I’m omitting details from his discussion that don’t affect my argument. See Sect. 4.1.2 of Turner (2010).
I’ve actually modified the argument in one small respect. Turner’s version of the argument involves an extra symbol: ‘\(\delta\)’. I’ve changed the argument to remove the need for this extra symbol. I find the plethora of new symbols involved in the argument already rather confusing. The discussion is not affected in any important way by this change. Sceptical readers should compare my presentation of the argument with Sect. 6.3.1 of Turner (2010).
Quine himself introduced a rather similar functor (though for very different reasons) in his Quine (1981).
This generalises immediately to the case in which \(F\) and \(G\) have arity greater than one, and to complex predicates.
The general case is a bit more difficult here. Suppose \(F\) is an n-place predicate. Then ‘\(\varDelta F\)’ is equivalent to \(\ulcorner {\sim }(F \subseteq {\uparrow }^{n-1}\Lambda ) \urcorner\), where the superscript numeral represents repeated application in the obvious way. The generalisation to the case of complex predicates is trivial.
See Tarski and Givant (1999) for details.
One can’t avoid the problem by using second-order Peano arithmetic—the comprehension axiom for second-order logic has infinitely many instances.
From Hirsch et al. (2002).
For my purposes, it doesn’t matter whether this label is historically appropriate.
See Burgess (1998) for some discussion.
From the preface to Sider (2011).
Dasgupta (2009).
Dasgupta’s discussion can be reformulated so as to avoid the reification of facts—but to keep things simple I will follow Dasgupta by indulging in this reification.
Dasgupta also makes an epistemic point: absolute velocities cannot be measured, if \(NGT\) is true—and so \(NGT_{A}\) posits facts which are epistemically inaccessible to us. To save on space I don’t discuss this idea.
See in particular Quine (1951).
He does this, for example, in his chapter on the philosophy of time. Sider claims that tense operators (like ‘it will always be the case that’ and ‘it was once the case that’) are not elite; he defends this claim by saying that these operators will not occur in any of the most virtuous theories because one can ‘describe temporal reality without them—by quantifying over past and future entities and predicating features of them relative to times’ (p. 241). His most sustained discussion of ideological simplicity is on p. 219.
A parallel question can be raised about ‘\(\forall\)’ and ‘\(\exists\)’.
Similarly, he argues that both ‘\(\forall\)’ and ‘\(\exists\)’ are elite.
I say that his answer is ‘somewhat tentative’ because on p. 220 Sider parenthetically suggests a view on which the only elite truth functional connectives are conjunction, disjunction and negation.
Sider does not give such a careful definition of ‘ideological arbitrariness’. But I think it is clear that this is what he has in mind.
For example, in Unger (1984), Peter Unger defends modal realism on the grounds that it ‘minimizes arbitrariness’. The idea is that if there is only one concrete universe, at least some of its contingent features will be inexplicable. This argument has nothing to do with ideological arbitrariness in my sense. In Horgan (1993), Horgan briefly discusses a ‘principle of non-arbitrariness’ that is widely presupposed in debates about mereology. Again, the relevant kind of arbitrariness is not ideological arbitrariness.
I say ‘or almost all’ for the following reason. A connective ‘\(\rightharpoondown\)’ which is such that \(\ulcorner ( \alpha \rightharpoondown \beta ) \urcorner\) is equivalent to \(\alpha\) should surely be omitted, as should a connective ‘\(\leftharpoondown\)’ which is such that \(\ulcorner ( \alpha \leftharpoondown \beta ) \urcorner\) is equivalent to \(\beta\).
In his textbook on logic, Tarski was very explicit on this point:
We \(\ldots\) attempt to see to it that the system of primitive terms is independent, that is, that it does not contain any superfluous terms, which can be defined by means of the others. Often, however, one does not insist on [this principle] for practical, expository reasons \(\ldots\)Tarski (1994), p. 122.
Sider might defend himself by appealing to an analogy. It’s sometimes said that when stating one’s fundamental physical theory one should avoid mentioning any particular unit of measurement (e.g. the kilogram, or the metre). The argument is that to mention any one such unit would be objectionably arbitrary. [See for example Field (1980, p. 45 ), or Liggins (2012)]. This provides some precedent for Sider’s claim that ideological arbitrariness is a theoretical vice.
I’m afraid that I don’t have space to discuss this important issue in any depth, but for the record my position is this. I endorse the methodological point that we should if possible avoid mention of any particular unit of measurement in our fundamental theory. However, the ‘\(\in\)’/‘\(\ni\)’ example strongly supports the view that ‘ideological arbitrariness’ is not an epistemic vice. So some other explanation of the methodological point is necessary. I think that an alternative explanation of this claim can be found in the theory of measurement [Suppes and Zinnes (1963) is a useful introduction] but I do not have space to rehearse this explanation here.
Thanks to an anonymous reviewer at Philosophical Studies for suggesting this criterion.
It is perhaps worth pausing briefly to explain why condition (ii) is necessary. Suppose it turns out that some of the most virtuous theories are incompatible with one another—so that at least one of the most virtuous theories is false. In this case, we would presumably not be justified in thinking that all of the most virtuous theories are couched in elite vocabulary.
Strictly speaking this is not implied by my characterisation of elitism in the opening of this paper, but it is hard to believe that any elitist would reject this claim. Sider certainly accepts it—see his discussion of ‘conformity to the world’ on p. 62.
See Chap. 13 for this claim.
Roughly speaking, ‘metaphysical laws’ are simple, powerful generalisations in elite terms. See Sect. 12.5 of Sider’s book for discussion.
Notice that my term ‘structurally complete’ has nothing to do with Sider’s term ‘completeness’ in section 7.1 of Sider’s book. What Sider calls ‘completeness’ is the claim that ‘every nonfundamental truth holds in virtue of some fundamental truth’ (p. 105).
Of course, a similar point can be made about \(T_{LFOPL}\). This theory is structurally incomplete because it does not contain, for example, the term ‘\(\subseteq\)’. And it omits some metaphysical laws, such as ‘\(V \subseteq V\)’.
The pluralist might try to resist the claim that \(T_{NQ}\) fails to depict the world’s quantificational structure, on the grounds that \(T_{LFOPL}\) does depict the world’s quantificational structure and the two theories are analytically equivalent. This is mistaken. From an elitist point of view, even analytically equivalent statements may differ in the extent to which they faithfully depict the structure of the world. To see the point, suppose that \(E\) is an elite monadic predicate (perhaps \(E\) means electron), while \(P\) is a plebeian monadic predicate (perhaps \(P\) means dove). Suppose that \(Q_{1}\) and \(Q_{2}\) are defined as follows, so that they are also plebeian:
\(\ulcorner \forall x [Q_{1} x \leftrightarrow (Ex \vee Px)] \urcorner\) \(\ulcorner \forall x [Q_{2} x \leftrightarrow (Px \wedge \lnot Ex)] \urcorner\)
Now contrast the following two sentences:
(1) \(\exists x Ex\) (2) \(\exists x (Q_{1} x \wedge \lnot Q_{2} x)\)
These two statements are analytically equivalent, but elitists will agree that (1) is nevertheless a better representation of reality’s structure, since (2) contains plebeian terms.
In his classic paper Neumann (1967), von Neumann presented an axiomatic theory of functions and explained how to interpret the theory of sets within it. Von Neumann’s axioms are less than ideal for my purposes, because some of his axioms contain the word ‘set’ (which was, for von Neumann, a defined term). His system also contained a rather large number of axioms. However, it is not difficult to axiomatise the theory of functions in a way that makes no reference at all to sets, and without excess complexity.
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Acknowledgements
Most of all I would like to thank Brian Weatherson, who read many drafts of this paper and discussed them with me. I would also like to thank the other members of my PhD committee—Ernie Lepore, Cian Dorr and Andy Egan—for their advice. Jonathan Schaffer helped me with section six. An anonymous reviewer at Philosophical Studies said that I should add section seven—and (s)he was quite right. Thanks to Ted Sider for defending his views in conversation with me. Thanks to Kate Manne for her detailed comments on an earlier draft. I have also benefited from discussions with Jenn Wang, Tobias Wilsch, Matthias Jenny, Josh Armstrong, Krista Lawlor, Alexi Burgess, Mark Crimmins, Ken Taylor and Anna-Sara Malmgren.
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Donaldson, T. Reading the Book of the World. Philos Stud 172, 1051–1077 (2015). https://doi.org/10.1007/s11098-014-0337-5
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DOI: https://doi.org/10.1007/s11098-014-0337-5