On Williamson and simplicity in modal logic

Abstract

According to Timothy Williamson, we should accept the simplest and most powerful second-order modal logic, and as a result accept an ontology of "bare possibilia". This general method for extracting ontology from logic is salutary, but its application in this case depends on a questionable assumption: that modality is a fundamental feature of the world.

Keywords:

• Williamson
• modality
• fundamentality
• necessitism

Acknowledgements

Thanks to Karen Bennett and Timothy Williamson.

Notes

1 Whether it has causes and effects depends on what we say about causation by absence.

2 See also Cresswell (1991).

3 This is the monadic case of a more general principle. X may not be free in A; A will in general contain x free, and may contain further free variables, which may be bound to the prefixing quantifiers.

4 See Lewis (1986, 59–60), who emphasizes the need to recognize both abundant and ‘sparse’ properties.

5 See Dorr (2005, 2007), Fine (2009), Schaffer (2009), Sider (2009, 2011, chapters 7–9), Cameron (2010).

6 On the latter claim see Schaffer (2009, 361), Sider (2011, 169, note 10), Bennett (2016, chapter 7).

7 Compare Fine (1977).

8 This general style of kind of argument — requiring different grounds for different facts in certain cases — is pursued by Dasgupta (2014, 573) in a different context. There is, though, a hard question of exactly when distinct facts should be required to have distinct grounds, since the requirement is not correct in general — most friends of ground would admit, for instance, that the fact that P can ground both and also . (This hard question, in my view, is symptomatic of a problematic feature of the ‘conditional’ notion of ground, as opposed to ‘biconditional’ level-connecting concepts. See Sider (2013b; a).)

9 Well, the fact is partially singular since it involves , but what’s important is that it is general insofar as it involves nonconcreta.

10 The rejection might be restricted to actually unanchored things. This would be in the spirit of ‘nonserious actualism’ (Plantinga 1983; Fine 1985; Salmon 1998), and would arguably not be ad hoc: given a’s actual concrete existence, we can single it out and make modal claims about it in particular — even the claim that it could have been nonconcrete. (If a had been nonconcrete, what fact would have grounded its existence? Answer: ‘The singular modal fact that a could have existed fundamentally!’ — more of the same nonseriousness.)

11 See Weatherson (2003) for a discussion of a number of related issues.

12 On metaphysical fundamentality see Sider (2011).

13 As Williamson pointed out in response to a draft of this paper.

14 See also Bennett (2016, chapter 7).

15 ‘But what if one doesn’t know how, or even that, chemistry is based in physics?’ — I’m thinking of the epistemology here in an ‘externalist’ way. The believer needn’t compute what she’s a priori entitled to believe about chemistry from some initial assumptions about physics. The picture is rather that certain possible worlds are a priori more likely, and that those worlds are in fact physically and hence chemically simple.

16 These concepts might be, but needn’t be, metaphysically fundamental, since the ultimacy is epistemic: the presumption of simplicity is underived from other such presumptions. For example, perhaps there are ultimate presumptions of simplicity for certain higher level special-science concepts. This wouldn’t require saying that special-science concepts are fundamental, though it would be naturally paired with the idea that special-science lawhood cannot be explained merely by appeal to physical lawhood.

17 Note that my account of necessity is limited in crucial ways — it does not cover iterations of modal operators, for example — and therefore would need to be extended in order to make contact with Williamson’s argument, which involves the logic of second-order quantified modal logic. But on any way of extending it, necessity would remain disjunctive in the relevant sense.

18 See Kripke (1972, 45, note 13), Hazen (1979).

19 See Williamson (2013, 16). Williamson’s classification of Lewis as a Necessitist might seem to ignore Lewis’s (1968) scheme for translating modal formulas. For example, Lewis’s translation of (an instance of the Barcan schema) into the language of counterpart theory is a false sentence saying that if some L-world (i.e. maximal spatiotemporally interrelated object) contains an F then there is an F in this L-world. Williamson would complain that the translation treats the quantifiers in the modal formula as restricted, whereas Necessitism concerns unrestricted quantification. Though I think that calling Lewis a Necessitist is unnecessarily confrontational (Lewis would insist that ordinary modal claims are to be interpreted in his way, and why fight about that?), I do think that the classification is fair insofar as the issue is what Williamson cares about most: access to full second-order modal reasoning. Since Lewis’s translation scheme is contingentist it will sometimes interfere with modal second-order reasoning, but then Lewis will be happy to suspend the translation scheme, as he always did when it caused trouble (Lewis 1986, 12–13).

20 See, for instance, Fine (1994, 2001, 2012), Schaffer (2009), Rosen (2010), Bennett (2016).

21 Note that Fine takes the quantifiers to be ‘possibilist’ in Fine (1995), his formal development of the logic of essence — ‘for the sake of simplicity’, he says (244).

22 See, for example, Stalnaker (1986).

23 See Sider (2011, section 12.12) on a related issue. For the two approaches to counterfactuals alluded to in the text, see Stalnaker 1986 and Lewis 1973 on the similarity approach, and Lycan 2001 and von Fintel 2001 on the contextualist approach.