Modal science

Abstract

This paper explains and defends the idea that metaphysical necessity is the strongest kind of objective necessity. Plausible closure conditions on the family of objective modalities are shown to entail that the logic of metaphysical necessity is S5. Evidence is provided that some objective modalities are studied in the natural sciences. In particular, the modal assumptions implicit in physical applications of dynamical systems theory are made explicit by using such systems to define models of a modal temporal logic. Those assumptions arguably include some necessitist principles.

Notes

1. See Kratzer (1977, 2012, 49–62), Portner (2009, 144–184), and Vetter (2016), for instance. It is not denied that the same word can express an objective modality in one context and an epistemic modality in another: compare ‘She could run a marathon in three hours’ (objective) with ‘Goldbach’s Conjecture could be true’ (epistemic).

2. There is a plausible argument that, in propositional modal logic, if metaphysical modality obeys at least the principles of S5, then it obeys at most the principles of S5 (Williamson 2013a, 111).

3. Contingentists may wish to insert a qualification ‘if x exists’ within the scope of the necessity operator to handle the possible non-existence of the objects. What matters is that x = y licenses the inter-substitution of the free variables ‘x’ and ‘y’ in objective modal contexts; the same object is at issue under different guises.

4. See Strohminger (2015) for a detailed development of the case for perceptual knowledge of nonactual possibilities. This strikes at the Humean assumption that impressions are non-modal in content. Roca-Royes (Forthcoming) makes a more empiricist argument for inductive knowledge of nonactual possibilities via their similarity to perceived actualized possibilities: if you have seen cups break, and thereby know that they can break, you may infer that a similar unbroken cup can break (though that does less to confront empiricist worries about how we come to understand ‘can’ in the first place).

5. It is worth noting that some philosophers of mathematics interpret the language of mathematics itself as implicitly modal: mathematics becomes a science of possible structures (Putnam 1967; Hellman 1989). Despite taking such views seriously, for present purposes I prefer not to rely on philosophical interpretations so distant from the way mathematicians explicitly talk and think. More recent modal interpretations of the language of set theory, such as Linnebo (2013) and Studd (2013) (see also Parsons 1983, 298–341; Fine 2006), have been motivated by a (laudable) desire to avoid Russell’s paradox for sets without ad hoc restrictions; that too is quite far from the concerns of most working mathematicians. Moreover, the latter motivation requires a (so far somewhat obscure) non-objective reading of the modal operators, because even the hierarchy of pure sets must involve such a modal aspect (since Russell’s paradox arises even for pure sets, and it is generally agreed that the existence of pure sets is metaphysically non-contingent). Non-objective modal interpretations of the language of mathematics are not strictly relevant to the concerns of this paper.

6. The principle in the text is a form of single-premise closure. We do not assume the multi-premise closure principle that if a formula of L supports the formulas β₁, … , β_n of L⁺, and β₁, … , β_n jointly entail γ, then also supports γ. Unlike multi-premise closure, single-premise closure is consistent with an interpretation of ‘support β’ as ‘confer a probability above the threshold c on β’, where 0 < c < 1.

7. See Williamson (2007a, 293–299), for relevant background on the logical relations between counterfactuals and metaphysical modality.

8. As already hinted, it is controversial how much weaker than metaphysical necessity nomic necessity really is (see also Section 8). It is also controversial how much of natural science really aims at nomic necessity. The present remarks about nomic modality should be taken in the spirit of a first approximation.

9. Some Lewisians may object to the assumption because it clashes with the ‘small miracle’ conception of subjunctive conditionals, but the assumption is nonetheless very plausible.

10. Allowing infinitesimal probabilities does not solve the problem (Williamson 2007b).

11. Let Ω be the set of all possibilities. If we define □E = Ω if E has probability 1 and □E = {} otherwise, and ♢E = Ω if E has nonzero probability and ♢E = {} otherwise, then in the finite case in the text □ and ♢ satisfy the principles of the propositional modal system S5 (=KT45); in the infinite case they satisfy only the principles of the weaker modal system KD45, where the T principle (□E E) is weakened to the D principle (□E ♢E).

12. If we used evidential or subjective probabilities, the putative explanation would at best show that the explanandum ‘was to be expected’. But to show that an outcome was to be expected is not to explain why it occurred, in the relevant broadly causal sense. To revert to the example at the beginning of Section 1, it was trivially to be expected that n would number the inhabited planets, since ‘n’ was defined to name their number. For instance, given that n = 29, to explain causally why there are exactly n inhabited planets is to explain causally why there are exactly 29 inhabited planets, but the trivial ‘was to be expected’ explanation does not advance the latter project. The difference between the two non-obviously co-referential names ‘n’ and ‘29’ is epistemically relevant but causally irrelevant.

13. One may question the assumption that merely possible states of a physical system are abstract objects (Malament 1982, 533; Lyon and Colyvan 2008, 233). On the approach of Williamson (2013a, 7), their non-concreteness does not make them abstract. But that point is irrelevant to the present argument.

14. For some purposes we might require directions to be non-negative, which would require only forwards determinism.

15. Of course, this is not the only way of interpreting modal operators over a dynamical system. If a topology is defined over the states, one can interpret □ as the topological interior operator, which is a much more ‘local’ form of necessity (though it does not involve an accessibility relation between worlds). It yields an S4 modal logic, whereas the present ‘global’ interpretation of □ yields an S5 modal logic. The two interpretations are not rivals; they simply pick out different aspects of the system for study. The global interpretation is more general, because it does not depend on what kind of mathematical structure is defined over the states. For more on topological interpretations of modal logics on dynamical systems see Artemov, Davoren, and Nerode (1997) and Davoren and Goré (2002).

16. They are unrestricted in the sense of ranging over all states of the system; as already noted, they are typically not equivalent to metaphysical necessity and metaphysical possibility.

17. This ‘diamonds are forever’ principle is reminiscent of, but not equivalent to, the principle defended by Dorr and Goodman (Forthcoming); the latter concerns metaphysical possibility and a more standard reading of the tense operators.

18. For an introduction to tense logic that explains the relevant background see, for instance, Müller (2011).

19. In unquantified S5, every formula is equivalent to one without such embedded occurrences of modal operators. That is not in general so for quantified S5, even with constant domains; see Fine (1978, 146–151), for the case of first-order S5.

20. See Williamson (2013a) for a discussion in more depth. I am assuming that the rest of the model theory is more or less standard.

21. For reasons explained in Williamson (2013a, 254–261), reading higher-order quantifiers such as those in NNE_P as first-order quantifiers restricted to objects of a special sort (propositions) is ultimately inappropriate: semantically, the difference between name position and sentence position runs deeper than a difference between objects of one kind and objects of another, or between objects in general and objects of a special kind (where every object can in principle be named). Nevertheless, for present purposes the talk of propositions as objects is a harmless over-simplification.

22. For discussion of the biological case see Gunawardena (2009).

23. The complaint on pp. 286–288 of Williamson (2013a) that various contingentist comprehension principles for second-order modal logic are too weak to serve the purposes of ‘modal mathematics’ relates to just this point. Consider the free application of non-modal mathematics to an implicitly modal subject matter, as in reasoning about dynamical systems or Kripke models intended for some objective modality. When the intended modal content of the application is made explicit model-theoretically in the manner of Section 5 (which can be extended to other forms of quantification), unrestricted modal comprehension principles such as Comp_P can be proved valid in the model theory by non-modal mathematics (for instance, set theory). Since the free application of non-modal mathematics to the modal subject matter is committed to all such valid modal formulas, in particular it is committed to those unrestricted modal comprehension principles. That is fine for necessitists but not for full-blooded contingentists such as Stalnaker (2012).

24. See Cussens (2014) for a recent discussion of Leibniz’s failure to contribute to the mathematics of probability.

25. For a related application of probability to an issue in modal metaphysics see Kment (2012), although I doubt that Kment would endorse the conclusions of this paper.

26. The material in this paper has evolved over several years. Various parts of it were presented as the Ruth Manor Lecture at Tel Aviv University, the Saul Kripke Lecture at City University New York and the Wade Lecture at St Louis University. Earlier versions of the material were presented as talks at conferences on the epistemology of modality at Belgrade University, Aarhus University (where Daniel Dohrn provided a detailed response) and Stirling University, a conference on logic and metaphysics at the University of Southern California, a workshop on modal metaphysics in Montreal, and the Universities of Athens, Connecticut (Storrs), Michigan (Ann Arbor) and Oxford. Embryonic predecessors were presented to workshops at the Centre for the Study of Mind in Nature in Oslo and the Institute of Philosophy in London. I am grateful to all the participants at those events who helped me develop the material with their questions and comments, and for discussion or correspondence on the issues to Kit Fine, Peter Fritz, Peter Godfrey-Smith, Jeremy Goodman, Lloyd Humberstone, Matthias Jenny, Øystein Linnebo, Maurico Suárez and Trevor Teitel. I believe that Saul Kripke envisaged an analogy with states in phase space early in his thinking about possible worlds.