Continuity as a Guide to Possibility

Abstract

I propose a new guide for assessing claims about what is possible. I offer examples of modal claims that are, in a certain intuitive respect, ‘continuous’ with one another. I then put forward a general, defeasible principle of modal continuity that can account for our intuitions about those examples. According to this principle, statements that differ by a mere quantitative term don't normally differ with respect to being possibly true. I offer a precise statement of the principle, and then I consider exceptions in an effort to find a more nuanced continuity principle that is more reliable and still sufficiently general. Next, I offer a possible explanation of why modal continuity tends to hold and why exceptions occur where they do. Although my primary purpose is to introduce a new technique for modal reasoning, I showcase the power of the principle by applying it to a philosophical dispute concerning parts and wholes: the principle, if true, reveals a new cost of the thesis that composition is restricted. Furthermore, I point out examples of other philosophical inquiries (in philosophy of mind, metaphysics, and philosophical cosmology) that may benefit from a principle of modal continuity. The principle gives us a new tool for assessing a wide variety of modal claims.

Keywords:

• possibility
• conceivability
• modal scepticism
• modal continuity

Notes

¹ Here is the argument: (i) suppose it is possible that there is something N, such that it is necessary that it exists; then (ii) it is necessary that N exists (by S5); (iii) if it is necessary that N exists, then it is not possible that N does not exist; therefore (iv) if it is possible that N exists, then it is not possible that N does not exist.

² Alternatively: if propositions M₁ and M₂ are of the form possibly, A, and if they differ by a mere quantity, then M₁ is true iff M₂ is true. I offer this formulation for those philosophers, especially existentialists, who deny the inference from <possibly, A> to <the proposition that A is possibly true>.

³ Distance is determined by a distance function d on C, where d is a function, C × C → R, which satisfies the set-theoretic conditions of being a distance function. Since the mapping is to the real number system, I am limiting my investigation of unified classes (sets) of degreed properties to those whose cardinality does not exceed the cardinality of the set of real numbers. I am not concerned with whether or not there might be modal gaps in a class that has infinitely distant members.

⁴ Interestingly, we may deduce (M₂) from the seemingly more modest principle, (M_M) that a unified class of degreed properties has multi-member modal gaps if it has any modal gaps at all. To see this, let G be a modal gap of any size in a large class C. Then, by definition, C has at least one member c that can be instantiated. Now G either has exactly one member or has more than one. Suppose it has one, which I’ll call g. Since C is a unified class of degreed properties, any class C* that contains g and c is also a unified class of degreed properties. But G is a one-member class that is a modal gap in C*, which contradicts (M_M). Suppose, then, that G has more than one member, and let g* be one such member. Then there is a unified class of degreed properties C# that contains only g* and c. But the class containing g* alone is a one-member class that is the only modal gap in C#. And that contradicts (M_M). Therefore, if (M_M) is true, then no class of degreed properties has a modal gap of any size.

⁵ I owe the polygon examples to Peter van Inwagen.

⁶ I am assuming that it is strictly logically impossible for there to be a fractional number of things: for instance, it is strictly impossible for there to be exactly 4.5 things—given the meaning of ‘thing’. But if we drop that assumption, then we may instead explicitly add to (M₃) the condition that no member of C is such that it is strictly necessary that if it is exemplified, then there is a fractional number of things.

⁷ But, as Nathan Ballantyne suggested to me, the identity conditions of Lehrer might be rather flexible, such that he could be any size. Then why not think Keith Lehrer can eat any amount of cat food?

⁸ I owe this example to van Inwagen, who offered it to me in correspondence.

⁹ When considering candidate counterexamples, we should bear in mind that broad logical possibility is broader than nomological possibility: so, for example, even if electron orbits are discontinuous given our laws, there may be other laws that allow for orbits that occupy the gaps.

¹⁰ I am grateful to an anonymous referee for bringing these sorts of questions to my attention.

¹¹ For an elaboration and defence of particularism, see Chisholm [1973].

¹² I am grateful to Ballantyne for suggesting this way of developing the proposal.

¹³ Why work with (M₄), then, rather than with a principle that is restricted to one axis of specification? I have two reasons. First, I worry that there are breaks even in one-dimensional modal statements: for example, there can be one concrete object but not ½ concrete objects. Second, there are multi-dimensional statements that don't seem to have breaks, and (M₄) accounts for many of them. That said, a ‘one-axis’ continuity principle might serve our purposes in certain contexts. I don't have a master principle to offer that is maximally ideal for all contexts.

¹⁴ I shall limit the argument's scope to material (spatially situated) objects.

¹⁵ I am not suggesting that premise P2 is undeniable or completely uncontroversial. See, for example, Merricks [2005] for dissent. Nevertheless, advocates of restricted composition commonly assume a premise like this, and so it is useful to see how it may be used in an argument against restricted composition. Note also that the premise is especially plausible on the standard ‘bottom-up’ materialist picture according to which all properties of material objects are ultimately determined by the lower-level properties and relations instantiated by basic units of matter.

¹⁶ Ned Markosian, a defender of restricted composition, will agree. He accepts that the spatial profile of a world suffices for its compositional profile. See Markosian [forthcoming: 6–7].

¹⁷ Efird and Stoneham [2009] discuss and defend the subtraction argument.

¹⁸ Cf. Koons [1997].

¹⁹ I am grateful to Nathan Ballantyne for his extremely generous comments on many previous drafts of this paper. I also thank Peter van Inwagen and an anonymous referee for their comments.