¹ For the classic formulation, see ‘Frege's, Timothy Smiley's “series of natural numbers”’, (Mind 97 (1988): 583–4)Google Scholar, which traces the central idea back to Alan Ross Anderson's ‘The formal analysis of normative systems', (Technical Report Technical Report No.2, U.S.O ce of Naval Research, Group Psychology Branch, New Haven, 1956). For some discussion of the problems with the classic formulation, and alternative proposals, see Humberstone, I. L., ‘Relative Necessity Revisited’, Reports on Mathematical Logic 13 (1981): 33–42Google Scholar; Humberstone, , ‘Two-Dimensional Adventures’, Philosophical Studies 118 (2004): 17–65Google Scholar; van Fraassen, B. C., ‘The Only Necessity is Verbal Necessity’, The Journal of Philosophy 74 (1977): 71–85Google Scholar; and Hale, and Leech, , ‘Relative Necessity Reformulated’, Journal of Philosophical Logic (2016): 1–26Google Scholar.

² The notion of absolute necessity might be characterized in other ways. For example, we might think of it as a kind of limiting case of relative necessity: absolute necessities are those which hold relative to any body of propositions whatever. Or we might take them to be those propositions whose negations are in no relevant sense possibly true. In my Necessary Beings: An essay on Ontology, Modality, and the Relations between them (Oxford: Oxford University Press, 2013)Google Scholar, ch. 4, I discuss these alternatives and argue that under plausible assumptions, they coincide in extension with the generalized counterfactual explanation suggested here. The idea of explaining necessity and possibility operators in terms of the (strong) conditional goes back at least to Stalnaker, Robert (‘A Theory of Conditionals’, American Philosophical Quarterly, monograph series: 98–112 (1968)Google Scholar) and has since been taken up by Timothy Williamson (‘Modal Logic within Counterfactual Logic’, in Hale, B. and Hoffmann, A. (eds), Modality: Metaphysics, Logic, and Epistemology, 81–96 (Oxford University Press, 2010)CrossRefGoogle Scholar). The formulation using the generalized counterfactual is suggested by Ian McFetridge in his posthumously published essay on logical necessity (McFetridge, ‘Logical Necessity: Some Issues' in J. Haldane and R. Scruton (eds), Logical Necessity and Other Essays, volume 11 of Aristotelian Society Series, chapter VIII, 135–154, Aristotelian Society (1990)) and is explicitly adopted in my ‘Absolute Necessities', Nous Supplement: Philosophical Perspectives, 10(30) (1996): 93–117Google Scholar and my Necessary Beings (2013).

³ Kripke, Saul, ‘Naming and Necessity’, in Davidson, Donald and Harman, Gilbert (eds), Semantics of Natural Language (Dordrecht, The Netherlands: Reidel, 1972)Google Scholar; Kripke, , Naming and Necessity (Oxford: Basil Blackwell, 1980)Google Scholar.

⁴ See, for example, Naming and Necessity (1980), page 99 ‘… characteristic theoretical identifications … are not contingent truths but necessary truths, and here of course I don't just mean physically necessary, but necessary in the highest degree – whatever that means).’

⁵ This rough and ready statement of the argument skips over some important complications. One is that since we are defining absolute necessity by □p =de f ∀q(q□→p), it needs to be proved, on the basis of this definition together with a suitable semantics for □→, that □p is true at a given world iff p is true at every world accessible from that world. Another is that, since I reject the standard worlds semantics in favour of a version of possibility semantics, in which possibilities, in contrast with worlds as usually understood, are incomplete in the sense that they typically do not settle the truth-values of all propositions, the underlying semantics cannot be the standard world-based semantics (q.v. Stalnaker, ‘A theory of Conditionals' (1968) and Lewis, David, Counterfactuals (Basil Blackwell, 1973)Google Scholar, but must itself be adjusted to work with possibilities. A fuller statement of the argument, ignoring the second complication, is given in my Necessary Beings (2013), 5.4. As Christopher Menzel subsequently pointed out in correspondence, the argument there stated assumes, in effect, that propositions are defined as sets of worlds. Since I prefer to avoid reliance on that assumption, I cannot wholly endorse that formulation of the argument. However, as I claimed in a footnote (Necessary Beings (2013), 129, fn.19), the argument can be given, avoiding that assumption, in the version of possibility semantics described later in the book (ibid, ch.10).

⁶ Dummett, M., ‘Wittgenstein's Philosophy of Mathematics’, Philosophical Review 68 (1959), 169CrossRefGoogle Scholar.

⁷ The point was, I think, first brought into prominence by Paul Benacerraf, in connection with mathematical truth (Benacerraf, P., ‘Mathematical Truth’, The Journal of Philosophy 70 (1973): 661–80CrossRefGoogle Scholar). Here, as Benacerraf argued, we face a dilemma: the most natural and attractive account of mathematical truth sees it as grounded in the properties and relations of numbers, sets, and other abstract entities – but, given their lack of causal or other natural connection with us, this can seem to block any believable account of how we may know such truths; on the other horn, epistemologically more tractable accounts of the subject matter, such as those which assimilate truth to provability, lack credibility as accounts of mathematical truth. As Christopher Peacocke has subsequently emphasized, a parallel dilemma confronts us in many other areas of philosophy, including the metaphysics and epistemology of modality; Peacocke calls this generalization of Benacerraf's dilemma the Integration Challenge. (See Peacocke, C., ‘Metaphysical Necessity: Understanding, Truth and Epistemology’, Mind 106 (1997): 521–574Google Scholar; Peacocke, , Being Known (Oxford: Oxford Clarendon Press, 1999)CrossRefGoogle Scholar.)

⁸ See Blackburn, S., ‘Morals and Modals’ in McDonald, G. and Wright, C. (eds) Fact, Science and Morality: Essays on A.J. Ayer's Language, Truth and Logic (Oxford: Blackwell, 1986)Google Scholar reprinted in Blackburn, , Simon, , Essays in Quasi-Realism (Oxford University Press, 1993)CrossRefGoogle Scholar. For some detailed critical assessment of the proposed dilemma, see Hale ‘The Source of Necessity’, Nous Supplement: Philosophical Perspectives 16 (2002): 299–319Google Scholar; Hale, , Necessary Beings (2013), 91–7Google Scholar; and Cameron, , Ross, , ‘On the Source of Necessity’, in Hale, and Hoffmann, (eds), Modality: Metaphysics, Logic, and Epistemology, 137–52 (Oxford: Oxford University Press, 2010)CrossRefGoogle Scholar.

⁹ See previous footnote.

¹⁰ Hale, Necessary Beings (2013).

¹¹ See, for example, Ayer, A.J., Language, Truth and Logic (Victor Gollancz Ltd, 2nd edition, 1946), 16–18Google Scholar, 71–86.

¹² This approach has been developed and defended in detail by Alan Sidelle, originally in his Necessity, Essence, and Individuation (Ithaca, New York: Cornell University Press, 1989)Google Scholar. There are useful critical reviews by Yablo, Stephen (‘Review of Alan Sidelle, Necessity, Essence, and Individuation’, Philosophical Review 101 (1992): 878–91)Google Scholar and Mackie, Penelope (‘Review of Alan Sidelle, Necessity, Essence, and Individuation’, Mind 99 (1990): 635–37)CrossRefGoogle Scholar.

¹³ See Quine, W.V.O., ‘Truth by Convention’, in Feigl, H. and Sellars, W. (eds), Readings in Philosophical Analysis (1949)Google Scholar (reprinted from Lee, O.H. (ed.) Philosophical Essays for A.N. Whitehead (Longmans, New York, 1936), 250–73Google Scholar).

¹⁴ Dummett, ‘Wittgenstein's Philosophy of Mathematics' (1959).

¹⁵ Quine's and Dummett's objections, if well-taken, show that conventionalism cannot account for all necessities, not that it cannot account for any. For a fuller discussion of these objections, and an attempt to show that conventionalism cannot support even the weaker claim, see my Necessary Beings (2013), 116–27.

¹⁶ Lewis, David, On the Plurality of Worlds (Blackwell, 1986), 88Google Scholar. Lewis elaborates and qualifies the principle in a number of ways, but the finer details of his theory will not matter here. Other worldly individuals may, according to Lewis, be exact copies, or duplicates, of our individuals, but they need not be – they may be what he calls alien individuals. Another world may contain many duplicates, perhaps infinitely many, of any one of our individuals. So not all other worlds, in Lewis's view, are simply re-arrangements of duplicates of our individuals.

¹⁷ Combinatorialism is separable from Lewis's extreme form of realism about worlds. The combinational theory presented in On the Plurality of Worlds (Blackwell, 1986) assumes only a moderate form of realism about worlds. The basic entities in Armstrong's theory are not smallest parts (mereological individuals) as in Lewis's theory, but what Armstrong calls ‘fundamental properties' and ‘thin particulars', and his principle of recombination asserts that for any combination of fundamental properties and any thin particulars, there is a world in which those properties are co-instantiated in those particulars. I shall not discuss Armstrong's theory separately. As I observe below, my main objection to Lewis's theory applies, mutatis mutandis, to Armstrong's.

¹⁸ Lewis, , Counterfactuals (Blackwell, 1973), 86Google Scholar.

¹⁹ Lewis, , On the Plurality of worlds (Blackwell, 1986), 133Google Scholar.

²⁰ Lewis, , On the Plurality of worlds (Blackwell, 1986), 113Google Scholar.

²¹ Although his combinatorialism differs in details from Lewis's, Armstrong likewise endorses this principle – Hume distinctness, as he calls it – with the result that his theory likewise forecloses against many essentialist claims.

²² …which I take to include logical necessities.

²³ See, for example, Posterior Analytics Book A, ch. 4 (Barnes, J. (ed.) Aristotle Posterior Analytics (Oxford: Clarendon Press, 2nd edition, 1993), 6–8Google Scholar), where Aristotle argues that what is said of something in itself (kath’ hauto), or in what it is, is necessary.

²⁴ Fine, K., ‘Essence and Modality’, Philosophical Perspectives 8: Logic and Language (1994): 1–16Google Scholar.

²⁵ The idea that both individuals and general kinds have essential properties, of course, received strong support two decades or so earlier from Kripke in Naming and Necessity, especially lecture III. But while Kripke embraces essential properties, he does not discuss, much less endorse, the idea that necessities might be explained as grounded in the essences or natures of things.

²⁶ The use of x, y, … as universal variables over entities of all types involves a departure from Frege's doctrine that the reference of a singular term must always be an object, and more generally that ontological categories match up perfectly with logical types of expression. For further discussion, see my Necessary Beings (2013), ch. 1.

²⁷ Fine (1994), ‘Essence and Modality’.

²⁸ This assumes that there are no necessities the explanation of which requires invoking the natures of infinitely many entities. The assumption is certainly not obviously correct. For further discussion and defence of it, see Necessary Beings (2013), 6.4.3.

²⁹ Blackburn, ‘Morals and Modals' (1986), 121.

³⁰ The grounds for taking the logic of absolute necessity, briefly sketched above, seems to me compelling. The latter course has indeed been advocated, for quite different reasons, by Salmon, Nathan (‘The logic of what might have been’, Philosophical Review 98 (1989): 3–34)Google Scholar. But I am not alone (see Roca-Royes, Sonia, ‘Peacocke's principle-based account of modality: “Flexibility of Origins” Plus S4’, Erkenntnis 65 (2006): 405–26CrossRefGoogle Scholar, as well as Necessary Beings (2013), 128, fn.18) in thinking that Salmon's argument begs the question. Further, powerful arguments for taking the logic of metaphysical necessity to be S5 have been given, most notably by Williamson, Timothy (Modal Logic as Metaphysics (Oxford University Press, 2013)CrossRefGoogle Scholar, ch.3).

³¹ Applying (Necessity) directly to (c) would give us only ∃x□_x□∀z(z=a→Ha) which, in contrast with (d), is unobjectionable. Since □_a∀z(z=a→Ha) entails □□_a∀z(z=a→Ha), one can also obtain ∃x□□_x∀z(z=a→Ha), which again seems harmless.

³² My claim that properties may exist but be uninstantiated involves a clear break with the Aristotelian doctrine of universalia in res, which allows only instantiated properties. More controversially, perhaps – as my example of elephants and the property of being an elephant makes clear – it also puts me at odds from the quite widely held view that natural kind terms are extension-involving. At least, it does so, on the assumption that the predicates corresponding to some such terms are purely general. In this respect, I may also be in disagreement with Aristotle again, who requires that we must first establish that a general term φ is instantiated before we can enquire after the nature or essence of φs. The issue demands more careful discussion than a footnote permits. Very roughly, I think Aristotle is right in practice, at least in regard to essences discovered a posteriori, in the sense that it is invariably by investigating instances that we can get to know the essence. But that is an epistemological point which is consistent, in principle, with our being able to form a general predicate which defines the kind in question without extrapolating from instances, or even knowing whether there are any.

³³ Cf. Frege, Gottlob, Die Grundlagen der Arithmetik – Eine logisch mathematische Untersuchung über den Begri der Zahl Breslau (Wilhelm Koebner, 1884)Google Scholar, §83. Frege's term for a natural number is ‘finite number’ (endliche Anzahl). The finite numbers are those among the cardinal numbers which stand in the ancestral of the relation of immediate succession to 0. Frege proposes to define ‘n is a finite number’ to mean ‘n belongs to the natural series of numbers beginning with 0’. J.L. Austin renders Frege's German ‘n gehört der mit 0 anfangenden natürlichen Zahlenreihe an’ as ‘n is a member of the series of natural numbers beginning with 0‘, but this is badly misleading, as was first observed by Timothy Smiley (‘Frege's “Series of Natural Numbers”’ (1988)).

³⁴ For a less compressed development of this answer, see Necessary Beings (2013), 9.4.

³⁵ It is also worth noting that even if, in view of the way or ways in which species and other biological taxa are used in modern biology, essences cannot be purely general properties but must be conceived as relational properties involving, say, descent from earlier species members, this need not trouble the essentialist. To be sure, the existence of the relevant relational properties will no longer be necessary, but contingent. But that is no problem, given that any ‘new’ objects must be of kinds which already exist.

³⁶ This issue is also discussed in Necessary Beings (2013) (see especially 9.4.5), but I have since come to think that the answer there proposed is less than satisfactory. The answer proposed here drastically abbreviates a longer discussion in an as yet unpublished paper on existence and essence (‘Essence and Existence’, Revista de Filosofia de Costa Rica (forthcoming)).