1. J. N. Findlay ‘Can God's existence be disproved?’, in Antony Flew & Alasdair MacIntyre (eds) New Essays in Philosophical Theology (New York NY: Macmillan, 1955), 49–55.

2. Alvin Plantinga The Nature of Necessity (New York NY: Oxford University Press, 1974), ch. 1, passim; Saul Kripke ‘Naming and necessity’, in Donald Davidson & Gilbert Harman (eds) Semantics of Natural Language (Dordrecht: D. Reidel, 1972), 253–355.

3. So, e.g. Plantinga Nature of Necessity.

4. My conventions are these: letters like ‘P’ abbreviate sentences. Italicized sentence-abbreviations like ‘P’ and italicized sentences, like ‘God exists’ name the propositions those sentences express.

5. Richard Swinburne The Christian God (Oxford: Oxford University Press, 1994).

6. As noted below, Swinburne believes only in sentences, not propositions, and so I follow his usage in speaking of sentences as truth-bearers.

7. These considerations should, inter alia, suggest that Swinburne's account is far too simple – i.e. that any successful account broadly along its lines will have to include clauses dealing with these. The account probably needs time-indexing as well.

8. Swinburne has recognized the need for an ‘only if’ in correspondence, so I supply it here.

9. As of course are ‘suitable period’, ‘right reasons’, etc.

10. So, e.g. Timothy Williamson Vagueness (London: Routledge, 1994).

11. See e.g. Machina, Kenton ‘Truth, belief and vagueness’, Journal of Philosophical Logic, 5 (1976), 47–78.CrossRefGoogle Scholar

12. For supervaluationism see Kit Fine ‘Vagueness, truth and logic’, in Rosanna Keefe and Peter Smith (eds) Vagueness (Cambridge, MA: MIT Press, 1996), 119–150. For the ‘gap’ approach see Michael Tye ‘Sorites paradoxes and the semantics of vagueness’, in Philosophical Perspectives, 8 (1994), 189–206.

13. Private correspondence.

14. Some might reply, ‘but by coming up with new language, we render new things conceivable, and so it is up to us what we can conceive’. But there are limits on what we could ever use language to say. These are set by our natures and powers.

15. Whether anything other than the laws' existing falls into this category depends on what the true story about laws is. If they are abstract descriptions generalizing actual behaviour, for instance, there are no laws about light unless there is light. Again, if they are relations between properties and properties exist only if instanced, there are no laws about gravitation unless there are bodies to exert gravitational force.

16. Merely possible natural laws would yield a possible set of physical necessities, but if s1 states a merely possible physical law, either (1) does not express the claim that it actually is physically necessary that P or (1) is false.

17. See G. E. Hughes & M. J. Cresswell An Introduction to Modal Logic (London: Methuen, 1968), 21.

18. This is not to deny that if we suppose AB false, it winds up the case that AB is both true and false. But if it is true and false, it is true.

19. At least, that R is necessary is plausible. But see C. L. Hardin Color for Philosophers (Indianapolis IN: Hackett, 1988), 124–126.

20. A necessary truth is one that would hold no matter what. ‘No matter what’ in effect means ‘no matter whether this or that or that … occurs’. Different sorts of necessity differ by having a different ‘no matter what’ – because there is a different range of situations such that they'd hold no matter whether this or that or that of them occurred. What is nomically necessary, for instance, would be true in any possible situation whose physical laws were just like ours. To say that a claim would be true no matter whether S1 or S2 or S3 obtains is to say that it would be true in all of them. In effect, ‘necessarily’ is a universal (‘all’) quantifier over possible situations, saying that the sentence it governs would express a truth in all of these, or that the predicate it governs would be satisfied in all of these. The more situations the quantifier quantifies over, the stronger the necessity it expresses. So if every situation with our laws is an R-situation, but there are R-situations with laws different from ours, the necessity that R is stronger than nomic.

21. Private communication.

22. See Hardin, Color for Philosophers.

23. At least to those who consider them truths at all. Swinburne considers abstract particular numbers fictional entities (CG, 7) and calls himself a mathematical fictionalist (CG, 239), which suggests that he does not think mathematical statements true. (Fictional statements aren't true: just ‘true in the story’.) But he recognizes numerical properties (CG, 7). One can give an ontology and semantics for mathematics as well in property-terms as in terms of abstract individual numbers, as Russell first intimated and George Bealer has shown; Bealer Quality and Concept (Oxford: Oxford University Press), 1983. So while Swinburne calls himself a mathematical fictionalist, it is not clear to me why. He has enough ontology on hand not to be. The fact that any four-membered group is divisible without remainder into two two-membered groups seems a perfectly good thing for ‘2+2=4’ to be stating. And fictionalism is not intuitive. Here is a two-membered group of apples. That's fact, not fiction (CG, 7). Here are two more. Count them all: now we have a four-membered group, again as fact, not fiction. Even if the act of counting caused one apple to disappear before we got around to counting it, we'd know that there had been four there. Now repeat the trick with pears, peaches, plums: it works every time. Put one case of this into logical form, perhaps with special defined ‘numerical’ quantifiers, universally generalize, and we get a logical translation of the claim that 2+2=4: properly understood, to assert this is just to assert a universal generalization of the trick we've done. Swinburne, it seems, manages to believe that this generalization isn't true. This is a remarkable feat. I admire it but cannot emulate it.

24. For a selection of testimonials to this effect see Stephen Pollard Philosophical Introduction to Set Theory (Notre Dame IN: University of Notre Dame Press, 1990), 1.

25. It contradicts the axioms, of course. But if it sufficed to render P necessary that ~P contradict P, every proposition would be necessary.

26. That I call this the best available account does not imply that it does not have well known defects.

27. Christopher Menzel ‘Temporal actualism and singular foreknowledge’, in James Tomberlin (ed.) Philosophical Perspectives, V (Atascadero CA: Ridgeview Publishing Co., 1991).

28. On an alethic account of entailment, nothing can entail nonsense. But on Swinburne's, this seems allowable.

29. Swinburne may seem to offer an account which does not when he says that the naturally necessary ‘is the fully caused’ (CG, 117). But a full cause just is a cause a statement of whose activity entails the occurrence of its effect by way of statements stating appropriate natural laws. It is a fully explanatory cause, and explanation involves entailment.

30. Robert Adams points to this as a problem case – see his ‘Divine necessity’, in Thomas Morris (ed.) The Concept of God (Oxford: Oxford University Press, 1987), 43 – but develops the problem differently.

31. I assume that a purely syntactic account of entailment is not a live option.

32. Plantinga The Nature of Necessity, ch. 1, passim; Kripke ‘Naming and necessity’, 253–355.

33. Richard Swinburne ‘From mental/physical identity to substance dualism’, in Peter van Inwagen and Dean Zimmerman (eds) Persons (Oxford: Oxford University Press, 2007), 147.

34. Swinburne has another argument against divine necessity: Richard Swinburne The Coherence of Theism, rev. edn (Oxford: Oxford University Press, 1993), 274–275. It too rests on the move from coherence to possibility discussed below, and so does not require separate consideration.

35. Leibniz Monadology, secs 43–46; idem Theodicy, secs 180–185, 335, 351, 380.

36. Idem On Nature's Secrets.

37. See my ‘Aquinas on God and modal truth’, The Modern Schoolman, 82 (2005), 171–200.

38. Swinburne Coherence of Theism, 273.

39. Brian Leftow ‘Necessity’, in Charles Taliaferro and Chad Meister (eds) The Cambridge Companion to Christian Philosophical Theology (Cambridge: Cambridge University Press, 2010), 15–30.