Higher-order free logic and the Prior-Kaplan paradox

Abstract

The principle of universal instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior’s paradox and Kaplan’s paradox and the debate between necessitism and contingentism. We outline a distinctively free logical approach to the intensional paradoxes and note how the free logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramificationist approaches to higher-order logic. Our assessment of both approaches is largely pessimistic, and we remain reluctantly inclined to take Prior’s and Kaplan’s derivations at face value.

Keywords:

• Modal logic as metaphysics
• Prior’s paradox
• Kaplan’s paradox
• higher-order modal logic
• free logic

Notes

1 See Kaplan (1995), Kripke (2011), Ramsey (1960), Russell (1908) and Tucker and Thomason (2011).

2 Note that there are many different ways of implementing the ramificationist approach (it’s very natural to only allow types where k is greater than the levels of ). In what follows, we shall be following Church (1976) in considering only cumulative ramified type theories: informally, a relation that accepts arguments of a given level, accepts arguments of any lower level. For the connection between this and the various versions of Russell’s original theory, see footnote 2 of Church (1976).

3 We assume that applying the logical connectives does not take us to a higher level.

4 Provided B contains no free occurrences of p, and no free variables that would become bound if substituted for p in A, in what follows, we will take this condition to be tacitly assumed.

5 Without loss of generality, we give the proof of one direction (up to a change of variables): (1)    (instance of UI for .) (2)    (by universal generalisation for .) (3)    (by the principle when p is not free in A.)

6 See also Sher and Tieszen (2000) for a discussion of the second-order variant.

7 It’s arguable that the rejection of UI is also the key part to a Russell-style treatment of many other paradoxes, but we shall not defend the broader claim here.

8 Suppose that . Then, by EG, we can infer . We then have by necessitation, by universal generalisation and finally by necessitation again.

9 Whereas Prior’s original discussion is formulated in Polish notation and used the symbol d for the relevant unary operator, Kaplan used more standard notation and used the symbol Q. For the sake of typographical uniformity, we shall use Q throughout. Had Prior not been using Polish notaton, we suspect his paper would have entered the philosophical canon much earlier.

10 A complete axiomatisation of the logic of quantification into sentence position would include several other axioms governing the propositional quantifiers; however, this argument only requires UI, at least given certain non-trivial decisions about which truth functional connectives to take as primitive.

11 We say ‘roughly’ because we are using the English ‘if then’ to interpret the material conditional, we are using singular quantification instead of quantification that takes sentence position and finally, we are employing truth and falsity predicates that make no appearance in Prior’s actual theorem.

12 Obviously this isn’t the only way to reconcile Prior’s theorem with the apparent data. One could also insist that despite appearances, Kaplan wrote two or more things at midnight (see, e.g. Bradwardine 2010; Dorr 2012; Slater 1986). Or one might insist that while Kaplan didn’t say what he seemed to say, he succeeded in saying something else (Bacon 2014; Smith 2006). The challenge for these kinds of responses is to offer some guidance as to what the additional or alternative contents are.

13 A similar worry arises for the alternative and multiple proposition views. The alternative proposition view has to deny what seems obvious, namely: the antecedent of four, while the multiple proposition view must deny something else that seems obvious, namely: the negation of the consequent of four.

14 We should mention in passing another kind of paradox where denying universal instantiation might provide a path to a distinctive kind of solution. Here is a principle about propositional identity that is prima facie compelling, and particularly compelling if you think of propositions in a structured way: if Q and are propositional operators with different extensions, then the proposition that something is Q and the proposition that something is are not identical (more formally: ). But now a higher-order analogue of the Russell–Myhill paradox threatens to show that this is inconsistent. The derivation of this essentially involves the principle of universal instantiation into sentential position (Uzquiano 2015). It is beyond the scope of this paper to explore the promise of this response. However, we should note that the models in this paper are generally not useful for modelling structured propositions since, while they allow modally equivalent propositions to be distinct, they treat logically equivalent propositions as identical.

15 Both the first-order free logician described above, and the higher-order free logician will therefore have to be positive free logicians. That is, they will reject the inference of from even for atomic instances of .

16 One could also prove this by directly appealing to a general rule of necessitation; however, this rule is not uncontroversial in this context as it is relinquished, for example, in Kaplan’s logic of demonstratives in Kaplan (1977), and the status of necessitation in Williamson’s book is complicated. The above proof does not rely on necessitation – in general, most appeals to necessitation can be eliminated in favour of slightly stronger (necessitated) premises.

17 We are, of course, taking a few liberties here: for example, if the report is made after the writing, then we would use different tense morphologies.

18 Kaplan’s sympathy is not directed at the truth of this sentence, but at its logical coherence.

19 It should be noted that most of these authors, while attributing the paradox to Kaplan, did not have access to the details of Kaplan’s text since it had not been published.

20 Of course if you’re a hyperintensionalist, there won’t be any such thing as the proposition corresponding to a set of worlds. But if there are more propositions than collections of possible worlds, that only makes the situation worse for the principle.

21 Given universal instantiation, the latter definition satisfies schematic Leibniz’s law, but without it, we cannot derive it (we can derive quantified LL either way); thus, the latter definition only seems appropriate assuming UI.

22 This is a theorem under either definition of , as it amounts to and , respectively.

23 To be more precise, Williamson relies on a suitably necessitated version of a consequence of UI, namely: Comp. See below.

24 The proofs of these are well known (see Prior [1956]). In fact, much weaker axioms suffice for this purpose; for example, the principle (see Bacon 2011).

25 Notice that in Williamson’s system, there is a subtle distinction between comprehension principles for quantification into sentence position, and quantification into, say, monadic predicate position. The first is derivable from UI, but derivation of the second from UI requires additional assumptions about what counts as a predicate. In Williamson’s system, the only monadic predicates are simple monadic predicates (whereas it’s obviously not true that all sentences are simple), or at any rate, he doesn’t explicitly license a device for converting any old open sentence into a monadic predicate (such as -abstraction). Absent such a device, one cannot, for example, derive . And this is essentially because in order to use existential quantifier introduction to move from to , one would need to first convert the open sentence into a type suitable for monadic second-order existential generalisation. Given that the focus of this paper is on propositional necessitism, these subtleties are not very relevant to the main discussion.

26 That NNE, BF and CBF do not entail UI will be demonstrated by the models we consider in Section 4. To see that Comp does not entail UI, we must consider a model in which propositions are not individuated by necessary equivalence. For example, in the model , , , . , . Note that if propositions are individuated by necessary equivalence, then UI can be proven from Comp.

27 In (2013), Williamson focuses instead on abductive arguments for Comp which in turn guarantee many of the necessitist theses. There Williamson describes a sound (but incomplete) axiomatisation of higher-order modal logic. In that system, Comp is not an axiom, but a theorem derived in the above way from UI. This shift of focus from UI to Comp perhaps seems reasonable, given his simplifying assumption that higher-order entities are individuated by necessary equivalence.

28 It should be noted that it would also be inappropriate to restrict attention to universal accessibility relations – as is often done when, like us, one is dealing with models of S5 – but in the current context, that is unwise since we wish to allow models in which Q is hyperintensional.

29 A richer language might also include propositional letters, each of which denotes a set of points under an interpretation, but we will have no need for them here.

30 For example, one might take Frege’s ‘concept horse’ problem to be a result of not properly acknowledging this idea; see Williamson (2003) for this and other reasons for dissatisfaction with first-order models.

31 Note that this is a model of higher-order nominalism and moreover, we don’t use higher-order quantification in the articulation of the models. However, the model itself is of course not nominalistically acceptable in a broader sense because the models are constructed out of abstract objects, and so admit first-order objects that nominalists have historically rejected.

32 One could attempt to state this using substitutional quantification, but this may have limitations. Suppose the object language lacked Meinongian quantifiers, but the theorist had them. Then, the theorist could express the shortcomings by pointing out that there may be fictional things in the broad sense for which there were no names, and that imagining one of those alongside Pegasus would a counterexample to the intended meaning of the elliptical claim. Exactly the same point applies if one attempts to use substitutional quantification to capture the claim ‘I’m thinking that snow is white, but I’m not thinking that grass is green or ...’.

33 See McGee (2000). Roughly, a schema is understood open endedly if it holds under any extension of the language.

34 To satisfy K2, then there must be two distinct worlds that make and true, respectively. and are therefore both contingent and different. and require four different worlds as witness to the relevant instances of K2, but there are only three worlds to go around.

35 We know that such a bijection exists. Consider, for example, the bijection that maps each finite set X to where is the nth prime and is the nth largest member of X and is the size of X. And we map for a finite X to .

36 Given the bijection defined in footnote 35, and .

37 We shall show that expresses a finite or cofinite set whenever v assigns only finite or cofinite sets to the propositional variables (thus, every closed sentence expresses a finite or cofinite set relative to any assignment). Suppose and are finite or cofinite sets relative to every such assignment v. Then, any Boolean combination of them will be finite or cofinite. Given that R is universal, will be either W or and thus cofinite or finite, respectively. Similarly, for . If is finite or cofinite for every v assigning finite or cofinite sets, then is a singleton, as we noted earlier, which is finite (if is in the language, then is the same singleton). Finally, if is finite or cofinite for every such v, then where u and agree with v except for assigning W and to p, respectively. Since u and also only assign finite or cofinite sets to the propositional variables, , and thus are cofinite or finite.

38 Note that the result that expresses a different set of worlds for each n can be derived from K2 alone, or from K3 and the existence of and closure of existence under Q.

39 The crucial step is showing that if expresses a finite or cofinite set relative to any assignment that assigns finite or cofinite sets, expresses a finite or cofinite set. However, if the domain of propositions is infinite, potentially corresponds to an infinite intersection of sets of the form : even if the latter are all finite or cofinite, an infinite intersection of them needn’t be.

40 More formally, in the model currently under consideration, . This set is finite if and only if all but finitely many worlds , are members of . Thus, all but finitely many singleton sets must be paired with their members according to , which means that there are only finitely many worlds left for the rest of the finite/cofinite sets to be paired with, which is impossible. Similarly, it cannot be cofinite because it would mean that all but finitely sets of the form get mapped to x under , leaving only finitely many worlds to get mapped to the finite/cofinite sets.

41 It is relatively easy to tweak the model M so as to get K1 by brute force. Let be a bijection between and finite and cofinite sets given by . For , and . Assuming that the quantifiers range over finite and cofinite sets, we can show that is vacuously true at 0, and thus that , so this model satisfies K1 and K1. But there is no reason to think that it satisfies K2 or K2.

42 Suppose that where I is a set of nth stage possibilities (a set of n-tuples). Then, by construction, is the set of worlds where . (This is where the requirement that w(n) consists only of propositions not generated at earlier stages comes in.) So the set of -stage possibilities that contain I in their th component has as its completion, and so this intension belongs to the th stage.

43 We shall show that for every formula , there is some n such that is a completed stage n intension for every choice of v that maps variables to completed stage 1 propositions (and thus a closed sentence will always express a bounded proposition). The inductive hypothesis clearly holds for propositional variables. For each n, the completions of stage n intensions are closed under arbitrary Boolean operations. This means that if and are completed stage n propositions for each v mapping variables to completed sets of proto-worlds, and are also completed stage n intensions. Similarly, if is a completed stage n intension for each v mapping variables to completed stage 1 propositions, then is an intersection of completed stage n intensions, and is thus a completed stage n intension. and are always either or W, and thus completions of sets of stage 1 intensions. Finally, if is a completed stage n intension for each v of the required type, is a completed stage proposition.

44 One might think that fictional names are a counterexample to the first-order version of this generalisation. Even if Sally thinks that Pegasus flies, there is nothing Sally thinks flies, or so the objection goes. For his part, Williamson suggests that ‘we should distrust attempts to use fictional or mythological names to refute metaphysical or logical theses. Such terms have a confusing variety of uses [...]. The more such uses are studied, the less they seem to threaten classical theses of logic and metaphysics’. (See Williamson 2013, 153). Elsewhere, he suggests in a Fregean spirit that he is conducting his inquiry not in ordinary natural language but in a language free of certain defects that afflict natural languages. In this connection, a non-referring use of ‘Pegasus’ is cited as one such defect: ‘we should not distort our formal language by allowing for such a term’. (See Williamson 2013, 132).

45 For relevant discussion, see Williamson (1988).

46 Note that a language which could form disjunctions which contained a disjunct for every instantiation of a sentence of that very same language would not have a well-founded syntax since the instantiation of the disjunction itself would have to be a disjunct. One could countenance non-well-founded languages like this, but they are subject to their own liar-like paradoxes. For example, consider the sentence beginning with an infinite string of negations of ordinal type . Such a sentence would be identical to its own negation and paradox would quickly ensue.

47 See Chapter 6 of Williamson (2013).

48 This may be a bit quick, but it is beyond the scope of our paper to fully explore the implications of subtly different construals of the vicious circle principle.

49 The model we describe is not a regular model in the sense of Section 4 since the extension of Q is not given by a set of sets of points, but by a set of closed formulae of the language. Let , and , and let . Finally, let be a bijection between and . Truth at a world is defined recursively. For the most part, we follow the clauses in Section 4. is closed and . We’re not offering this as a model that’s metaphysically illuminating, but it does establish the consistency of Comp, K2 and K2.

50 We have discovered a very similar result in Myhill (1979), deriving a paradox from an unmodalised version of K4.

51 One could assert that no zero-level propositions are materially equivalent to by denying the existence of any zero-level propositions. Or one could go for a very radical form of ramification in which the truth functional connectives themselves have a hierarchical structure.

52 Note that sometimes philosophers adopt the convention of calling a schema true if its universal closure is true (where each schematic letter is bound by a quantifier of the requisite type), but that won’t help in the relevant setting since it will send us headlong into the first problem of insufficient generality.

53 Note that when this is spelled out fully, an unary operator (for example) is indexed by a pair of numbers: one that signifies the level of the input proposition and the other that signifies the level of the output proposition. Recalling the notation from the introduction, we would represent this by the ramified type (() / n) / m.

54 We note in passing that some implementations of Tarski’s stratificational approach to the liar adopts this second tack: for example, he endorses the schema when contains a truth predicate of level n or higher.

55 The most straightforward definition of sharing a grammatical category has to do with substitution preserving grammaticality. Unfortunately, this is not available to someone who takes type mismatch to induce ungrammaticality since, for example, switching names of different types may turn a grammatical sentence into an ungrammatical one, even though all names belong to the same grammatical category. Here is an alternative definition that is available to such a theorist, at least if they opt for a cumulative approach: two expressions belong to the same grammatical category just in case there is some expression such that both are in the range of its significance. If instead one goes non-cumulative, it’s no longer clear how to make good on the vision that one is stratifying within a grammatical category since the notion of grammatical category is no longer clear. Such a view arguably collapses into Fregeanism.

56 As noted in the introduction, on standard ways of setting up the framework, the expressions of the form can be ordered by the natural numbers because the variable p can be assigned a ramified level – the expression itself is syncategorematic.

57 One typically pursues this idea by ramifying the variables instead of the quantifiers, by insisting that a variable of type be only assigned values in a particular domain, and treating quantifiers themselves as type neutral rather like Russellian negation (as noted above). One could also achieve a similar effect by having a single type of variable, but by ramifying the device, we use abstract. It is unclear to us what, if anything, is at stake between these variants.

58 This seems very much in line with a solution entertained, albeit in a compressed way, by Kripke to his own puzzle.

59 As argued in a lot more detail.

60 The ordering technique we have in mind is one according to which if ‘expresses’ expresses expressing, then expressing comes immediately below expressing in the ordering. Ruling out loops and infinite descending sequences together would still guarantee no more than a partial order; further arguments would be needed to secure the kind of well-order that would justify indexing by the natural numbers.

61 Even if one took Kaplan’s lessons to heart, this wouldn’t make trouble for the models that we employed in Section 4. Note, importantly, that Kaplan makes it definitive of possible world semantics that one treats the object-level propositional quantifiers as ranging over all sets of worlds, but note that the models of Section 4 therefore do not count as ‘possible worlds’ models in that sense. There is more to be said about the relation between possible world semantics, as Kaplan describes, and what typically goes under the name ‘possible worlds model theory’, but we are not going to pursue these issues further here.

62 Of course for reasons given earlier, hardcore syntactic constraints on truthfulness and meaningfulness adopted by some ramifiers do not sit well with coarse-grained theories of propositions that are often adopted within possible worlds semantics.

63 This could be achieved by stipulating that all propositional variables have the type () / 0: this effectively means quantification over propositions doesn’t come in levels, but the operators still get ramified types of the form (() / n) / m for and so cannot be iterated. If we are going for a falsity interpretation of level mismatch, we would not require any special syntactic restrictions.

64 Much of what we have to say can be adapted to the view that there is a hierarchy of thinking relations, but no straightforward syntactic scope constraint concerning when attitudinal verbs can be embedded within one another to produce truths.

65 For example, let , , .

66 For what it’s worth, we do recognise that there may be something a little unstable about a ramification strategy that imposes austere syntactic constraints on attitudinal verbs, but does not ramify the variables or quantifiers. One effect of this combination is that one can have syntactically well-formed instances of the premises of Leibniz’s law, such as Qp and , but ungrammatical instances of the conclusion, such as QQq. But as we see it, there is no similar instability attaching to a view that espouses a hierarchy of relations but imposes no syntactical prohibitions. In other respects, the view might make similar moves to the more traditional ramifier, for example, with the traditional ramifier, it may think that our inclination to the hypothesis ‘it’s possible to uniquely think that everything you’re thinking is false’ or ‘it’s possible for it to uniquely seem as though you’re saying that everything it seems as though you’re saying is false’ is based on a blindness to the underlying structure of our mental states and speech acts.

67 On the interpretation on which type mismatch induces ungrammaticality, this interpretation of K2 is especially natural as we don’t in general consider ungrammatical substitution instances of schematic principles.

68 We let the quantifiers range over bounded propositions, and the extension of is the extension of Q (as defined in Section 4.3) intersected with the n-level propositions. To see that this model works, one first shows that if the highest type in is n, then expresses an n-level intension. Thus, every sentence whose type does not exceed n expresses an n-level intension, and so is at some world.

69 For our part, we don’t see any special appeal to a view that says K2 is obviously logically coherent, but not the conjunction of K2 with a supervenience thesis. Why not say instead that the logical mastermind would pour scorn on both on account of Prior’s theorem?

70 We might note in passing that, although not in the Russellian tradition, one might want to sort first-order variables into types with disjoint domains (see McDaniel 2009). One might also envisage cross-cutting domains – a picture encouraged by some of the literature on quantifier variance. Talk of ‘domains of quantification’ here and in the text is of course a bit of cheat (see footnote 7.1 below).

71 Even this characterisation of how the quantifiers relate to each other is something of a cheat since it pretends to be able to talk all at once about all the levels, yet we have no quantifier that can do this. This kind of consideration motivates Williamson (2003) to think that absolutely general first-order quantification is unavoidable; arguably, these considerations generalise to quantification into sentence position.

72 In this framework – where operators and connectives aren’t typed – it’s not so obvious whether any type mismatches can occur since a variable cannot be in the scope of another. Thus, the exact treatment of type mismatches doesn’t matter so much in this context.

73 Such a manoeuvre surely would not constitute special pleading, for even the Fregean would not countenance ‘is a horse is not a concept’ as an instance of the first-order generalisation ‘nothing is a concept’.

74 This thought turns on the cumulative aspect of the hierarchy, but as noted in footnote 55, it’s hard to properly distinguish the Fregean from the non-cumulative ramifier since we wouldn’t want the distinction to turn on very superficial syntactic categories.