Relative Necessity Reformulated

Abstract

This paper discusses some serious difficulties for what we shall call the standard account of various kinds of relative necessity, according to which any given kind of relative necessity may be defined by a strict conditional - necessarily, if C then p - where C is a suitable constant proposition, such as a conjunction of physical laws. We argue, with the help of Humberstone (Reports on Mathematical Logic, 31, 33–421, 1981), that the standard account has several unpalatable consequences. We argue that Humberstone’s alternative account has certain disadvantages, and offer another - considerably simpler - solution.

1 Introduction

Attributions of necessity and possibility are often qualified. We may assert, not that something is necessary or possible simpliciter, but that it is logically necessary or possible, or physically, or mathematically so, for example. It is also natural and plausible to suppose that at least some of these kinds of necessity are not absolute, but relative, in the sense, roughly, that what is said to be necessary is not being said to be necessary outright or without qualification, but only on—or relative to—certain propositions taken as assumptions, or otherwise held fixed.¹ Thus on one common view, physical necessity is a matter of following from the laws of physics, and physical possibility is compatibility with them. Varying the body of laws, or other propositions, gives us other forms of relative necessity.

How, in more precise terms, should the idea that a kind of necessity (possibility) is relative be explained? In broadest terms, it seems that the most promising approach will involve taking some kind of non-relative, or absolute, modality as one’s starting point, and explaining other, relative kinds as in some way relativizations of that basic kind. But how should this be done? What kind of necessity and possibility should be taken as the starting point, and what exactly is it for another kind of necessity and possibility to be a relativized form of that kind?

Interest in these questions may be prompted, and answers to them shaped, by at least two quite distinct considerations. Our goal may simply be to achieve a better understanding of the contrast between absolute and merely relative forms of necessity and possibility. This may go with a view to the effect that some kinds of necessity are absolute, and others merely relative, but it need not. One might still seek to elucidate the contrast, even if one thought it empty on one side. For one might suppose that while, if one kind of necessity is to be explained as a relativized form of another, the second kind must not itself be a relativized form of the first, it does not follow that the more basic kind must itself be absolute. To be sure, it might be thought that unless at least one kind of necessity is absolute, we shall be involved in an infinite regress. But it is at least not obvious that any such regress must be both infinite and vicious.²

We may be more ambitious. Our hope may be that, by showing that an ostensibly large and varied range of kinds of relative necessity can be exhibited as restrictions or relativizations of a single underlying kind of necessity—logical necessity, perhaps—we can achieve a conceptual reduction of ostensibly different kinds of modality to a single kind. Here we shall simply observe that this is a further, independent aim. As we shall see in due course, there is room for serious doubt whether it can be accomplished. But even if it cannot, that need not preclude an account of relative modalities that casts light on the contrast between absolute and merely relative necessities, and so answers to the first goal. Our aim in this paper is to cater to the first goal, without committing ourselves to the second.

We shall proceed as follows. First, we introduce the standard account of relative necessity, and discuss some putative problems for it, most notably propounded by Lloyd Humberstone. Having resolved some of those problems, we move on to discuss Humberstone’s diagnosis and solution to the remainder, and highlight some shortcomings with his approach. We then explore some alternative remedies, ultimately unearthing a much deeper problem with the standard account and some of its amended versions. In response, we offer our own proposed account, and discuss some of its consequences, technical and philosophical. We close with a summary of our findings.

2 Relative Necessity: The Standard Account

The standard account defines each kind of relative necessity by means of a necessitated or strict conditional, whose antecedent is a propositional constant for the body of assumptions relative to which the consequent is asserted to be necessary. Thus in a now classic treatment, Timothy Smiley wrote:

If we define O A as L(T ⊃ A) then to assert O A is to assert that T strictly implies A or that A is necessary relative to T. Since the pattern of the definition is independent of the particular interpretation that may be put on T we can say that to the extent that the standard alethic modal systems embody the idea of absolute or logical necessity, the corresponding O-systems embody the idea of relative necessity—necessity relative to an arbitrary proposition or body of propositions. They should therefore be appropriate for the formalisation of any modal notion that can be analysed in terms of relative necessity. [21] p.113.³

This kind of formulation of relative necessity has been quite widely endorsed in subsequent work.⁴ The standard account takes the kind of absolute necessity in terms of which different forms of relative necessity are to be explained to be logical necessity, so that it is relatively necessary that p just when, as a matter of logical necessity, C materially implies p, where C is a proposition of a certain kind—briefly, □(C → p)); and it is relatively possible that p just when p’s truth is not ruled out by C, i.e. when it is not the case that, as a matter of logical necessity, C implies not-p—briefly, ¬□(C → ¬p)), or equivalently ♢(C∧p).

Before considering those features which are genuinely problematic, it is worth briefly surveying some more or less obvious peculiarities of the notion of relative necessity, as formulated on the standard account. First, in the absence of any restriction upon the choice of C, each proposition p will be relatively necessary in an indefinite number of ways or senses. Any conjunction of propositions which numbers p as one of its conjuncts strictly implies p, as does p itself. Forms of relative necessity come cheap, and most of them are entirely without interest. This is not—or not obviously—a crippling drawback. A defender of the standard account can reply that it simply underlines the point that we should not expect interesting, non-trivial kinds of relative necessity to result unless the choice of C is restricted in interesting ways—to the laws of physics, say, or those of mathematics.

Second, since every logical necessity is strictly implied by any proposition whatever, every logical truth is C-necessary—necessary relative to C—no matter how C is chosen. But this, again, is not problematic. In particular, the fact that what is logically necessary is automatically relatively necessary in any sense you care to specify is in no tension with the plausible claim that logical necessity is absolute. The important contrast is not with relative necessity, but with merely relative necessity, where it is merely relatively necessary that p if it is, say, C-necessary that p but absolutely possible that ¬p.

Thirdly, since C always strictly implies itself, it automatically counts as C-necessary. So on the standard account, assuming that physical necessity, say, is to be analysed as a form of relative necessity, the laws of physics themselves automatically—and so, it would seem, trivially—qualify as physically necessary. But while physical necessity may not be absolute, it may be felt that there is more to the kind of necessity attaching to the laws of physics than their mere self-strict-implication.⁵ This may be felt to be a more serious objection and we shall have a little more to say about it below. First, we should see why the standard account appears to be in far deeper trouble, for reasons to which Lloyd Humberstone drew attention over three decades ago.⁶

3 Humberstone’s Problems

Humberstone raises the following problems for the standard account.⁷

3.1 Modal Collapse

One might suppose that there are several distinct kinds of relative necessity, and that many of them are factive, in the sense that, where □ _C is our relative necessity operator, □ _C p → p, for every p. In other words, the characteristic axiom of the quite weak modal logic T holds for □ _C . In particular, one would expect any kind of alethic necessity operator to be factive.⁸ For example, it might be held that both biological and physical necessity are two distinct kinds of necessity which are both relative and factive. However, Humberstone argues that, on very modest assumptions about the logic of the absolute modality operator in terms of which, on the standard account, they are to be defined, there is at most one factive kind of relative necessity. The modest assumption is that the logic of □ is at least as strong as K—the weakest normal modal logic. Let $\Box _{C_{1}}A$ and $\Box _{C_{2}}A$ be defined by □(C ₁→A) and □(C ₂→A) respectively. Then, Humberstone claims, it is readily proved that $\Box _{C_{1}}A$ and $\Box _{C_{2}}A$ are equivalent.

(1) □(C 1→p)→p	T-axiom for $\Box _{C_{1}}$
(2) □(C 2→p)→p	T-axiom for $\Box _{C_{2}}$
(3) □(C 1→C 1)→C 1	(1) ×C 1/p
(4) □(C 2→C 2)→C 2	(2) × C 2/p
(5) □(C 1→C 1)	propositional logic × Necessitation Rule
(6) □(C 2→C 2)	propositional logic × Necessitation Rule
(7) C 1	(3), (5) × modus ponens
(8) C 2	(4), (6) × modus ponens
(9) □(C 1⇔C 2)	(7), (8) × propositional logic,
	Necessitation Rule
(10) □(C 1→q)⇔□(C 2→q)	(9) × obvious reasoning in K
(11) $\Box _{C_{1}}q\leftrightarrow \Box _{C_{2}}q$	(10) × Def. □C 1, Def. □C 2

The key steps in this proof purport to show that when any relative modality □ _C defined in the standard way satisfies the T axioms, one can prove C. By Necessitation, □C, so that for any p, □ _C p entails □p. But □p obviously entails □ _C p.⁹ So relative modalities satisfying the T axioms collapse, not only into each other, but into logical modalities—relativization is a waste of time.

3.2 Unwanted Interactions: Imposing S4

It seems reasonable to suppose that our absolute necessity operator, □, satisfies the S4 axiom. If □ does obey the S4 axiom, every relative necessity operator □ _C must do so as well. The proof is simple. It is a theorem of S4 that □A → □(B → □A). An instance of this theorem, substituting C → p for A and C for B is □(C → p)→□(C → □(C → p)). But this is simply the S4 axiom for a relative necessity operator □ _C defined by □ _C p = _{d e f} □(C → p). So, if we assume that our absolute necessity operator, □, satisfies the S4 axiom, it immediately follows that our relative necessity operator, □ _C , does as well, i.e. □ _C p → □ _C □ _C p. Yet it may seem both odd and implausible to suggest that if it is, say, physically necessary that p, then it is physically necessary that it is physically necessary that p. More generally, we should expect to be able to define kinds of relative necessity in terms of □ which do not satisfy the characteristic S4 principles. The standard account is either overly restrictive, or defective, in only allowing one to define kinds of relative necessity that satisfy the S4 axiom.

Second, assuming □ is S4-like, one can show in a similar way that, where $\Box _{C_{1}}$ and $\Box _{C_{2}}$ are any two different forms of relative necessity defined in the standard way, the bizarre ‘mixed’ S4 principle $\Box _{C_{1}}p\rightarrow \Box _{C_{2}}\Box _{C_{1}}p$ holds. Again, it is a theorem of S4 that □A → □(B → □A), so in particular: □(q → p)→□(r → □(q → p)). Hence for any C _i : □(C ₁→p)→□(C _i →□(C ₁→p)). Yet it appears quite implausible that if something is, say, physically necessary, then it is biologically (or morally, and so on) necessary that it is physically necessary. To give another example, one might define drawer-necessity relative to truths about items in NN’s top desk drawer. For example, it is drawer-necessary (but not, say, physically necessary) that all the pencils in NN’s drawer are blunt. However, the consequences of the standard account include that if it is drawer-necessary that p, then it is physically necessary that it is drawer-necessary that p (and also that if it is physically necessary that p, then it is drawer-necessary that it is physically necessary that p). But surely both results are absurd. It is not a matter of the laws of physics that NN keeps only blunt pencils in his top desk drawer. Nor should it be a matter of drawer-necessity what does or doesn’t follow from the laws of physics.

3.3 Out of the Frying Pan?

To the modal collapse proof as it stands, there is a fairly obvious objection. The problem concerns the application of Necessitation at step 9. This rule allows us to to necessitate a proposition only if it has been established as a theorem, and if □ expresses logical necessity, the proposition to be necessitated must be a theorem of logic. But in this case, the proposition to be necessitated, viz. C ₁⇔C ₂, is no theorem of logic, since it depends upon 1) and 2), which are non-logical axioms of a system for relative necessity. Consider 1): its inner antecedent, C ₁, will be some non-logical proposition. For example, if we are considering an analysis of physical necessity, it will be a proposition we take to be a physical law, or perhaps a conjunction of such laws. Thus it might be, say, the proposition that force = mass × acceleration, so that 1) will assert that if the proposition that p is strictly implied by the proposition that f = m×a, then (it is true that) p. True or not, this is evidently no theorem of logic. It would seem, then, that the crucial step of Necessitation is illicit, so that the proof does not, after all, lead to modal collapse.

A similar flaw occurs in other relevant proofs. In his original paper, Humberstone argues that the standard account is afflicted by two other problems. First, on the assumption of a relative necessity operator □ _C satisfying the K, T and S4 axioms, one can show that the S4 axiom must also be satisfied by □. However, this relies on the modal collapse argument to establish the crucial result, □C, and so is also invalid.¹⁰ Second, he claims that, on the assumptions that one relative necessity operator $\Box _{C_{1}}$ satisfies the K axioms, the axioms B and D, but not T, while another $\Box _{C_{2}}$ satisfies the K axioms and also the T axiom, we can prove that $\Box _{C_{1}}$ must satisfy the T axiom after all. We have chosen not to discuss this problem in detail not only because it is a rather less intuitive combination of modal properties, but also because the proof fails for the same reason as that of problem 1: illicit use of Necessitation.

Is the standard account thus vindicated? No. Whilst some of Humberstone’s problems, in the form presented, can be dissolved, those of the unwanted imposition of S4, and ‘mixed’ S4, remain. Hence, there is still good reason to explore alternatives to the standard account. Moreover, in so doing we must take care to ensure that any alternative does not accidentally revive those problems now deemed solved. More importantly, in developing an alternative account, we will discover a far deeper problem for the standard account.

4 Humberstone’s Solution

Although, as we have observed, some of Humberstone’s problems may be set aside as relying on illicit steps of Necessitation, others remain. We wish now to consider his proposed solution to them. To understand it, it is useful to review his diagnosis of the source of the problems he takes to be fatal to the standard account.

He writes:

[T]he difficulties we have become entangled with result from making substitutions too generally. To see this, let us consider what happens semantically with the idea of relative necessity. We are given an arbitrary modal operator and asked to code it up as some combination of a fixed operator ‘ □’ and a propositional constant. But a propositional constant, if it is to be of the same category as the usual propositional variables, is (in effect) assigned a set of worlds in the Kripke semantics whereas the operators we are trying to use these constants to express correspond instead to binary (accessibility) relations between worlds: thus there is bound to be a loss of information in the translation. ([10], p.36. See also [11], p.51)

His thought, in other words, seems to be as follows: modal operators are quantifiers over worlds, and in particular, relative modal operators are restricted world quantifiers. Thus whereas an absolute operator, □, quantifies unrestrictedly, so that □p says that p is true at every world, a relative necessity operator, □ _C , quantifies only restrictedly, so that □ _C p says that p is true at each of a restricted range of worlds. The question is: how is this restriction to be captured? The standard account seeks to capture it by relativizing to a propositional constant—a constant which expresses a proposition true at just those worlds in the restricted range intended. Instead of interpreting the relative necessity claim as that p is true at each of a restricted range of worlds, it interprets it as the claim that the conditional C → p, where C is the propositional constant, is true at all worlds. The trouble with this, Humberstone thinks, is that it loses vital information about the distinctive accessibility relation in terms of which a relative modal operator needs to be understood. What distinguishes one kind of relative necessity from another, he thinks, is that each corresponds to a different accessibility relation. Thus what is physically necessary at a given world is what holds true at every world that is physically possible relative to that world, whereas what is epistemically necessary is what holds true, rather, at every world that is epistemically possible with respect to that world. These are quite different relations, but the standard account does not do justice to their difference.

In more detail,

if R is a binary relation on some set W, and S is a subset of W, we define the range-restriction of R to S symbolized R _S thus: x R _S y iff x R y and y∈S. …translating (or defining) O A as □(C → A) amounts to taking the accessibility relation for “ O” to be the range-restriction of that for “ □” to the set of worlds at which C holds. Thus instead of being able to cope with an arbitrary collection of modal operators, we are forced to deal only with collections whose accessibility relations are range-restrictions of a single relation, and it is this circumstance which underlies the difficulties … ([10], p.36)

In terms of this diagnosis, one can, he points out, give a straightforward explanation of the difficulties. Thus defining the relative necessity operators $\Box _{C_{1}}$ and $\Box _{C_{2}}$ by: $\Box _{C_{1}}A$ iff □(C ₁→A) and $\Box _{C_{2}}A$ iff □(C ₂→A) amounts to taking their associated accessibility relations R ₁ and R ₂ to be range-restrictions of the underlying accessibility relation R associated with □, so that they are just sub-relations of R with the same domain W but (proper) subsets S ₁ and S ₂ of R’s range W as their ranges. But then the supposition that $\Box _{C_{1}}$ and $\Box _{C_{2}}$ both obey the T principle is just the supposition that R ₁ and R ₂ are both reflexive—whence, since each has domain W, it must have the whole of W as its range after all. So the supposition of reflexivity undoes the range-restriction— R ₁ and R ₂ both end up with the same range as R, and hence as each other, and since they share the same domain, they are just the same relation by different names. Hence collapse. Similar explanations, he suggests, may be given for the other difficulties. See [10], pp.36–7.

It does not, of course, follow from the fact that some of Humberstone’s arguments rely on illicit Necessitation steps that there are no formally correct arguments to the same conclusions; so it does not straightforwardly follow that there must be something amiss with his purported diagnosis. But in fact there is a questionable assumption on which it rests. For if the addition of the T-schemes for $\Box _{C_{1}}$ and $\Box _{C_{2}}$ is to enforce reflexivity on the their corresponding accessibility relations R ₁and R ₂,Humberstone must assume that those schemes are logically valid. The assumption of validity, as opposed to mere truth at a world, is crucial. □p → p could be true at a world w without the accessibility relation being reflexive—for we might have w R w but ¬w ^′ R w ^′. v(□p → p, w) = 1 requires that if v(□p, w) = 1 then v(p, w) = 1 also, and since w R w, the truth of this antecedent requires the truth of this consequent. Further, we can suppose that v(□p, w ^′) = 1 while v(p, w ^′) = 0—there is no incompatibility here, since the truth of this antecedent does not require v(p, w ^′) = 1, since ¬w ^′ R w ^′. What cannot be the case is that v(□p → p, w) = 1 is true for all w, but R is not reflexive. But the assumption that the T-schemes are logically valid is simply the model-theoretic counterpart of the equally problematic proof-theoretic assumption—that $\Box _{C_{1}}p\rightarrow p$ and $\Box _{C_{2}}p\rightarrow p$ are logical axioms—which, as we have seen, vitiates the formal proofs for problems 1 and 3.

Prescinding from our reservations, Humberstone’s proposed remedy should come as no surprise given his diagnosis—since the propositional constants central to the standard treatment lose essential information about the distinctive accessibility relations which characterize different kinds of relative necessity, they must be replaced by a new type of constant which encodes the lost information. As he puts it, ‘the constants must do some of the “relational” work themselves’ (p.37). To this end, he assumes a two-dimensional framework, in which formulae, including the new type of ‘relational’ constants, are evaluated with respect not to single worlds, but pairs of worlds:

[W]e want, for A a formula and “ ⊧” a truth-relation determined by a model 〈W, R, V〉, to make sense of not the usual ⊧ _x A but rather ${\models _{y}^{x}}A$ (where x, y∈W), so that when it comes to evaluating one of our special constants, which I shall now write as R instead of C, to emphasize the relationality, we can say: ${\models _{y}^{x}}\boldsymbol {R}$ iff x R y. ([10], p.37)

The idea is that a given such R is to be true with respect to the pair of worlds x and y just when x bears the relevant accessibility relation, R, to y. Thus

If we have in mind a formalization of physical necessity, we might read “${\models _{y}^{x}}\boldsymbol {R}$” as “the laws of x are true in y” ([10], p. 38)

Using constants of this new type—semantically interpreted as ‘dipropositions’—Humberstone’s revised definitions of relative necessity operators have the same surface form as in the standard account. That is, where O is a relative necessity operator, we have

$$\mathbf{O}A=_{def}\Box(\boldsymbol{R}\rightarrow A) $$

But, crucially, dipropositional constants are not substitutable for normal propositional variables:

The propositional variables really do range over propositions, but the sentential constants R _i cannot be substituted for them because the latter are not propositional constants. In the terminology of [Humberstone (1981)], they are semantically interpreted not as propositions but as dipropositions—sets of (ordered) pairs of worlds. ([11], p. 53)

Clearly, since each of problems depends upon substitution of the relevant propositional constant C _i for a propositional variable, this restriction effectively blocks all of them. Crucially, this restriction blocks the problems that, on closer examination, are still valid and hence still pressing. In particular, the schema for the S4 axiom for a relative necessity operator is:

$$(\text{S}4) \Box(\boldsymbol{R}\rightarrow A)\rightarrow\Box(\boldsymbol{R}\rightarrow\Box(\boldsymbol{R}\rightarrow A)), \text{provided}\,\, \boldsymbol{R}\, \,\text{does not occur in} A.</p><p class="noindent">$$

5 Shortcomings

Humberstone’s solution comes at a price. In this section we highlight two disadvantages, which seem to us sufficiently serious to warrant looking for an alternative solution. The first disadvantage concerns the modal logic of the underlying absolute necessity operator, □. It is usual, and it seems to us overwhelmingly plausible, to take □ to express logical necessity. And it is, further, commonly supposed, and again very plausible, that the modal logic of □, interpreted as expressing logical necessity, should be S5.¹¹ But Humberstone cannot both take his absolute necessity operator □ to express logical necessity and agree that logical necessity satisfies the S5 principles. It is less than totally clear which claim he means to deny. Noting Smiley’s suggestion that we should take absolute necessity to be logical necessity, he comments that this is

a suggestion which is not entirely easy to evaluate. Most people believe that logical necessity satisfies at least the T axiom and the S4 axiom; however, neither □A → A nor □A → □□A is valid for arbitrary A, though both are valid for R-free A. … it may be held that since “ □” does not satisfy all instances of the familiar schemata, it cannot be regarded as expressing logical necessity. This matter cannot be settled here. ([10], p.40)

Since denying that the T and S4 axioms hold unrestrictedly for logical necessity is hardly an attractive option, it may seem that Humberstone’s best course is to accept that his absolute necessity operator does not express logical necessity. But this, too, has its disadvantages. Setting aside the absence of any plausible alternative candidate, it is independently plausible that the kind of necessity in terms of which various forms of relative necessary are to be explained should be logical necessity. Further, if that r$\hat {o}$le is assigned to some other kind of (absolute) necessity, logical necessity would, if not simply a restriction of that kind of necessity, have to be treated as a form of relative necessity.¹² But it is unclear how logical necessity could be merely relative, if logical necessity is stronger than the kind of necessity of which it is supposed to be a relativization (as it would be, if it obeys unrestricted T and S4, and plausibly S5, axioms). And in any case, even if the absolute necessity is not identified with logical necessity, it may be argued that its modal logic should be S5.¹³ All told, if there is an alternative formulation of relative necessity that can avoid making difficult claims about logical necessity—either that it does not satisfy the S5 principles, or that it is not absolute necessity—so much the better. Our aim below is to offer such a formulation.¹⁴

A second, and in our view more fundamental, disadvantage of Humberstone’s solution is that it is inextricably reliant on the assumption that modal thought and talk is to be understood as, and analysed in terms of, thought and talk about possible worlds. Our concern is not simply that the proposed solution presupposes the two-dimensional framework, which in turn appears to require acceptance of a plurality of worlds. Nor is the complaint that the solution requires acceptance of some form of extreme realism about worlds, such as Lewis’s—for there is no reason to suppose that it does so. The point is rather that, while there is no denying the enormous utility of possible world semantics in model-theoretic treatments of modal systems, it is one thing to hold that the truth-conditions of modal propositions can be usefully modelled in terms of systems of worlds, and quite another to claim that such propositions are fundamentally propositions about worlds and the relations between them—that understanding such propositions requires understanding them as making claims about worlds, and relations between them. The standard account, for all its faults, made no controversial demands on the metaphysics of modality, and took no stand on disputed questions about the nature and basis of necessity. This, it seems to us, is a virtue which a better account should preserve. Humberstone’s proposal, by contrast, requires us to accept that modal propositions—or at least propositions asserting relative necessity—are really propositions about relations between worlds. This comes out most forcefully in the new dipropositional constants. Humberstone does not spell out just what, under his proposed analysis, we are saying when we assert, for example, that such-and-such is physically possible, or physically necessary—how, exactly, R is to be understood, when it is the constant for physical necessity. The closest he comes to doing so is his suggestion that ${\models _{y}^{x}}\boldsymbol {R}$ might be read as “the laws of x are true in y”. This does not tell us how to interpret R in the context □(R → A), but it seems that there will be no alternative but to construe it as asserting something about worlds which are physically accessible from, or physically possible relative to, a given world. What is lacking is a way to account for our understanding of claims about relative necessity which does not require us to understand them as claims about relations between worlds. There may be some world-free way to construe dipropositions, but we take the onus to be with our opponent to offer such a construal.

Humberstone’s is not the only possible two-dimensional treatment.¹⁵ Alternative two-dimensionalist solutions may be able to avoid these disadvantages. But it is, we think, of interest to see whether a satisfactory solution can be developed which does not draw on a two-dimensional framework, and this will be our course in what follows.

6 Remedies

6.1 Adding a Conjunct

Before we present our preferred alternative treatment of relative modalities, it will be useful briefly to discuss another remedy which has been proposed. The added conjunct stategy, in its simplest form, consists in expanding the definiens for □ _C p by adding the propositional constant C as a conjunct.^{16 , 17} That is, for any form of relative modality for which the T axiom is to hold, we define:

$$\Box_{C}p=_{def}C\wedge\Box(C\rightarrow p) $$

Whilst this strategy blocks Humberstone’s modal collapse argument, that argument itself collapses anyway, so that there is no need for any further measure to block it. More importantly, this added-conjunct strategy does nothing to solve Humberstone’s S4 problem, as the reader may easily verify. If □ satisfies the S4 axiom, so must any form of relative necessity—adding a conjunct does not block this result, because the S4 axiom for $\Box _{C_{i}}$ under the revised definition, i.e.

$$(C_{1}\wedge\Box(C_{1}\rightarrow p))\rightarrow(C_{1}\wedge(\Box(C_{1}\rightarrow(C_{1}\wedge\Box(C_{1}\rightarrow p))))) $$

is fairly obviously still a theorem of S4 for □.¹⁸ The ‘mixed’ S4 problem, however, is now solved.¹⁹

6.2 A Better Diagnosis

We are now in a position to draw two crucial lessons from the failings of the standard account, and attempts to remedy those failings. In brief, (1) crucial information has been lost, but (2) that information must be reintroduced in a suitably general form, if the account of relative necessity is to have any plausible application.

First, then, independently of these logical shortcomings, the additional conjunct strategy suffers from another defect. Recall Humberstone’s complaint that the standard account founders because it loses vital information. We agree with Humberstone on this point—but not on what vital information is lost. Here is our alternative diagnosis. When we adopt the standard account of a relative modality—physical necessity, say—we proceed as follows: ‘Let C be a conjunction of the laws of physics. Then to say that it is physically necessary that p is just to say that p is a logical consequence of C—in symbols: □(C → p).’ The key point here is that, in adopting this formalization, we simply leave it to be understood that C is a conjunction of physical laws—nothing in our definiens actually records that that is so. So that information is lost. And it is, surely, vital information. Nothing explicit in the definiens distinguishes between a strict conditional purporting to express the physical necessity of its consequent, and one which purports to express some other kind of relative necessity.

One moral we might draw from this is that a better definition needs explicitly to record the relevant information about the status of the antecedent C. The simple added-conjunct strategy does no better in this respect than the standard account. An obvious way to remedy this particular shortcoming would be to employ an additional operator, π say, which might, in case we are seeking to define physical necessity, be read as ‘it is a law of physics that …’.²⁰ Amending the simple added-conjunct analysis to:

$$\text{It is physically necessary that}\,\, p \qquad{}\text{iff}\qquad{}\pi(C)\wedge\Box(C\rightarrow p) $$

not only restores the lost information, but actually provides a solution to the S4 problem. This analysis preserves what is good about the added-conjunct strategy—blocking the second ‘mixed’ version of the S4 problem—and it deals with the simpler S4 problem, which eluded the simpler strategy. For now, the S4 principle for □ _C1 is:

$$(\pi(C_{1})\wedge\Box(C_{1}\rightarrow p))\rightarrow(\pi(C_{1})\wedge(\Box(C_{1}\rightarrow(\pi(C_{1})\wedge\Box(C_{1}\rightarrow p))))) $$

and a simple calculation reveals that this is no theorem of S4 for □. Crucially, while antecedent and consequent here share the same left conjunct, π(C ₁), this conjunct is not necessitated—it is allowed that the laws of physics might, logically, have been otherwise. Consequently, there is a simple two world counter-model in which π(C ₁) is true along with C ₁ and p at w, but π(C ₁) is false at w′, even though C ₁ is true there. The counter-model exploits the fact that being true is necessary, but not sufficient, for being a law of physics—so that there are possible circumstances in which the propositions which are the actual laws of physics would still be true, but only accidentally, and not as a matter of physical law.

This brings us to our second crucial point. The additional conjunct strategy, whether in its simple or revised form, remains afflicted with a defect which was all along sufficient to justify rejecting the standard account, quite independently of the problems with which we have been occupied, but which appears to us, remarkably, to have escaped notice in previous discussion. As the use of C and π(C) makes plain, each of these accounts simply assumes that we are able actually to state, say, the laws of physics; neither provides an analysis we could advance, unless were we actually able to do so. At least, this is so, on the charitable assumption that the intention behind these accounts is that what is physically necessary should be what is required by the true laws of physics, and not just by what we take to be the laws. We do not, of course, mean to deny that it may on occasion reasonably be claimed that something is physically necessary, or physically impossible. Someone who makes such a claim may believe, on good grounds, that such-and-such propositions are laws of physics, and that they require, or rule out, the truth of this or that further proposition. Or, without being confident about any specific candidates to be laws of physics, she may believe that whatever precisely the relevant laws are, they will be such as to require, or rule out, the truth of certain propositions—such as that a human being should move by its own unaided effort at 200mph. Our point, at bottom, is that someone may understand the claim that, or the question whether, such-and-such is a physical necessity or (im)possibility without knowing, or claiming to know, what the laws of physics are, even approximately. Whatever the question is, that such a person is considering, it cannot be the question whether certain specified propositions are laws of physics which strictly imply that such-and-such. Of course, she may put her question this way: ‘Do the laws of physics require that such-and-such?’. But her question need not concern certain specific (candidate) laws—it can still be a perfectly general question: ‘Are there physical laws which require that such-and-such?’

6.3 Existential Generalization

There is an obvious way to do justice to this point—replace our propositional constant by a variable and existentially generalize through the position it occupies. That is, we should define:

$$\text{It is physically necessary that}\,\, p \qquad{}\text{iff}\qquad{}\exists q(\pi(q)\wedge\Box(q\rightarrow p)) $$

where, as suggested above, π(q) may abbreviate something like ‘It is a law of physics that q’. It is not assumed that the laws are known; hence π(q) could be read as saying that q is one of the actual physical laws, whatever they are, or may be—but not ‘whatever they could have been.’²¹

For any kind of relative necessity, then, our proposal is to define it in line with the following schema:

$$\text{It is}\; \varPhi\text{-necessary that}\; p\qquad{}\text{iff}\qquad{}\exists q(\varPhi(q)\wedge\Box(q\rightarrow p)) $$

where, Φ(q) abbreviates ‘It is a Φ-proposition that q’. The condition captured by “ Φ” may be more or less interesting, more or less restrictive. We will continue to use the example of the laws of physics, i.e. π(q), as a representative, and philosophically interesting, example.

Such a proposal captures quite naturally, or so it seems to us, the main idea of relative necessity. It introduces explicitly the information lost by the standard account. Not only does it specify the kind of proposition to which the necessity is relative, e.g. laws of physics, but also does so in a plausibly general way. If one claims, for example, that it is necessary relative to the laws of physics that p, one need not be claiming that p is necessary relative to some particular laws that one is able state, but only that there are some such laws that necessitate p. So what might at first glance appear to be an unnecessarily complex formulation, as contrasted with the standard account, is in fact simply recording explicitly the information that was all along assumed by the standard account, and in a suitably general way.

Is this proposal subject to any of the problems Humberstone took to afflict the standard analysis? Setting aside the arguments which we saw to be flawed by faulty Necessitation steps, it is easy to see that the arguments central to the remaining problems—those involving unwanted imposition of S4-like properties—no longer go through, since they both depend upon substituting a special propositional constant or constants for propositional variables in the S4 theorems □(q → p)→□(q → □(q → p)) and □(q → p)→□(r → □(q → p)), both of which are derivable in S4 from the so-called paradox of strict implication □A → □(B → A), and we no longer have any such propositional constant(s) to substitute.

Might there be some other route by which some or all of these problems could be re-instated? In the case of Humberstone’s first problem, the question is whether, when a T-axiom governing □ _π , defined as now proposed, is added to an underlying modal system, say K or T, to obtain a system S, we can derive □p from □ _π p. The distinctive axiom is ∃q(π(q)∧□(q → p))→p. With this available, we could argue:

$$\begin{array}{llll} 1 & \qquad(1) & \qquad\exists q(\pi(q)\wedge\Box(q\rightarrow p)) & \\ 1 & \qquad(2) & \qquad p & \qquad\text{ by T-axiom for }\Box_{\pi} \end{array} $$

But to get □p, we would need to apply the Rule of Necessitation or a rule of □-introduction. We cannot use the Rule of Necessitation, as this only permits us to necessitate theorems, but p is no theorem. And we cannot use the □-introduction rule, as this allows us to necessitate a formula only if it depends only on suitably modal assumptions, and (1) is not suitably modal. There is no other obvious strategy for a derivation, and we are confident there isn’t one to be found. Proving this—assuming we are right—is a more substantial task than we can undertake here. But we can at least sketch how we think things would go.

Giving a model to show that $\exists q(\pi (q)\wedge \Box (q\rightarrow p))\nvdash _{S}\Box p$ would require giving a semantics for the system S, and that will require, inter alia, a semantics for propositional quantification. Perhaps the most obvious method is (following [5]), to interpret propositions as sets of worlds, and take propositional variables to range over subsets of the set of worlds of the model. We would need to show that S is sound with respect to the semantics, i.e. that Γ⊩ _S A only if $\Gamma \vDash _{S}A$. As well as dealing with propositional quantifiers, the semantics will include a clause governing the π-operator (and more generally, for any relative necessity system, a clause governing the Φ-operator). We may presume that this will ensure that π(p) is true at a world w only if p is so: the laws of physics are, apart from anything else, true.

Since ∃q(π(q)∧□(q → p))→p is an axiom of S, we require this formula to come out true in every S-model. To see, informally, why it will do so, let $\mathcal {M}$ be any S-model. $\mathcal {M}$ will have a set of worlds W as its principal domain, so that propositional variables range over subsets of W. If ∃q(π(q)∧□(q → p))→p is to evaluate as true at each world w, it must be that ∃q(π(q)∧□(q → p)) is false at w or p is true at w. If p is true at w, so is ∃q(π(q)∧□(q → p))→p, just as required. So suppose p is false at w. Then we require that ∃q(π(q)∧□(q → p)) be false at w. In effect, a propositionally quantified formula ∃p A(p) will be true in a world in a model iff A(p) is true at that world for some replacement of the variable p by some propositional constant p ₀. So ∃q(π(q)∧□(q → p)) is false at w iff (π(q ₀)∧□(q ₀→p)) is always false at w, no matter how q ₀ is chosen. Pick any q ₀. We can suppose that q ₀ is true or false at w. If q ₀ is true at w,then since p is false at w, q ₀→p is false at w, so that □(q ₀→p) is false and hence (π(q ₀)∧□(q ₀→p)) is false as required. If q ₀ is false at w, then given our assumption about the clause for π, π(q ₀) will likewise be false at w, so that again (π(q ₀)∧□(q ₀→p)) is false as required.

To see that $\exists q(\pi (q)\wedge \Box (q\rightarrow p))\nvdash _{S}\Box p$, we shall need to show that in some S-model, for some w, ∃q(π(q)∧□(q → p)) is true at w while □p is false at w. Intuitively, this is clearly possible. Suppose □p is false at w. For ∃q(π(q)∧□(q → p)) to be true at w, we require that π(q)∧□(q → p) is true at w for some choice of q. Since we are taking π to be factive, q must be true at w. And □(q → p) must be true at w. This means that p must be true at w (we are assuming S includes the T-axiom for plain □, so that the accessibility relation is reflexive). But this is entirely consistent with p and q being false at w ^′ for some w ^′ accessible from w, as required for truth of □(q → p) and falsehood of □p at w.

We are reasonably confident that with further work, this informal sketch can be turned into a rigorous model-theoretic proof, and that we shall be able to show, along similar lines, that our proposed definition of relative necessity does not succumb to any of the other problems discussed in Section 3.

7 Complications and Refinements

The system S, lightly sketched in 6.3, clearly calls for more rigorous formulation and development. We cannot undertake a full-dress presentation here. Our purpose in this section is rather to draw attention to some signficant aspects of the system we intend, and to deal with some complications noted in Section 6.2 (see fn. 20). Here we have benefitted from very helpful discussion of an earlier version with David Makinson and some observations made by an anonymous referee, to whom we are much indebted. The complications, and in some cases, refinements, concern the following facts. (i) □p → □ _Φ p, i.e. the principle that logical necessity implies relative necessity, which Makinson calls Down, is not a theorem of S. (ii) The converse of Down, □ _Φ p → □p, which Makinson calls Up, is not a theorem of either S or S + Down. (iii) S appears to lack the conjunction property, i.e. that □ _Φ A,□ _Φ B⊩□ _Φ (A∧B). We give a brief statement of the logical facts, then comment on the issues to which they give rise.

• (i) $S \nvdash Down$

  Let our relative necessity operator be □ _Φ , defined as ∃q(Φ(q)∧□(q → p)). Interpret Φ so that v(Φ A, w) = 0 for all w∈W. Then v(∃q(Φ q, w) = 0 for all w∈W. Hence v(∃q(Φ q∧□(q → A)), w) = 0 for all w∈W, i.e. v(□ _Φ A, w) = 0 for all w∈W. So, vacuously, v(□ _Φ A → A, w) = 1 for all w∈W. That is, the T-schema holds with this interpretation of Φ.

  Now let ⊤ be any tautology, so that v(⊤, w) = 1 for all w∈W. Hence v(□⊤, w) = 1 for all w∈W. But v(□ _Φ ⊤, w) = v(∃q(Φ q∧□(q → ⊤)), w) = 0 for all w∈W. Hence v(□⊤→□ _Φ ⊤), w) = 0 for all w∈W. That is, Down is not derivable in S.

• (ii) S (S + Down) $\nvdash$ Up

  Let W = { w ₁ , w ₂ } and R = W × W = { 〈 w ₁ , w ₁ 〉 , 〈 w ₁ , w ₂ 〉, 〈 w ₂ , w ₁ 〉, 〈 w ₂ , w ₂ 〉 } so that R is an equivalence relation on W. Interpret Φ so that v(Φ A, w) = 1 iff v(A, w) = 1 for all w∈W.

  It follows that v(□ _Φ A, w) = 1 iff v(A, w) = 1 for all w∈W. It immediately follows from this last that v(□ _Φ A → A, w) = 1 for all w∈W—since if v(□ _Φ A, w) = 1, then v(A, w) = 1 for all w∈W. So the T-schema is validated by this model. But this model falsifies Up, if we stipulate (as we may), for some p, that v(p, w ₁) = 1 but v(p, w ₂) = 0.

• (iii) $\Box _{\varPhi }A,\Box _{\varPhi }B\nvdash \Box _{\varPhi }(A\wedge B)$

  Under our definition, the premises are ∃q(Φ(q)∧□(q → A) and ∃q(Φ(q)∧□(q → B). As far as these premises go, there need be no single proposition q such that Φ(q) which strictly implies both A and B. So we appear unable to infer ∃q(Φ(q)∧□(q → (A∧B)), as required for the conjunction property.

7.1 Comments on (i), (ii) and (iii).

For reasons which will rapidly become apparent, it makes best sense to begin with point (iii).

7.2 The Conjunction Property

What, in essence, appears to block the derivation of □ _Φ (A∧B) (i.e. ∃q(Φ(q)∧□(q → (A∧B)))) from □ _Φ A and □ _Φ B (i.e. ∃q(Φ(q)∧□(q → A)) and ∃q(Φ(q)∧□(q → B))) is the fact that we cannot infer from the premises that there is a single proposition r such that Φ(r)∧□(r → (A∧B)). As our anonymous referee points out, this problem would be solved if we could generally infer Φ(A∧B) from Φ(A) and Φ(B), since we could then conjoin the possibly distinct propositions q _i , q _j such that Φ(q _i )∧□(q _i →A) and Φ(q _j )∧□(q _j →B), whose existence is guaranteed by the premises, to obtain Φ(q _i ∧q _j )∧□((q _i ∧q _j )→(A∧B)), whence ∃q(Φ(q)∧□(q → (A∧B))). As a matter of fact, we make just this assumption for our operator π above (see footnote 20). However, an objection to this remedy, also proposed by our referee, is that it re-opens and reinforces the concern, to be discussed in section 8, that a Φ-operator is really nothing but a thinly disguised duplication of the relative necessity operator which we are seeking to define. In that case, rather than working via the conjunction property for Φ, we might as well directly stipulate that (□ _Φ A∧□ _Φ B)→□ _Φ (A∧B).

The problem raised is a special case of a wider issue, i.e., whether relative necessity operators, according to our proposed formulation, are closed under logical consequence, that is, if A ₁,..., A _n ⊩B then □ _Φ A ₁,...,□ _Φ A _n ⊩□ _Φ B for n≥1²². We need a solution that does not simply beg the question. And indeed, there is a simple revision of our proposal from which closure under logical consequence follows, without the complication of needing to specify and justify a conjunction property for Φ-operators.

Instead of defining □ _Φ A by a simple existential quantification ∃q(Φ(q)∧□(q → A)),we may simply use a finite string of quantifiers:

$$\Box_{\varPhi}A=_{def}\exists q_{1},...,\exists q_{n}(\varPhi(q_{1})\wedge...\wedge\varPhi(q_{n})\wedge\Box(q_{1}\wedge...\wedge q_{n}\rightarrow A)) $$

Clearly we obtain closure under logical consequence now, without needing to insist that a conjunction of Φ-propositions must itself be counted as a Φ-proposition.²³

7.3 Down

Makinson’s own view is that the failure of Down is a defect—without it, the system is ‘rather weak’. On the contrary, we take there to be good reason why we should not have Down. Down is, obviously, closure under logical consequence for the limiting case where n = 0. It is bound to fail if, as seems reasonable, we allow that the laws of physics, say, might have been otherwise, and that it is a contingent matter whether there are any such laws at all, for then □B may be true, but □ _Φ B (i.e. ∃q ₁,...,∃q _n (Φ(q ₁)∧...∧Φ(q _n )∧□(q ₁∧...∧q _n →B))) false, because there are no true Φ-propositions. Of course, under the hypothesis that there is at least one true Φ-proposition, Down will hold; where there are, say, some laws of physics, their logical consequences will, trivially include all the logical necessities, which will harmlessly qualify as physical necessities.

That Down should be validated is, no doubt, just what one would think, if one thinks about relative necesity in essentially world-terms: that is, so that what is physically necessary, for instance, is essentially just what is true throughout a restricted range of all logically possible worlds. For then, since anything logically necessary (i.e. true throughout the whole unrestricted range) must be true throughout any restriction of it, it must also be physically necessary. Our answer to this is that it is just a mistake to think of forms of relative necessity as fundamentally to be understood in world terms. If we drop that prejudice, then it can seem entirely natural and correct to characterize or define a form of relative necessity in such a way that logical necessity does not automatically ensure relative necessity.

7.4 $S (S+Down) \nvdash \,Up$

Evidently this is a welcome result from our point of view; Makinson’s proof confirms what we claim in 6.3.

8 Further Issues

In this closing section, we anticipate and respond to an objection to our preferred account, and draw attention to a limitation on its range of application.

The objection focuses on the sentential operators, such as π which play a key r$\hat {o}$le in our definition scheme. These operators serve to demarcate propositions to which specific kinds of relative necessity and possibility are relative. Because we wish to avoid assuming that we are able to actually state explicitly the relevant propositions, we need to characterize them in general terms, as propositions of a certain kind. Thus, as we suggested, π(q) might be read as abbreviating something along the lines of ‘it is a law of physics that q’, where this is to be understood as making a non-specific reference to the laws of physics, whatever they are, rather than to what we happen presently to take to be laws. Put bluntly, the objection says that by making essential use of operators so understood, the account simply gives up on the reductive explanatory aspiration which informs the original Anderson-Kanger-Smiley project. For what drove that project—and what gave it its interest—was the prospect of showing that, contrary to appearances, we do not need to recognize a great variety of independent notions of necessity, because we can explain each relative form of necessity using just a single ‘absolute’ notion (probably logical necessity). This explanatory aim is completely undermined by our appeal to laws of physics, for example, because the notion of a law of physics itself involves the idea of physical necessity.²⁴

A first point to be made in response to this objection is that, whatever force it may possess, it does not tell selectively against the explanation of relative necessity we have proposed. For essentially the same objection could be brought against Humberstone’s two-dimensionalist proposal. Of course, he makes no use of an operator comparable to our π operator. But for each distinct kind of relative necessity, his proposal will require a distinctive dipropositional constant, R, and this will need to be explained. The only kind of explanation he suggests is that we might understand ${\models _{y}^{x}}\boldsymbol {R}$ as ‘the laws of x are true in y’. As it stands, of course, this is hopelessly vague—there are many different kinds of laws, and he presumably does not mean that

they should all hold. What he intends, presumably, is that two-dimensionally understood, the diproposition will say, for example, that x’s physical laws are true in y. So his proposal, too, essentially involves an appeal to the notion of physical law, and so is no less vulnerable than ours to the envisaged objection. Thus if it should prove impossible satisfactorily to characterize physical laws without appeal to the notion of physical necessity, his account will be no better placed to subserve the reductionist aims of the standard account than our own.²⁵ It is true enough that the objection does not apply to the standard account itself. But as we have seen, there are good reasons to explore alternatives. Moreover, it seems inescapable that the deficiencies of the standard account have their source in its suppression of vital information. It is quite unclear how the requisite information could be incorporated without making use, in the analysis of physical necessity, say, of the notion of a physical law, or something near enough equivalent to it.

Second, even if it is granted that the notion of a law of physics, say, involves an implicit appeal to the notion of physical necessity or possibility, it is not clear that this need be objectionable. Whether it is so or not depends upon what the proposal sets out to accomplish. As we observed at the outset, one might hope that an analysis of relative necessity would subserve a reductionist programme. That is, one might hope to show that, contrary to appearances, there is no need to recognize a variety of independent kinds of necessity—physical, mathematical, etc.—by showing that each of these ostensibly different kinds of modality may be fully explained using just one single kind of modality, such as logical modality. If this were one’s aim, then the objection, if sound, would indeed be fatal. However, as we saw, there is another, no less important, aim one might have in pursuing analyses of forms of relative necessity—that of achieving an improved understanding of the contrast between merely relative and absolute kinds of necessity. The achievement of that aim is in no way compromised by the irreducibility—if such it is—of the various kinds of putatively relative necessity to a single absolute necessity.

Third, the claim that the notion of physical law cannot be understood without bringing in that of physical necessity may be challenged. There are various ways in which one might attempt to characterize laws of physics that make no overt appeal to the notion of physical necessity. One might characterize laws of physics without any appeal to (familiar) modal notions at all. For example, Maudlin [18] takes law-hood to be a primitive status, and indeed proposes a definition of physical modality in terms of laws.²⁶ Or, more commonly, one might characterize laws of physics making use of some modal notion other than physical necessity. For example, Lewis’s “best deductive system” account of laws of nature arguably only draws on the logical necessity built into the notion of a deductive system (see Lewis [15–17]). Or so-called “necessitarians” take the laws of physics to be metaphysically necessary (see [3] for a representative summary). These views would avoid the troublesome circularity of defining physical necessity in terms of physical laws, in turn defined in terms of physical necessity. The latter would of course introduce a further kind of modality—metaphysical—to be treated as relative or absolute, but the objection presently under consideration would be avoided. The success of this response to the objection depends upon the success of one of these alternative accounts of the laws of physics, but we will not be able to adjudicate on that matter here. It is certainly not obvious that these will all prove to involve a more or less thickly disguised appeal to the notion of physical necessity, and so to be unavailable as independent characterization of laws of physics.²⁷

Our claim in this paper is that our proposal is the preferable treatment of alethic kinds of relative necessity, such as physical necessity or mathematical necessity. There are potential problems for applying it to non-alethic relative modalities. Whether these problems can be overcome, we leave as further work to be carried out elsewhere. But we will briefly survey two key difficulties for treating non-alethic relative necessities.

The first difficulty arises from the relation between relative necessity and logical necessity. On the standard account, since every logical necessity is strictly implied by any proposition whatever, every logical truth is C-necessary—necessary relative to C—no matter how C is chosen. Similarly, although, as we have seen, Down does not in general hold for our alternative account, if it is C-necessary that p just when ∃q(C(q)∧□(q → p)), then, so long as the existence condition is fulfilled (there is a C-proposition), every logical truth will be C-necessary. This seems to be a reasonable result for alethic necessities: it would certainly be strange to claim that, although it is logically necessary that p, it is nevertheless possible, relative to existing physical laws, that ¬p. However, it has implausible consequences for non-alethic necessities for which, intuitively, even if some relevant C-propositions exist, it is not always the case that if □p, then □ _C p, such as epistemic or deontic modalities. Consider kinds of necessity defined relative to a conjunction of known propositions, or a conjunction of moral precepts. The current proposal for treating these necessities would yield the result that any logical truth is thereby epistemically necessary and morally necessary. However, it seems that we should leave room for the epistemic possibility that a proposition whose truth-value is as yet undecided should turn out to be false, even if in fact it is a logical truth. It also seems wrong to take logical truth to be a matter of moral obligation—we might not think that the world would be a morally worse place if a contradiction were true in it. Indeed, one might think, to the contrary, that if ‘ought’ expresses moral obligatoriness, ‘It ought to be the case that p’ implies that it is at least metaphysically (and so logically) possible that ¬p.

A different kind of problem arises from kinds of relative necessity where one might expect to find inconsistent q, although C(q), such as doxastic and legal modalities. One might expect that some conjunctions of beliefs are inconsistent, or that some conjunctions of laws of a given state are inconsistent. However, for any C-necessity defined in terms of an inconsistent proposition (i.e. where the existential condition is fulfilled by an inconsistent proposition), this would have the unfortunate result that everything would be C-necessary, given the inference rule that everything follows from a contradiction. But again, just because the statute books are likely to contain strict inconsistencies, does not mean that everything is legally required. That would be quite bizarre.²⁸

9 Conclusion

Our leading question has been: How should we best formulate claims of relative necessity? We have considered several answers

First, we reviewed the standard account: □ _Φ = _{d f.}□(C → p). This, we argued, falls foul of the S4 and mixed S4 logical problems as presented by Humberstone. Moreover, it omits important information about the nature of the proposition (i.e. C) to which the necessity is relative. Further, and crucially, the account assumes that, in making a claim of relative necessity, one is able to state all of the relevant propositions; for example, to make a claim of physical necessity in accordance with the standard account, one would have to be able to state the laws of physics. But such an assumption is too demanding—whilst we may be unable to answer the question whether it is physically necessary that p without some knowledge of the laws of physics, no such knowledge is required merely to understand the question; nor, accordingly, should it be presupposed by a good explanation of claims about physical necessity. These reasons led us to explore alternatives to the standard account.

Second, then, we considered Humberstone’s two-dimensional alternative. We had two main concerns with this approach: first, it is, we argued, unclear how best to interpret the □-operator of absolute necessity, without making unpalatable claims about the logical properties of logical necessity; and, second, it appeared to us implausibly to require not just that claims about relative necessity can be modelled in terms of worlds, but that they are in fact to be understood as claims about worlds.

We then considered a series of amendments to the standard account. The simplest of these consists in adding the proposition C itself as an extra conjunct in the analysans. This does indeed block the mixed S4 problem, but the simple S4 problem remains. So also does the crucial problem of understanding: the account still requires that one be able to, for example, state the laws of physics in order to make a claim of physical necessity. Adding a more complex conjunct, including the information of what kind of proposition is involved (e.g. specifying that C is a law of physics), solves the remaining S4 problem, but does nothing to resolve the problem of understanding, for the proposed analysans is still one which we can offer only if we are able to state the relevant laws (e.g. the laws of physics).

We then introduced our own, preferred, account.

$$\Box_{\varPhi}A=_{df.}\exists q_{1},...,\exists q_{n}(\varPhi(q_{1})\wedge...\wedge\varPhi(q_{n})\wedge\Box(q_{1}\wedge...\wedge q_{n}\rightarrow A)) $$

This captures the information lost by the standard account and the simple added conjunct account—that it is Φ-propositions relative to which things are necessary—but, in contrast with the more refined additional conjunct account, it does so in a suitably general way. The proposal avoids the logical problems (imposing S4, etc.), and does so without any intrusive use of worlds semantics in the analysis. There is no longer the problem that one must (be able to) state the Φ-propositions if one is to give the analysis—the analysans involves only the non-specific requirement that there be some propositions of that kind. Finally, we aired some technical and philosophical issues that may arise for our proposal. We have not had space here to resolve every issue, but we hope to have shown that pursuit of this account for relative necessity is at the very least a promising alternative to what, in our view, are some rather less promising accounts.²⁹

Appendix: An Alternative 2-Dimensional Solution?

In an article to which Humberstone refers ([7]) Bas van Fraassen proposes a rather different two-dimensional analysis of relative (e.g. physical) necessity. The analysis replaces the strict conditional □(A → B) by a two-dimensional conditional A ⇒ B (van Fraassen uses →, but this risks confusion with our use of the same symbol for the truth-functional conditional), defined as follows: [A ⇒ B](α) = {β:[A](β)⊆[B](α)}. Sentences within square brackets denote the propositions they express, which are identified with the sets of worlds at which they are true. So this says that the proposition expressed by A ⇒ B at world α is the set of worlds β such that the proposition A expresses at β is included in the proposition B expresses at α. In other words, what A ⇒ B says at α is true at β iff every world at which what A says at β is true is a world at which what B says at α is true. This contrasts with the strict conditional, which in the two-dimensional setting is true at α iff every world at which the proposition expressed by its antecedent at α is true is one at which the proposition expressed by its consequent at α is true. The conditions coincide when the world of evaluation is the same as the world of utterance, but diverge when they diverge. ‘It is physically necessary that A’ is then defined as R ⇒ A where R is a special constant which does double duty, both expressing ‘the appropriate relation of relative physical necessity’ and the corresponding proposition, defined [R](α) = {β:α R β}, i.e. the set of worlds physically possible relative to α (op. cit., p.82). Van Fraassen tells us that R—the law sentence, as he calls it—‘may say that the laws of α hold, or that they are laws, or that they are the only laws.’

Van Fraassen’s leading idea is that the relative character of physical necessity is best understood indexically. We can approach this as follows. When we hold that it is physically necessary that p, we commit ourselves to the claim that, even if things had been different in a whole host of ways, barring some (miraculous) suspension of the laws of physics, it would still have to be the case that p. We are not just claiming that, given the fact that things are the way they actually are (right down to the last detail), it has to be that p; we are allowing that circumstances might have varied in all sorts of ways, and claiming that even so, the laws of physics require that p. We can understand van Fraassen’s new conditional as designed to capture this. It seeks to do so by allowing the context for the antecedent to vary in any ways consistent with its still saying something true, but keeping the actual context for the consequent fixed. Thus we might read R ⇒ A as something like ‘In any circumstances in which the laws of physics hold it will be that A’—we can do so, if we may read A ⇒ B as ‘Any circumstances in which A, as said in those circumstances, would evaluate as true, is one in which B, as said here and now, would also be true’. In sum, the idea is to give a way of understanding claims about physical necessity which captures the fact that they make a claim whose truth has a high degree of independence from the actual circumstances in which they are made—so long as the physical laws hold, then although things can differ in all sorts of other ways, it will still be that p. Obviously, when we envisage different circumstances here, we are restricting attention to different circumstances in which the laws of physics as they are would not be altered—that is, we are keeping the actual laws, not talking about (more radically divergent) circumstances in which there would be different physical laws. That is, we are using ‘the laws of physics’ rigidly. The same is true of our own account—we require π(q) to say that q is a statement of the laws of physics as they are—we aren’t claiming (falsely) that no matter what the laws of physics might be, they strictly imply p.

The obvious questions are: Does this account avoid the kind of problems which beset the standard account? Is it otherwise satisfactory?

As far as we have been able to see, the answer to the first is that it does. In van Fraassen’s system, validity can plausibly be understood in one of two ways. A strong requirement would be that validity requires that ∀α∀β T r([A](α), β), i.e., A is valid just when, for any worlds α, β, what A says at α is true at β. A weaker requirement would be that ∀β T r([A](α), β) when α is taken to be the real world, i.e., A is valid just when what A says at the actual world is true at any world β. Whichever notion of validity one operates under, it appears that, due to the unconventional behaviour of van Fraassen’s conditional, neither A ⇒ (B ⇒ A), nor the especially relevant instance for the S4 problem, (B ⇒ A)⇒(B ⇒ (B ⇒ A)), is valid. Hence, van Fraassen’s system appears not to be subject to the S4 problems that afflict the standard account.³⁰

As regards the second question, we have some doubts. A first point is that A ⇒ A is not a law of van Fraassen’s logic. For since R is indexical, there may be a world, other than the actual world, such that what R says there is true at some world at which what R says at the actual world is not true. Whence, on the proposed semantics for the conditional, R ⇒ R is not valid.³¹But the law of identity, ‘If A then A’, is often taken to be fundamental, and has as strong a claim to be definitive of the conditional as the other principles which van Fraassen mentions (op. cit., p.82) as ‘earmarks’ of the concept. It is true that he views it as a ‘cluster concept’, so that some may go missing without destroying a connective’s claim to be a conditional (or implication connective—van Fraassen makes no distinction here). It remains a serious cost, and one that we, at least, are reluctant to incur.³²

A further concern is that the account may suffer from the same drawback as we found in Humberstone’s, i.e. that it does not just exploit the two dimensional framework in the model theory, but builds talk of worlds into the very content of claims about, say, physical necessity. The key question here is how the special propositional constants, such as van Fraassen’s R, are to be understood. That there is unwanted worlds content is certainly suggested by his own proposed readings of R, which ‘may say that the laws of α hold, or that they are laws, or that they are the only laws.’ It is not obvious that the world variable is dispensable without unsuiting R for its purpose. That van Fraassen thinks, to the contrary, that there is no essential reference to worlds is perhaps suggested by his official agnosticism about them (‘The items in the models, such as possible worlds, I regard with a suspension of disbelief, as similar to the ropes and pulleys, and little billiard balls that were introduced in nineteenth-century physics’, p.74). But if, more generally, we ask ourselves: ‘What is the parameter with respect to which talk of physical necessity is supposed to be indexical?’ it is not clear that the answer can be anything other than ‘Possible Worlds’.

Perhaps these doubts can be answered. If so, there is another, and better, two-dimensional solution to Humberstone’s problems. We see nothing inimical to our own proposal in this. For it would remain the case that if we are right, those problems can be solved more simply and economically, without any recourse to the two-dimensional framework.