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So, let us make the proposal explicit by running an example with a common family of relative (alethic) modalities, namely nomological possibility and necessity. Let us, once more, take our cue from Lewis:

“Thus it is nomologically necessary, though not unrestrictedly necessary, that friction produces heat: at every world that obeys the laws of our world, friction produces heat. It is contingent which world is ours; hence what are the laws of our world; hence which worlds are nomologically ‘accessible’ from ours; hence what is true throughout these worlds, i.e. what is nomologically necessary.” (Lewis 1986a: 7)

So, we learn that the basis of our restriction in the case of the nomological modalities is accessibility, i.e. similarity, with respect to the natural laws of the base world, usually ours:

(PN) It is nomologically possible at w0that A iff there is some world w1 which is

similar to w0 with respect to its natural laws N, and at w1, A.181

This analysis of nomological possibility, informally taken here from Lewis, will be non- modal only insofar as the laws of the base world w0 can be non-modally specified. In

particular, we need first a non-modal definition of the notion of a natural law. Then we

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must say a little about what similarity between two worlds with respect to their natural laws amounts to. Let us take these in turn.

5.3.1 The Notion of a Natural Law

The notion of natural law best suited for this purpose is Lewis’ own regularity or ‘best-system’ theory of natural laws, according to which:

“...a contingent generalisation is a law of nature iff it appears as a theorem (or axiom) in each of the true deductive systems that achieves a best combination of simplicity and strength. A generalisation is a law at a world i, likewise, if and only if it appears as a theorem in each of the best deductive systems true at i.” (Lewis 1973a: 73)

Thus, what separates laws of nature from accidental generalisations, in essence, is not the modal status of the former, but the fact that unlike accidental generalisations, the laws feature as theorems in our strongest and simplest systematisations of particular matters of fact. Of course, unlike the accidental generalisations, the laws do have special modal status, but this is reducible to their theoretical role in our best systems. It is no straight- forward matter of course to decide which statements should be the laws. The general guidelines, Lewis elucidates, are somewhat as follows:

“I take a suitable system to be one that has the virtues we aspire to in our own theory- building, and that has them in the greatest extent possible given the way the world is. It must be entirely true; it must be closed under strict implication; it must be as simple in axiomatisation as it can be without sacrificing too much information content; and it must have as much information content as it can have without sacrificing too much simplicity. A law is any regularity that earns inclusion in the ideal system. (Or, in the case of ties, in every ideal system.)” (Lewis 1983d: 367)

Lewis goes on to discuss this proposal at a little more detail.182 He notes, for instance, that the primitive vocabulary of the best systematisations had better refer only to perfectly natural properties, in order to avoid difficulties brought on by artificially perverse formulations of the relevant facts. (Lewis 1983d: 366-68) Giving the guidelines that

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Besides Lewis (1973a) and (1983d), see also his (1986c). See also Beebee (2000) for a defence of the best-systems view of natural laws.

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“simplicity without strength can be had from pure logic, strength without simplicity from (the deductive closure of) an almanac” (Lewis 1973a: 73), he notes that it is the purpose of scientific theory, with its existing (often vague and pragmatic) standards for evaluating competing systems, to decide which axiomatisation is to count as simplest and strongest and so which axioms are to count as the laws. We may additionally assume that if there are many non-overlapping systematisations, or an infinite ascending series of better and better systematisations, it may be a pragmatic matter which set of axioms to choose as the best, in the sense of being overall the most useful in scientific theorising. And we may allow that there are lawless worlds that cannot be systematised. (We may even remain open to a somewhat realist conception of natural laws that takes the axioms that constitute the laws to capture complex structural relations between sets of the world’s individuals (i.e. instantiated properties)).183

The important feature about this account of natural laws is that the laws are

individuated, not by virtue of being those generalisations that are necessarily true, but non- modally, by virtue of their offering the best – simplest and strongest – systematisation of particular matters of fact at a world. This allows us to characterise the natural laws of a world w non-modally and thereby fix the similarity, i.e. accessibility, relation applicable to nomological modalities without recourse to primitive modal notions. An analysis of nomological modality based on similarity with respect to natural laws can thus be deemed fully reductive. And the necessary status of the natural laws can be reduced to their truth at all nomologically accessible worlds.

5.3.2 Nomological Similarity

Now, let us turn to the question of nomological similarity between different worlds. We can choose alternative strengths of similarity. In the weakest case (for alethic modalities), where the accessibility relation is only reflexive, all we need for the accessed world to bear the requisite similarity to the base world, is for the former to simply render

true all the laws of the latter. In the stronger case, where accessibility can also be deemed transitive, we may demand that the laws of the base world are also laws at the accessed

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This modest attempt at realism, involving higher order relations between properties is still a far cry from the Dredske (1977)-Tooley (1977)-Armstrong (1978) understanding of natural laws as relations of

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world. And in the strongest case, adding symmetry, we may demand that the base and accessed worlds share exactly the same laws.184

There is one final important consideration here to note, namely that whatever else nomological accessibility may be, it ought never reach beyond worlds that are logically

accessible from a base world. This is only to be expected given that the axioms that are to count as the natural laws at a given world are, after all, determined only upon a prior acceptance of a particular logical system that allows us to close the set of truths at a world under strict implication. Then, to specify the natural laws of a world we may need the notion of ‘the best systematisation’ to involve acceptance of a particular logical system L

operative at w,with respect to which that systematisation is achieved. While I see no reason to deny that there may be cases that involve logical variations, it seems that the default position ought to be that nomological accessibility lies within the confines of logical accessibility. With these comments in mind we can think of nomological similarity in any of the following terms:

(i) A world w1is nomologically similar to a base world w0 if and only if w1renders

true the natural laws w0.

(ii) A world w1is nomologically similar to a base world w0 if and only if the natural

laws of w0 are also natural laws at w1.

(iii) A world w1is nomologically similar to a base world w0 if and only if w0 and w1

share exactly the same natural laws.

While (i) seems rather weak to capture the physical accessibility relation, it might bear further discussion whether the physical accessibility relation should only be reflexive and transitive (ii), or whether it should be an equivalence relation (iii). I will not enter such a discussion here.185 Additionally, we arguably want to nest nomological modalities within logical modalities in order to avoid having the explosion world, where every proposition is true, be nomologically similar (e.g. under (i)), hence possible. We might do this by only

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It may be, for instance, that two worlds render true the same universal generalisations yet one of them admits of a simpler systematisation than the other. Or, we may have further nuances and degrees of similarity by considering the accessed worlds as rendering true, or obeying, almost all the laws of the base world. See Pargetter (1984: 337-339) for a discussion of these matters.

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evaluating for nomological similarity worlds that are deemed logically accessible (in a sense to be explained) from the base world.

While the details of this picture merit deeper discussion, what we have here is a strategy, which agrees with the existing Lewisian conception of restricted modalities, and by which to give an appropriate answer to our original question, namely what we should take to be the basis of the relevant restrictions that demarcate the possible from the

impossible. In the uncontroversial case of the nomological (relative) modalities, the basis of the relevant restriction is similarity with respect to natural laws. So, given an

appropriate conception of a natural law, we get an analysis of one kind of relative

modalities, namely the nomological modalities, to be properly non-modal. Interestingly, under this picture, the nomologically necessary status of the natural laws simply reduces to truth at all nomologically accessible worlds. This means that it is nothing over and above the axiomatic status of certain truths in our best scientific theories that (rather

uninterestingly) guarantees their nomological necessity.186 Now, the idea is to apply the same strategy in order to specify the relevant restrictions for the broadly logical modalities.