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535 Notre Dame Journal of Formal Logic Volume 38, Number 4, Fall 1997 Impossible Worlds: A Modest Approach DANIEL NOLAN Abstract Reasoning about situations we take to be impossible is useful for a variety

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535 Notre Dame Journal of Formal Logic Volume 38, Number 4, Fall 1997 Impossible Worlds: A Modest Approach DANIEL NOLAN Abstract Reasoning about situations we take to be impossible is useful for a variety of theoretical purposes. Furthermore, using a device of impossible worlds when reasoning about the impossible is useful in the same sorts of ways that the device of possible worlds is useful when reasoning about the possible. This paper discusses some of the uses of impossible worlds and argues that commitment to them can and should be had without great metaphysical or logical cost. The paper then provides an account of reasoning with impossible worlds, by treating such reasoning as reasoning employing counterpossible conditionals, and provides a semantics for the proposed treatment. 1Introduction Some things just can t happen. Reasoning about such impossibilities, however, seems perfectly possible, and indeed important: I find myself doing it often. Some people have taken the legitimacy of reasoning about impossibilities to show that our logic must be weakened it is thought that logic must not only cater for relatively well behaved worlds like ours, but must be suitable for dealing with the more logically unruly cases we nevertheless have to consider. 1 I think such reasons are not good reasons to tamper with our logic we can, for example, keep even classical logic while making adequate room for thinking about impossibilities. In this paper I will outline what I take to be a modest approach to impossibilities, and impossible worlds. Of course, one person s modesty is another s extravagance and often a third s cowardice, so I do not expect this approach will appeal to everyone. It should however serve as a challenge to both those who are suspicious of impossible worlds and those who embrace impossible worlds more thoroughly than I do, by revising their logics to welcome them: since impossible worlds can be had comparatively cheaply, why not accept them? On the other hand, why pay more? In this paper, I will begin by discussing some of the uses impossible worlds have, and why I find reasoning about cases which are not even possible so important. Then I will present my modest proposal and argue that the advantages of impossible worlds can be had rather cheaply. Finally, I will explore some of the important difficulties which remain. Received November 20, 1997; revised December 16, 1997 536 DANIEL NOLAN 2Why we need impossible worlds There are a variety of areas in which it is useful to be able to reason about impossible situations and to do so in a nontrivial way (so that it is not good enough to just throw up one s hands and say that everything follows). The mere fact that we can think about what is impossible does not commit us to impossible worlds, any more than the mere fact that we claim that some claims are necessary or possible commit us to possible worlds. But just as it is a natural way to cash out our talk of necessity and possibility in terms of possible worlds, it is tempting to talk about impossible worlds, or situations, or ways things couldn t be. My defense of impossible worlds, then, will mostly be a defense of the need to nontrivially reason about claims or theories which we think cannot possibly be correct. If this is established, then the same sorts of reasons that encourage people to move from talking of possibility and necessity to possible worlds can be applied, mutatis mutandis,tomove from talking of impossibility to impossible worlds. 2 One of the first reasons which springs to mind for trafficking in the impossible is the understanding of logics which one takes to be incorrect. In one sense, to understand a logic different from the one(s) that one prefers does not require any special reasoning about the impossible: treating a logic as a set of symbols, or as merely representations of a set-theoretic semantics, does not require any suppositions about the impossible (especially when the metalanguages of such logics, and the logics governing the semantics, are not at all unusual: a classical logician can happily experiment with rival logics whose metalanguages are classical, and for which the set theory governing their models is classical). However, when we take rival logics to be claims about what really follows from what, or what inferences are licensed using connectives much like the English and, or, and not (and other connectives), then we do get into the realm of serious disagreement. Nobody denies that there is a formal system where p p fails to be a theorem. But most nonintuitionists balk at the suggestion that there is (or even possibly is) a proposition p such that it is the case that not-not-p but it is not the case that p. One strategy which has some popularity is to suppose that logicians who disagree with one are really talking about something else (they have changed the subject, as Quine would say ([26], p. 70): so intuitionists are really talking about provability, or some such; quantum logicians are really only talking about set-theoretic operations of various sorts in Hilbert spaces; truth-value gap logicians are talking about sentences rather than propositions, or whatever. And some of the time this strategy is the correct one: some people exploring or applying a logic may not take it to tell us anything about what really does follow from what, or how the logical connectives of natural language work (or work, given suitable idealizations). But sometimes this strategy seems to me to be clearly inapplicable: there are genuine disputes in logic, and people do often mean what they appear to mean: some intuitionists, dialetheists, and classical logicians really do disagree with each other about negation (among other things), Aristotelians really do disagree about nonsyllogistic inferences, S4-ers and S5-ers really do disagree about which modal inferences are acceptable, and so on. Another strategy is, of course, to take one s logical opponents to be talking incoherent nonsense, and so there is nothing useful to understand. Hardline intuitionists who claim that classical logic, insofar as it goes further than intuitionistic logic, is AMODEST APPROACH 537 incoherent and nonsensical, or makes meaningless assertions, are an example, as are those hardline classical logicians who dismiss deviance as gibberish. These strategies are much less common these days, as far as I can tell, and rightly so: the theory that rival logical schools are saying things that are utterly meaningless, only seem to communicate between themselves, and make no claims which are even assessable for plausibility, seems to me to border on a conspiracy theory. Surely the simplest explanation is that they have a theory: perhaps one which could not even possibly be correct, but one which is at least intelligible. Finally, there is one more approach to rival logical systems which would allow them to be understood and evaluated without entertaining things one takes to be impossible. Some logical systems have only a proper subset of the axioms or rules of inference of others. When one accepts a strictly stronger logic, it is comparatively easy to understand weaker logical systems, since all of the inference rules of such systems are acceptable: they simply do not exhaust the acceptable inference rules. So, for example, propositional logic does not capture important quantification-involving inferences, but nobody finds the propositional fragment of their preferred logic mysterious or incoherent on that account (even though, for example, there will be models where All men are mortal and Socrates is a man will hold, but Socrates is mortal will not). It is possible to consider weaker logics as merely leaving out some of the principles which could have been added. This approach to weaker logics is not entirely satisfactory in all cases either defenders of some relevant logic as the one true logic, for example, are genuinely disagreeing with a classical logician in a way that someone who was primarily interested in the propositional fragment of classical logic would not be. If these are genuine, meaningful disagreements, and at most one of the parties to the various disputes can be correct, then it seems that the other parties are reasoning about, and even believing in, impossibilities (albeit without knowing it). Furthermore, even those who have the good fortune to be correct, if they understand and can draw out the implications of their rivals views, will need to be able to consider and usefully reason about logical impossibilities. This nontrivial reasoning about impossible worlds is a central feature of much logical debate: and we seem to be quite proficient at it, even those who, in their philosophical moments, claim that such reason and understanding is impossible, or very limited, when it comes to situations or theories which could not even possibly obtain or be correct. A second area where people reason about the impossible, and where we may have to examine the commitments of impossible theories, is mathematics. One aspect of this reasoning about impossible situations has received a lot of notice: the apparent nontriviality of reductio proofs. When I start from supposing some premises which are in fact inconsistent, there seems all the difference in the world between a proof to the negation of one of the premises via derivation of a reductio, and a proof with the same premises and conclusion, but without the intervening steps (or with intervening steps which seem totally irrelevant: p, q, r,sothe moon is made of green cheese, so not-r, or some such). I do not think that these cases need the invocation of a distinction between trivial and nontrivial reasoning in the presence of inconsistency in order to distinguish them, however. The difference between an acceptable reductio proof and an unacceptable proof from the same premises can be found elsewhere. Besides 538 DANIEL NOLAN having a proof which is formally valid, we typically require that a proof is obviously formally valid, or at least is such that the rule used at each step is reasonably obviously formally valid. If a proof lacks this feature, then it is not as useful for the purposes to which we put proofs: convincing ourselves of the conclusions, or making sure of our results, or offering pieces of reasoning which we hope will convince, or any of these purposes. A proper reductio, as opposed to a chaotic jumble with steps like the moon is made of green cheese, is one where the steps from the supposition to the explicit contradiction (or whatever other absurdum which is employed) are obviously (or reasonably obviously) formally valid. So I think that reductio proofs do not show the need for nontrivial reasoning about impossibilities which many take them to, since the reason why some steps of derivation are acceptable and others are not can be explained in terms of the obviousness of the validity of some deductions but not others, rather than a distinction between the validity of different possible deductive steps per se. There is another sort of case which worries me more. In set theory, for example, there are axioms which some accept and others reject: the continuum hypothesis and the axiom of choice are the two most famous. There are also debates about areas of set- and setlike theory: whether there are non-well-founded sets, whether there are proper classes, whether category theory is about a new domain of mathematical objects, or whether its real interpretation is to be found in set theory with extra large-cardinal axioms, and so on. Positions in such debates are often inconsistent with each other (obvious enough when we examine systems, one of which has the continuum hypothesis (or generalized continuum hypothesis) as an axiom, and the other the negation of the continuum hypothesis as an axiom) yet it is not obvious that either side of such disputes is incoherent: in fact, with many of these issues it has been proved that such systems are consistent if standard set-theoretic systems are (both the axiom of choice and its negation are relatively consistent with Zermelo- Fraenkel set theory, for example that is, if ZF is consistent with itself, then it is consistent with the axiom of choice, and it is consistent with the negation of the axiom of choice). However, a very common view, and one which I share, is that the truths of mathematics have the same necessary status as the truths of logic: 3 so it seems that at least one of the sides in these debates is committed to a theory which could not even possibly be true. Some people at least think that there is a sensible question about what mathematical system is correct: for example, many people do not believe in proper classes, whereas I do. My opponents will typically admit that there are interpretations of the axioms of a theory of proper classes which have set-theoretic models which they do believe in, and so in that sense at least are as consistent as the theory in which they have models (see e.g., Fraenkel, Bar-Hillel, and Levy [9], p. 141). If the realm of mathematics is a realm of necessary truths, then it seems, prima facie, one of us holds a belief which is necessarily incorrect. But both sides in this disagreement are perfectly adept at reasoning about what would be true if our opponent s view was correct (as well as reasoning about what would be true if our own view was correct). Others might think that the debate as set up here is misguided. Perhaps the appearance of dispute is caused by an illegitimate Platonism, and that what is going on is not a dispute about the existence of sorts of objects (or the objective truth of AMODEST APPROACH 539 mathematical propositions) at all. Or perhaps our Platonism is not generous enough: perhaps there are not only Zermelo-Fraenkel sets, but New Foundations sets, and non-well-founded sets, and Gödel-Bernays-von Neumann classes, and Kelly-Morse classes, and categories, and systems of sets with large cardinals, and systems without large cardinals, and some systems with a generalized continuum hypothesis, and some systems without...ifallofthemathematical systems happily lived together in Platonic heaven our dispute would not be as it seemed either. Or perhaps the disputes are not as they seem because we are misled into thinking that it is a once-andfor-all issue what sets are like, or how sets and setlike objects (like classes) really are: perhaps the true picture is some sort of modal story, or a structuralist story, or both, or something other again, so that it turns out that there are not real disagreements between rival systems, since they each capture in an equally valid way one of the systems which mathematics could appropriately describe. Maybe the battle lines in these disputes are drawn in an unobvious way perhaps the simple picture I put forward of partisans of different axiom systems being in the uncomfortable position of at most one being right is not the right picture. Nevertheless, when partisans of rival systems do disagree disagree about what system is right, rather than merely which one is more convenient, or which one is the most interesting, or which lends itself to interesting applications then still, at most one can be right. For even if, say a partisan of Zermelo-Fraenkel and a partisan of New Foundations could both be right, when each condemns the other (perhaps by saying your axioms, taken together, are false ) still at most one can be correct indeed, if they are both correct as far as their positive views go (say, because each system is present separately in Gödelian heaven, or because there are appropriately interpretable structures for each), then both will be incorrect in their rejection of the truth of their rival s system. Indeed, regardless of the exact truth of the matter, provided only that people are coherently disagreeing with each other, then some will be coherently operating with a mathematical framework which is necessarily incorrect, should mathematics be necessary. 4 A third area which I find it important to have the ability to coherently discuss impossibilities is in metaphysics. Many metaphysical views seem to be such that if they are true at all, they are necessarily true, and if false, necessarily so: yet rivals understand each other, and we metaphysicians flatter ourselves that we are engaging in real debates, where argument and invocation of considerations are important: we are not babbling mere nonsense, even when some of our number (or many of our number) fall into necessary falsehood. The metaphysics of modality and possible worlds is only the most obvious example: when a metaphysical picture commits one to claims about the nature of possible worlds, and modal claims as a result about what is and is not possible (like Lewis s denial that there could be several disconnected spacetimes not otherwise connected by some special natural external relations 5 ), it is often involved in commitments that are necessarily false if certain of its rivals turn out to be true instead. Nevertheless, debate over modal questions continues, and exploration of systems of modal metaphysics other than the sort one accepts is a standard part of such investigation, both to see if one can put one s finger on what one finds unattractive, and to see whether one should switch one s views to a rival which proves more plausible. 540 DANIEL NOLAN As well as the debate about modality and possible worlds themselves, there are closely related debates with modal implications. Consider the debate about identity. There are views according to which I am not identical to the sum of my parts, but am only constituted by it; views according to which I am identical to the aggregate of my parts, but only contingently; views according to which I am essentially identical to the aggregate of my parts; and more besides. It seems plausible that if one of these alternatives is true, the others are necessarily false. 6 Yet weneed to reason about what would follow from these various theories if we are to hope to discover which one is the most plausible. There are other debates which seem to me to have modal implications: the debate between a realist about properties who claims that there must be a property of redness if a rose is to be red (and indeed that properties and relations are needed to underwrite all, or at any rate virtually all, predication), and a nominalist who claims that roses are red without there being any such property (and indeed that predication never needs to be underwritten by properties or relations), does not seem to be a debate about a matter which may be true in some worlds and false in others. Similarly the debate about whether normativity could reduce to dispositions seems to be one where many positions, if they are right at all, are necessarily so. I find myself, if I am to seriously and sympathetically understand and evaluate rival positions, forced to consider a range of options, many of which are impossible (though we will argue about which are the impossible ones, of course). It seems I must think about, and distinguish between, ways the world could not have turned out, as well as ways that it could if I am to best work out which are the impossible ones and which one is the actual one, or which of a small handful of the alternatives are genuinely possible. In all of these areas, discussing alternatives to the ways things could possibly be seems important. Of course, one could balk at moving from discussing different impossibilities to acceptance of the existence of different impossible worlds: just as one could balk at the equivalent move of going from talk of what is possible to talk of different possible worlds. But it certainly simplifies matters to be able to talk, for instance, about a world where Leibniz s metaphysics is correct and it seems proper to do so even while the question of whether such an account is even possibly correct

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