Exploring Ontology

It's all about the deep questions.

Second Principle on the Structure of Possible Worlds: A Solution to Hume’s Problem of Induction

1. An Allegorical Tale

Imagine that you have been taken prisoner. The judge of this imaginary prison, however, decides that your fate will be decided by chance in the following manner. Every morning during your stay at this prison a card will be drawn from a hat. Within this hat, 100 cards are placed. Furthermore, you are told that 99 of them have the word “death” written on them while 1 of them has the word “life” written on it. If one of the 99 are drawn on a particular morning, you will be killed that morning. If the “life” card is drawn, you will be spared. Not surprisingly, the very moment that you are told this, you expect to die the following day. Luckily, you don’t. This doesn’t take away your dread, however. The rest of the day is just as nerve-wracking as the previous one. However, the next morning, you are spared again. This cycle of dread and redemption continues on for days, then weeks. Gradually, the feeling of dread begins to wane and subconsciously you begin to expect to live to another day. Moreover, this feeling is justified. As long as you were not 100% confident that the judge was telling the truth with respect to the hats contents, the more days you live the more confident you are licensed to believe that the probability distribution of cards in the hat is not 99% chance of death and 1% life, but something much more skewed in the direction of life (this can be argued mathematically with conditional probabilities and Bayes’ theorem). As the months approach a year, you begin to wonder if there were any “death” cards in the hat at all. 

2. Preliminaries

Let us first start with some definitions. Call a possible world a Reasonable World (RW) precisely when that world, loosely speaking, is governed by reasons. This can be fleshed out in a number of ways. RWs can be said to have a “causal structure” where causes can be justly inferred from effects because of some internal necessity between events. These RWs can be said to have natural laws or nomological facts. These RWs are precisely the kind of world that science is justified in extrapolating its results into the future. Alternatively, call a possible world an Unreasonable World (UW) just in case it is not an RW. Now intuitively, a world which is in actuality a UW may resemble a RW , at least in part, “accidentally”. That is, a world with no natural law and no causal structure, can, from the point of view of an observer, look and behave as if it were a world with a causal structure. For another definition, a UW is said to be isomorphic to an RW just in case it behaves in this way. For our purposes, we will also need to similarly say that a UWis isomorphic until time t to a RWjust in case these two are indistinguishable to an observer at time t and at all times prior to time t, and after time t they are distinguishable. (Technically, under the Humean picture UWs that are completely isomorphic to RWs just are RWs. However, I do not need the existence for UWs that are completely isomorphic to RWs for my argument. I just need those UWs that veer off from RWs after some time to be conceptually coherent.)

3. Setting up the Analogy

David Hume’s problem boils down to the fact that no amount of sensory experience seems to be able to justify that we are not living in an isomorphic UW up to the present moment to the RW that science describes by extrapolating it’s results into the future. Sensory experience can’t do this since, by definition, these worlds are, at least up to the present, isomorphic to the world we suppose we inhabit. Now the allegorical story comes up. We are in precisely the same position as the prisoner. Except worse. With our current philosophical understanding, the actual world may be the RW that science posits or any one of infinitely many different isomorphic UWs until the present time. In almost all of these UWs, I will not be alive in the next second. Since in UWs future time states are not in any way constrained by past time states (if we are living in an isomorphic UW, time t could be our world, time t+1 could be a world consisting only of one floating pebble, and time t+2 could be a world filled with exploding supernovas) the chance that the aggregate of molecules that is me will continue to exist with all of the memories intact for one more second (in which an infinity of instances have past) is, basically, zero (or a hyperreal away). In this picture, an event that is next to impossible happens every instant: the “judge” draws the “life” card (the RW that science posits) out of all of the other UW isomorphic worlds and the RW. What should we conclude? Our conclusion must make all of these troublesome UWs not be genuine epistemic possibilities for the actual world. How do we go about doing that? By not letting these UWs be metaphysical possibilities. Furthermore, it is not enough restricting them to some finite number since we are “drawing out of the judges hat” infinitely many times a second. If the probability of picking the RW world described by science “out of the judges hat” is anything less than one on each drawing, our state of affairs quickly turns into an impossibly lucky state of affairs. There are no UWs in the judge’s hat.

4. The Argument

  1. If UWs are metaphysically possible, a miracle (or a violation of the correct laws of nature) would have occurred sometime in the past minute.
  2. A miracle, or a violation of the correct laws of nature, did not occur some time in the past minute.
  3. Therefore, UWs are not metaphysically possible.

(Note: Under the non-humean conception of laws there is a fact of the matter as to what laws govern the world. Under the humean conception of laws, there is a fact of the matter as to which set of statements that accurately describe the world are the most simple, economical, elegant, etc. for some explication of simplicity, elegance, etc.)

This argument proves the Principle of Reasonable Worlds (PRW): only RWs are metaphysically possible. What this argument concludes is what is called a strong necessity in the philosophical literature (for example, David Chalmers in his book The Character of Consciousness discusses the possibility of strong necessities as relevant to his conceivability argument for dualism), or alternatively a fundamental law of metaphysics. It is a necessary principle that is not known aprioribut can only be known, if it can be at all, by partly a posteriori considerations. Strong necessities constrain the space of metaphysically possible worlds alongside logical considerations. It can be easily shown that strong necessities are brute facts about reality. If they were to have some grounding to them, it wouldn’t be anything contingent since a contingent fact can not ground a necessary fact. Necessary logical facts cannot entail strong necessities since if they did strong necessities would be knowable a priori, which by definition they are not. Of course, if we accept PRW as a metaphysically necessary principle, which the argument argues for, David Hume’s famous Problem of Induction that has been plaguing the philosophy of science for centuries will have been answered. In defending the argument, premise 1 is doing the most work. Although it is conceivable to hold a view that miracles occur all the time in this world, I take very few (at least nonreligious) philosophers to hold this view. In what follows I will give a brief recap of why premise 1 is nearly certainly true, and then I will answer some potential objections. Finally, I will consider some ramifications of the principle.

4.1 A recap for the motivation behind premise 1

Imagine the initial allegorical tale except that the current philosophical opinion says that there are infinitely many cards that say death on them (corresponding to isomorphic UW worlds) and the “life card” (corresponding to the RW world in which all of scientific law carries on as usual). Furthermore, drawings of the hat occur every instant, infinitely many times in the span of a second.  Current philosophical opinion would say that we are just “lucky” that every single one of those cards turns out to be the life card. This amount of luck is, almost literally, impossible. If the prisoner is justified in believing the hats contents do not contain death cards, we should be justified in concluding that the metaphysical hat of all possibilities do not include UWs.

4.2 Objection 1: Are strong necessities impossible?

There is a strong tendency to shy away from strong necessities for understandable reasons. Up to now, no strong necessities have been recognized to be true. They are brute facts that constrain the realm of all possible worlds, and one would like to avoid brute facts whenever possible. However, strong necessities don’t seem to be epistemically impossible. After all, it is a substantive claim about reality that X, where X is some maximally consistent state of affairs, could have been the case. Put in other words, it is not analytically true that “For all maximally consistent state of affairs X, X could have been the case.” When we say something could have been the case we are saying something more than just X is logically consistent.

Another reason for thinking that strong necessities are epistemically possible is the epistemic possibility of Modal Deflationism. Modal Deflationism says that there is nothing in the world making it true that X, where X is a non-actual state of affairs, is metaphysically possible. So, strictly speaking, it is not the case that X is metaphysically possible under Modal Deflationism. Even the fact that intelligent philosophers can hold this claim to be true should make someone hold that “If X is logically possible, then X is metaphysically possible” with less than absolute certainty. For example, David Chalmers being able to entertain the possibility of strong necessities would be a reason for not holding that the existence of strong necessities have probability zero. Again, to be able to deny the argument, it is not enough to say that it is extremely unlikely that there exist states of affairs that are logically consistent but not metaphysically possible. One would have to hold this view with absolute certainty, with the same certainty as an analytic statement, for example. However, this view is not analytic.

What would an argument for the impossibility of strong necessities look like? Since neither any particular strong necessity or its negation follows from a priori logical considerations, and contingent facts in all likelihood do not bear on them, it is a similar brute fact about existence that there are no strong necessities (a brute fact similar to admitting that there are strong necessities). It would almost seem that the only way to avoid strong necessities is to posit a strong necessity barring all other strong necessities! In my opinion, there is no reason to believe that human beings have a sort of privileged access to the constraints of all causally detached metaphysically possible worlds, an access that is completely infallible. So, it is not surprising that some strong necessities may exist that are not self-evident.

Furthermore, we must weigh the epistemic considerations we have concerning the truth or falsity of PRW. On one side is the understandable reason to not admit brute strong necessities constraining possible worlds, but on the other side of the argument we literally have the most evidence possible for any non-tautological proposition. I personally think it is quite clear that the epistemic considerations in favor outweigh those against. Now, being a necessary principle, it either constrains all possible worlds or no possible worlds. Since the epistemic considerations for such a proposition is literally the highest which it can possibly be, we should believe in the metaphysical necessity of PRW, a claim that can be argued through Bayesian reasoning.

4.3 Objection 2: The argument is too ambitious in the scope of its strong necessity

Technically, this claim is correct. From the above considerations it does not immediately follow that I am entitled to claim that all UWs are metaphysically impossible. The problematic UWs are just those that are isomorphic to the RW that science attempts to describe. If those are metaphysically impossible and some others are, then that would be fine. Premise 1 should be revised to read “If isomorphic UWs to the RW that science describes are metaphysically possible,….”. The conclusion should then read “isomorphic UWs to the RW that science describes are metaphysically impossible.” Although strictly speaking this claim is true, the resulting strong necessity proven by the argument turns out to be somewhat awkward. The principle would rule out only those isomorphic UW with the actual world up to the present moment. The previously proposed principle, PRW, is much more unified, simple, and (if it counts for anything) philosophically elegant. Either way, however, the purpose of this paper is to solve the problem of induction, which either principle solves anyway.

4.4 Objection 3: Couldn’t you do this for anything that is improbable?

While it is correct to say that someone could do this procedure for anything that is improbable (i.e. banish the improbability by guaranteeing the improbable event in question with a metaphysically necessary principle), it wouldn’t be epistemically justified.  The argument I have spelled out could be couched in terms of Bayesian reasoning. The crucial point is this: no matter what non-zero prior probability you assign to strong necessities, my argument would go through with absolute certainty (technically, a hyperreal away from that). It also has to be the right kind of an event. If infinite people played the lottery and Bob won, a strong necessity shouldn’t be invoked to guarantee this event since you didn’t predict before hand that he would win. All in all, however, I think it is an interesting question for further analysis when someone is justified in invoking strong necessities.

5. Ramifications

PSW, and more generally simply the existence of strong necessities, not only solves the problem of induction but it also significantly impacts several philosophical debates. Every argument that uses conceivability to entail possibility (i.e. arguments in the philosophy of mind for dualism) must acknowledge that the existence of strong necessities not knowable a priori leads to conceivable situations that are not metaphysically possible (UWs). Not surprisingly, the study of modality in general would be dramatically change. The question of modal epistemology would be much more salient. PSW is also eerily familiar to Leibniz principle of sufficient reason, which could breathe new life into the argument from contingency in the philosophy of religion. Perhaps most importantly it would open the road for the examination and discovery of other strong necessities, if any more exist.


11 responses to “Second Principle on the Structure of Possible Worlds: A Solution to Hume’s Problem of Induction

  1. JF April 8, 2014 at 12:17 pm

    I appreciate the spirit of your argument, however I’m skeptical that you have ‘solved’ the problem of induction. An equivalent Bayesian formulation based on coin flips might go as follows:

    The RW hypothesis corresponds to a biased coin whereby p(Heads) = 1.0 (“all heads”) and the UW hypothesis corresponds to p(Heads) < 1.0 ("not all heads"). Thus, by flipping the coin a trillion times and observing a 'head' on each flip, the posterior probability of 'all heads' converges to 1.0 and the UW hypothesis converges to zero. So far so good. But Hume (and Goodman) might then object that this assumes a principle of uniformity of probability (that the actual bias of the coin does not change over time). Accordingly, the observation of a trillion heads supports the RW hypothesis 'p(Heads) = 1.0' only up to that point, after which the bias of the coin might still abruptly change, consistent with the UW hypothesis (which is isomorphic to the RW hypothesis for flips 1 through 1 trillion, but which diverges thereafter).

  2. tarrobread April 8, 2014 at 12:45 pm

    Interesting – I don’t think the objection succeeds, but I might be wrong. The crucial point would be about the metaphysics of modality. I take it that either there are no UWs, or they are absolutely everywhere. For example, if there is a possible world where green changes to grue at a specified time t, then for every time t there is a possible world where green changes to grue. So, those are the two hypotheses on the table. Otherwise, modal space would be utterly arbitrary if divergences can happen “late” in a world but not otherwise. Given this, I think your objection doesn’t work. We have two theories of the nature of modal space on the table. Given that we flip the coin to heads a trillion times, that is exactly what we should expect under the RW hypothesis, but definitely not what we should expect under the profligate UW hypothesis. Of course, it doesn’t make it absolutely impossible that this world is a UW where the coin diverges very late. I didn’t claim absolute certainty however. Just that purely empirically speaking, one theory of metaphysics is confirmed perhaps more than any scientific hypothesis! Since additionally I endorse the thesis that the likelihood of a divergence in any given minute given the profligate UW hypothesis is basically 1 or an infinitesimal away from that, we can be way more certain of the RW hypothesis (which would seem to solve induction) more than we are of any scientific hypothesis. That is a lot to say for a metaphysical theory! So, do you 1) agree with all of this 2) disagree that the two options are RW or profligate UW or 3) disagree that there being exactly these 2 options entails that your objection doesn’t succeed.

  3. JF April 8, 2014 at 1:26 pm

    Perhaps I’m not following your argument correctly, but let me try explaining my point this way.

    Let’s say that RW and UW hypotheses are described by the following series of coin flips:

    … etc

    Assume that each ‘H’ above represents a trillion flips of a coin, all coming up ‘heads’, and that so far we have observed 10 (trillion) coin flips, all showing ‘heads’. Based on Bayesian reasoning, the evidence observed so far equally supports all of the above hypotheses (the likelihood for each is the same). Thus, we have no basis (apart from habit or custom) for concluding that the next coin flip will come up ‘heads’, as ‘tails’ (T) coming up next is predicted by UW5, which has received equal support from the evidence observed thus far.

    It seems to me that the only way to break the tie between RW and UW5 (for instance) is to assign different priors to the above hypotheses, but this assignment appears arbitrary. However, perhaps this can be done in a principled way based on ideas from Kolmogorov complexity and algorithmic probability, as described in the following paper by Gilboa:


  4. tarrobread April 8, 2014 at 5:14 pm

    I’ll have to look through that paper – looks very interesting. I understand your point. I’ll try to relate your point to my prior comment. In my prior comment, I said we were considering two hypothesis about modal space, RW and profligate UW (pUW from now on). I predict the next trillion flips will be heads, and we called that prediction H. We all agree Cr(H given RW) = 1. Now, I believe Cr(H given pUW) is incredibly low, which will suffice for very strong confirmation of RW. I take your point to be denying this (If you accept this, my argument goes through). Your point says something along the lines of: what is the justification for thinking that if pUW is true, our world should be expected to diverge SOON? After all, under pUW, we could be in one of those worlds that diverge extremely late. In fact, given pUW, there are many worlds consistent with our past history. There is one RW world consistent with our history, and many many UW worlds. However, if “most” of the UW worlds that are consistent with the past veer off very late, then actually the Cr(H given pUW) may be rather close to 1. So, H is only weak evidence in favor of RW.

    Here is my response. To be more general, suppose we are doing an experiment with not just two outcomes like the coin flip but with n outcomes, numbered 1, 2, … , n. The RW world corresponds to 11111…. Any deviation from all 1’s is a UW. Now, let us consider the hypothesis of pUW, where every UW world is actually a possible world. I predict the next trillion experiments will be 1’s. I would claim that given pUW, the likelihood of any particular trillion results of those experiments (e.g. 39203857298137489231948749182…) is equiprobable. And why shouldn’t it be? GIVEN pUW we have absolutely no reason to favor any trillion number sequence over the other. This could be construed as a sort of principle of indifference, which I know is controversial, but in this finite case I can literally just count the number of worlds. Suppose I knew that, for example, every possible world ended after 2 trillion of these experiments is performed. Since every possible sequence of 2 trillion numbers is a possible world, and we pick uniformly at random among them, any initial trillion segment is equiprobable. From which it follows that Cr(H given pUW) is 1/n^trillion, which is very low as desired.

    • JF April 11, 2014 at 5:25 pm

      I certainly agree with you that p(HHHH…|RW) >>> p (HHHH…|UW). Thus, if the hypothesis space is partitioned in the way you describe (RW vs UW, in the generic sense), you would have indeed ‘solved’ the problem of induction- and I would congratulate you for putting to rest a philosophical issue which has plagued philosophers for centuries!

      However, if one chooses to partition the hypothesis space in a different way (e.g., with each hypothesis referring to a specific ‘history’ of the universe, as I have described above), one may arrive at inductive skepticism, such that, for instance, p(HHHH…|UW1) = p(HHHH…|RW). Unfortunately, I’m not sure if there is any principled way to decide which partition is the ‘correct’ one.

      • tarrobread April 11, 2014 at 5:35 pm

        The hypotheses are about what is possible, not about the history of the actual world. It seems clear to me that the only two options about what is possible is either NO UW is possible or ALL UW are possible (so either green can never become grue in any possible world or if green can become grue at some particular time in some possible world, then for every time there is a world where green becomes grue). So I think there’s a very strong case that really this is the only possible partition. Thanks for the comments – I’ve enjoyed this exchange!

  5. JF April 11, 2014 at 6:07 pm

    Perhaps I’m still misunderstanding something…

    If all worlds are possible, then HTTT… is one possible world, HHTT.. is another, HHHT… is another, and so on. HHHH…for all time (what you refer to as ‘RW’) is one possible world out of the set of possible worlds (modeled as trains of coin flips- or rolls of trillion-sided dice, if you wish). HHHH…with a T occurring at 12 am on Jan 1, 2050 is another possible world. Before Jan 1, 2050, both hypotheses are equally supported by the data. Of course, our current observation of constant H’s refutes those hypotheses which refer to possible worlds in which T’s have already occurred. But those in which T’s are to occur in the next or subsequent moments, whatever the current moment may be, are equally supported by the data we have observed up to this point.

    Do you disagree with this statement?

    (and thanks for the thoughtful replies- I agree this has been an interesting exchange!)

  6. tarrobread April 11, 2014 at 9:26 pm

    I agree with your statement but it is tangential to my argument. You have misunderstood me I think. There are two hypotheses on the table. The first is that all “arbitrary” predicates (like grue and bleen) are not joint-carving in ANY possible world. So, there is no possible world where grue is the fundamental predicate instead of green. In the coin example, there is NO possible world where ANY “T” occurs. If there is no possible world at all where there are any T’s, then of course the probability of all Hs in this world is 1. The other hypotheses is that ANY world with ANY combinations of H’s and T’s is a genuinely possible world. I take it those are the only two options. In the second option, we should have expected our observations to diverge from all H’s long ago. The hypotheses disagree on which worlds are in fact possible.

  7. JF April 15, 2014 at 10:07 am

    Thanks for the reply. Let me first confirm that I understand your argument. Your argument is basically that:

    p(a long stretch of H’s|UW) is so low as to be effectively impossible (it is arbitrarily close to zero) in any possible world that is an UW world. Thus, there will be no ‘gruesome’ UW worlds. Moreover, our world is almost certainly an RW world, because if it were an UW world, we would have observed a divergence in our ’emeralds’ by now.

    Have I got it right? If so, then I feel that your exposition might be simplified along the lines just described (for the benefit of us non-professional philosophers) by avoiding the use of abstruse philosophical terms/concepts like ‘modal deflationism’ and ‘strong necessities’, which may be difficult for many readers to understand and which seem unnecessary for your argument.

    Your argument is intuitively appealing and persuasive. This makes one suspicious that your solution to the problem of induction, which has somehow evaded the minds of the greatest thinkers for hundreds of years, is ‘too easy’.

    Accordingly, and unfortunately, I think the argument has a crucial flaw: RW and UW hypotheses do not exhaust the possibilities. There is a third possibility: namely, worlds which change from RW to UW. In these worlds the probability distribution of Hs and Ts itself changes. To use your prisoner analogy, it corresponds to the case where the hat out of which ‘life’ and ‘death’ cards are drawn is switched by the ‘judge’ at some later time during the prison sentence for a different hat which contains a different proportion of ‘life’ and ‘death’ cards. I submit that if the possibility of this ‘hat switch’ cannot be excluded (on either logical or probabilistic grounds), the inductive skeptic remains vindicated.

    • tarrobread April 15, 2014 at 10:30 am

      Yes – P(we observe a long stretch of H’s given that all UW’s are possible worlds) is effectively impossible. Our world is almost certainly an RW world since all possible worlds are RW worlds. There just are no gruesome worlds.

      And maybe your right about the terminology. I was just sticking with the literature since a strong necessity is just a necessary constraint on all possible worlds over and above merely a priori/logical considerations. It is not a priori that grue-worlds are impossible, but they are impossible via this strong necessity. The existence of strong necessities has lots of relevant impact outside this argument – like conceivability/possibility arguments.

      And I’m not entirely sure I understand your objection. I always think of UW worlds as just worlds where natural properties “veer off”. So electrons suddenly become positively charged or green suddenly turns to grue, etc. A world where the fundamental natural properties never veer off is an RW world. So, your third possibility: “worlds which change from RW to UW” would just seem to me to be worlds where the natural properties dont veer off for some time, but then do. By my definitions, those would just be UW worlds, and hence proved to be impossible by the argument. I would be interested to hear whether you think this reply works or if I misunderstood you or something.

      Lastly, I do want to point out that this argument alone doesn’t suffice to “solve” induction. But, I do think any answer to induction must buy the strong necessity I argued for – a restriction on possible worlds over and above a priori/logical restrictions. Since if it doesn’t buy this strong necessity, there is no hope for induction to be justified. However, this would just be a necessary component of any solution to the philosophical problem, rather than a sufficient solution. My argument already assumes we have a justified way to pick out which properties are natural or not gerry-mandered, and in virtue of that, we can define what an RW and an UW is in the first place. Without this notion of naturalness my argument can’t get off the ground. Of course, we all think it is intuitively obvious which properties are in fact gerry-mandered (grue and bleen as opposed to green and blue) but it’s notoriously hard to flesh out this “obvious” intuition. My argument in conjunction with an epistemology of naturalness would suffice to solve induction, I think.

  8. JF April 15, 2014 at 11:36 am

    I think your argument would indeed solve the problem of induction, provided that the third possibility I mentioned could be excluded. Alas, I don’t think this third possibility is just a variant of UW for the following reasons. RW assumes one type of unchanging probability distribution (e.g., all ‘life’ cards), UW assumes a different, though *unchanging*, distribution (e.g., ‘life’ and ‘death’ cards are equally likely), whereas possibility 3 assumes that the probability distribution changes over time (or space). Thus, the probability distribution itself is ‘gruesome’.

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