Accessible Worlds

[aleph_0]

I'm curious if anybody has read literature on modal logic, in particular, the claim that S4 is an invalid modal axiom. Kripke, the very man who invented the scheme of seeing possibility as worlds which are not actual, denied that all worlds are "accessible" to all other worlds. What this meant, or so he and those like him claimed, was that you may have some world W which we in the actual world will say is possible. But there is some world W'' which we say is impossible, but which people in world W say is possible. That is to say, just because something is "possibly possible" (i.e., relative to some world, a thing is possible) does not imply that the thing is possible.

Does anybody know arguments for and against this claim? It seems to me to no longer treat possibility as worlds at all, but as psychological or linguistic in nature. For if to be "possible" simply means for there to be some world in which the possible thing is actual, then the fact that a given world is inaccessible does not make it any less possible. So long as these worlds exist, they are possible. We just may not know that they exist. I do not take possible worlds to actually exist out there like Platonic ideals, but it is convenient short-hand and a good model for thinking of truths of possibility as truths concerning reality.

That is, unless possibility is not talk about actual, real-world things but rather talk of mental states (being able to think of some state of affairs, etc.). Yet this is not how the natural language seems to operate. When speaking in a natural language about some thing or other being possible, we do not intend thereby to make claims about our personal mental states. Does denying S4 then require that any natural language be contradictory, and how can any language be contradictory? They seem to be built on a practice of performing a given act in the presence of some objective state, and the act either corresponds or does not--how can any language be inherently contradictory?

[Edit for style]

Edited October 30, 2006 by aleph_0

[DMR]

Does anybody know arguments for and against this claim? It seems to me to no longer treat possibility as worlds at all, but as psychological or linguistic in nature.

Since I don't think that possibilites should be treated as a Lewisian bloated universe of worlds, I don't regard this as problematic. A better accound of possibilities was given by Michael Jubien, although he is a Platonist running-dog. For Jubien an object (and a state of affairs is just a really big object, namely the whole universe) is possible just in case it doesn't have contradictory properties. For example, a green ball is a possible object because there is no contradiction between the properties of greenness and ballness, but a round square is not a possible object because there is a contradiction between the properties of roundness and squareness.

You could of course cast this in a Lewisian/Kripkean framework and say that there is a (accessible) possible world containing a green ball and no possible world containing a round square, but the reason for this is the same as the one given by Jubien, so that the talk about possible worlds is so much wasted breath.

just because something is "possibly possible" (i.e., relative to some world, a thing is possible) does not imply that the thing is possible.

Of course, this is not the whole story of S4. Something is possibly possible just in case it is possible relative to some world which is itself possible. Denying S4 really amounts to denying that the accessibility relation is transitive. Kripke believed it was, although I don't know of any reason he gave other than that something which is possibly possible is obviously possible. But again, this is a problem that only has to come up if you take the Lewis/Kripke view seriously. The Jubien view seems to lead to the conclusion that "Possibly possible" just doesn't mean anything, or if it means anything, it means "possible." And after all, what possible use could the consideration of the "possibly possible" have to anyone other than a professional philosopher?

[y_feldblum]

I was under the impression that only this world exists. Equally, I was under the impression that possibility meant something in this world.

[Nate T.]

DMR,

an object ... is possible just in case it doesn't have contradictory properties.

Would you consider a cow that is able to jump over a 1000 foot skyscraper to be possible or impossible?

[DavidOdden]

Would you consider a cow that is able to jump over a 1000 foot skyscraper to be possible or impossible?

Impossible: although, a veterinarian might have data to show that I'm mistaken, but I think it's impossible.

[DMR]

Would you consider a cow that is able to jump over a 1000 foot skyscraper to be possible or impossible?

I assume you are getting out the fact that this isn't a straightforward A and not-A style contradiction. The answer is that, given the sort of mass and body a cow has (and as a background assumption, that the gravitational constant is unchanged), this is just as impossible as a round square, although it takes a lot more math to show it.

[aleph_0]

Since I don't think that possibilites should be treated as a Lewisian bloated universe of worlds, I don't regard this as problematic. A better accound of possibilities was given by Michael Jubien, although he is a Platonist running-dog. For Jubien an object (and a state of affairs is just a really big object, namely the whole universe) is possible just in case it doesn't have contradictory properties. For example, a green ball is a possible object because there is no contradiction between the properties of greenness and ballness, but a round square is not a possible object because there is a contradiction between the properties of roundness and squareness. You could of course cast this in a Lewisian/Kripkean framework and say that there is a (accessible) possible world containing a green ball and no possible world containing a round square, but the reason for this is the same as the one given by Jubien, so that the talk about possible worlds is so much wasted breath.

Well, I don't mind the wasted breath since I find it helps clarify matters.

(As a note, this view is not originally from Jubien but--as far as I can tell--first conceived by Wittgenstein when he says that the only possibility is logical possibility.)

Of course, this is not the whole story of S4. Something is possibly possible just in case it is possible relative to some world which is itself possible. Denying S4 really amounts to denying that the accessibility relation is transitive. Kripke believed it was, although I don't know of any reason he gave other than that something which is possibly possible is obviously possible.

Do you have a quote in which he affirms the transitivity of accessibility? I was under the impression that he denied it.

But again, this is a problem that only has to come up if you take the Lewis/Kripke view seriously. The Jubien view seems to lead to the conclusion that "Possibly possible" just doesn't mean anything, or if it means anything, it means "possible." And after all, what possible use could the consideration of the "possibly possible" have to anyone other than a professional philosopher?

I would argue, by the fact that one is perfectly interchangeable with the other, the problem arises in both. Now, by the Jubien account, something following '<>' just means that it contains no contradictions, so <><>p just means <>p contains no contradictions, which means p contains no contradictions. But, in making possibility psychological, something following '<>' would mean that it contains no apparent contradictions--i.e. that it's conceivable. And so denying S4 would amount to saying that just because something is conceivably conceivable doesn't mean that it's conceivable. I find this arguable at best. So I guess my question is, is this view in any way tenable?

Good to make your acquaintance, DMR.

[DMR]

(As a note, this view is not originally from Jubien but--as far as I can tell--first conceived by Wittgenstein when he says that the only possibility is logical possibility.)

Jubien was just the guy I heard advancing it.

Do you have a quote in which he affirms the transitivity of accessibility? I was under the impression that he denied it.

I've never read anything Kripke himself wrote on modal logic. I'm getting this from a secondhand account, a book on the model theory of modal logic. The author claims that Kripke held that S5 was obviously the correct modal propositional logic, since the accesibility relation was obviously an equivalence.

in making possibility psychological, something following '<>' would mean that it contains no apparent contradictions--i.e. that it's conceivable. And so denying S4 would amount to saying that just because something is conceivably conceivable doesn't mean that it's conceivable. I find this arguable at best. So I guess my question is, is this view in any way tenable?

Possibility, according to every philosopher I know of who has written on the subject, is a metaphysical condition. If you wanted to work out a modal logic of conceivability, you could, but it wouldn't be the modal logic of necessity that S4 is supposed to be, and so the same rules need not apply.

However, I nonetheless think that they would. What does it mean to be conceivably conceivable? It means that there are no apparent contradictions in supposing that there are no apparent contradictions in supposing some proposition. For S4 to be wrong, there would have to be some proposition which had apparent contradictions, but for which the claim that it had no apparent contradictions had no apparent contradictions. Since the contradiction in the proposition is apparent, this seems to be a contradiction.

Whew!

Good to make your acquaintance, DMR.

Likewise, I'm sure.

[aleph_0]

To take up the side denying S4, I can conceive of myself conceiving a round square. I imagine myself thinking to myself, and someone asks me, "What are you thinking about?" and I answer, "A round square. It has been on my mind much of late." And if I can conceive of something, it must be possible, for what else could be contained in the notion of "possibility"? It may be the case that something is possible but I cannot conceive of it, but the inverse simply cannot be.

To take up the side asserting S4, we cannot directly conceive of a round square, that must be known. We should all agree that a round square is impossible, and if impossible then inconceivable. Hence, since it is impossible to conceive of a round square, it must be inconceivable to conceive of a round square. To conceive of conceiving of a round square would require, not that you conceive of a person making gestures as if he were conceiving of a round square, but that you actually conceive him and his conceptions.

I take the later to have won. So I'm just wondering if anybody can offer up other (perhaps even pseudo-) objections to S4. For I'm having difficulty understand such a proposition, though otherwise non-stupid people like Nathan Salmon have argued this very case.

[Edit: By "this very case" I don't mean the psychological argument, necessarily. I just know that he has denied S4 for some reason.]

Edited December 4, 2006 by aleph_0

[y_feldblum]

It is certainly the case that there are possibilities that any given person at any given time with a definite context of knowledge is utterly incapable of conceiving, and it is certainly the case that there are impossibilities of which one can easily conceive.

Possibility, properly understood, is a component of the law of causality. The recognization of a possibility is the identification of a fact of reality.

It is just beyond me why anybody seriously discusses nonexistent alternate planes of existence in defining what possibility means.

[Nate T.]

I assume you are getting out the fact that this isn't a straightforward A and not-A style contradiction. The answer is that, given the sort of mass and body a cow has (and as a background assumption, that the gravitational constant is unchanged), this is just as impossible as a round square, although it takes a lot more math to show it.

Okay, that's right. It seemed like you were arguing that such a thing would be possible, because one can "conceive" of it. Just wanted to make sure that the analytic/synthetic dichotomy wasn't rearing its head here.

[Atlas51184]

Why is this question of interest or of use? I've been told that predicate logic is useful for computer programming and in linguistics. However, I fail to see of what significance the formalization and symbolization of language and logic is, at least of what help or significance it is to philosophy. While using counterfactuals can be helpful in understanding a principle, possible worlds methodology is not a path to discovering new truths.

[aleph_0]

It is certainly the case that there are possibilities that any given person at any given time with a definite context of knowledge is utterly incapable of conceiving, and it is certainly the case that there are impossibilities of which one can easily conceive.

The former I assert; the later I deny.

Why is this question of interest or of use? I've been told that predicate logic is useful for computer programming and in linguistics. However, I fail to see of what significance the formalization and symbolization of language and logic is, at least of what help or significance it is to philosophy. While using counterfactuals can be helpful in understanding a principle, possible worlds methodology is not a path to discovering new truths.

Specifically, I have in mind its implication in Chisholm's Paradox. However, I have since been able to side-step this particular issue, though it is still of some significance to pin down the precise meaning of "possibility" as it relates to many conversations, including essence and logic. The use of symbolization is, at times, complicated. But a generic and true answer is that it helps to reduce highly technical and convoluted conversation, with many fine-grained distinctions, to an easily perceived format. It is also an interesting subject of study in metalogic.

[y_feldblum]

I am surprised at your response. I would have thought it the opposite, or at least a double denial.

In any event, the first statement implies that there are also possibilities that nobody at all, at any given time, is capable of conceiving. Further, there are possibilities that nobody at all, at any definite point in time in the future, will be capable of conceiving. Would you concede, then, that there are possibilities which are unrelated to the act of cognition - in other words, possibilities which are facts of reality?

The second statement is easily seen. For example, as per the second law of thermodynamics, processes which can be characterized as entailing a reduction in total entropy (dS/dt < 0) are absolutely impossible. Some such processes are the backwards-in-time versions of similar processes entailing an increase in total entropy. In other words, while the second law of thermodynamics does not disallow a drop of water from falling and splashing onto a countertop, it absolutely disallows the reverse process of the splattered drop reforming on its own. And yet, conceiving of it is very easy: one can simply videotape the drop of water falling and splashing, and then play the tape backwards. Unless one considers as "possible" that which one knows can never happen....

[aleph_0]

Actually, I have no stead-fast beliefs about the nature of possibility. To the best of my ability to understand it right now, I will revise what I said and put it thus: By our ability to understand the objective world, the only thing that should count as a possibility is what is logically possible--but in a different sense than Wittgenstein meant it. By logical possibility, we take to mean that which does not violate any known identity in principle. So even though it probably will never happen that a man will be stuck on an inescapable island with a man who controls the island's food supply, it is merely the particular arraingment of circumstances that makes this unlikely, and nothing about the principle or identity of metaphysical existence makes it impossible. However, by our knowledge of what a cow is and what it means to jump, it is logically impossible for it to jump 1,000 feet. We might define a different kind of possibility the psychological possiblity--but this would, given what has been said, amount to literally nothing more than the conceivable and so why proliferate terms for the same concept?

This, however, has the somewhat (though not all together) dubious consequence that what we say is possible hinges on the state of science at a given time.

[y_feldblum]

The consequence is not dubious at all. The identification of a possibility, like all identifications of facts of reality, depends on the advancement of our abilities at identification.

What actually is possible never changes, because possibilities are facts of reality.

[aleph_0]

I'll buy it. Tentatively.

[aleph_0]

I thought about a potential problem with this account of possibility. For if we restrict possibility to our current convictions about physical laws, then we cannot propose counter-legals. Thus, we cannot say, strictly speaking, that any physics besides the current theory is possible. So while we do not have a unified theory of physics at the moment, we could (possibly) live in a society where the accepted theory of physics is so unified and can make predictions according to its own theory (such as that Newtonian physics, according to its own theory, requires that cold water comes to a boil quicker under the same heat than warm water). In buying such a theory, you could not even allow that another theory of physics is possible, since quite obviously, it contradicts the current beliefs in physics. So people who had accepted Newtonian physics--and indeed, they had every reason to--should not have considered the possibility of non-Newtonian physics?

I see a couple ways around this difficulty, bit it may be helpful to talk it out.

[y_feldblum]

We certainly can: provided doing so is reality-based. In other words, if you have some evidence even hinting at an unknown fact or an unconsidered context, then you have reason to look for a more general account or a new viewpoint. But evidence, to at least a minimal degree, is required to identify new possibilities.

[aleph_0]

Let me make explicit what it at least seems that you're saying: Something is not possible (and so by definition impossible) iff there is no evidence for it yet. If that is your claim, then quantum physics was impossible during Aristotle's time.

If that is not your claim, would you acquiess to the claim that Newtonian physics is possible? How about string theory, which is only, in a sense, "admitted" by mathematics, though not proven. I.e. it is shown consistent with the facts but not uniquely consistent.

[y_feldblum]

Not at all.

Something is not known to be possible if there is no evidence hinting at it whatsoever. Fantasies should be regarded as: fantasies. Not as possibilities.

The level of identification of what is possible does not determine what is in fact possible (an Aristotelian potentiality). I am here explicitly separating between knowledge of facts and the facts themselves.

The identification of quantum physics as a possibility during Aristotle's time was impossible. Quantum physics was, of course, always a possibility: it was a potentiality during Aristotle's time and now it is an actuality. People during Aristotle's time could not identify it; people during Bohr's time could identify it as a possibility; and people now can identify it as fact (at any rate, the mathematical formalism, if not the crazyland theories about nonidentities or multiple worlds).

String theory cannot currently be identified as a possibility: there is no evidence for it as a physical explanation of all existents.

Note that there are two alternatives to possible. The first is impossible, which means that there is evidence against. The second is arbitrary, which means that there is no evidence one way or the other. For both alternatives, however, the proper response is not to consider the matter further, and instead focus on what is known to be possible.

[aleph_0]

Well, Aristotle considered something impossible if it will never occur--I would think [hope] a different notion of possibility than what we are dealing with. In any case, I'm glad I decided to talk this out before presenting my two "outs". I'll think on this some more, but it so far seems consistent, practical, and conforms to the natural language.

[y_feldblum]

I wouldn't think so.

What things could Aristotle tell you are impossible? Those things about which he has evidence will never occur.

[aleph_0]

Assume that, in all of the world's existence, a 6' 11''-tall man never jumps into the Red Sea. Aristotle would consider it, then, impossible. I should think you and I would not.

[y_feldblum]

I am not familiar enough with Aristotle to discuss his philosophy.

I would say that the potential for such such a person to act in such a way exists. There is of course clear evidence of such a potential, though no evidence of its actuality (meaning, one knows it to be possible). The evidence is that 1) there are men that height, 2) there is no essential difference between men that height and other men when it comes to jumping into the Red Sea, 3) many other men do jump into the Red Sea.