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Recently, I have begun studying modal logic, using Brian Chellas's Modal Logic: An Introduction. Something keeping me from fully understanding the material is the idea of a possible world. They seem to be described as maximally-consistent sets of propositions, but what determines whether a possible world with a different set is accessible from the actual world, or any world, for that matter? If one was attempting to evaluate a modal argument, what would decide what was possible?

I apologize for the clumsiness of the question, and I would like to state up-front that I'm no philosophy buff.

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    probably related and a really helpful answer from Dennis aswell: philosophy.stackexchange.com/questions/6259/… – Lukas May 23 '13 at 19:02
  • @Lukas Haha, I forgot I answered that question. – Dennis May 23 '13 at 19:36
  • @Lukas : That was quite helpfull, actually. Thank you. – Zzig May 24 '13 at 5:03
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Accessibility is determined by the axioms of the modal logic you're working within. The axioms of S5, for instance, require the relation to be an equivalence relation and so have the result that every world is accessible from every other world.

In a system like K, however, there are no constraints placed on the relation. The only time you can tell if a world w1 is accessible from the initial world w0 is if you have a diamond statement true in w0. If that is the case, then to discharge the diamond you must "create" a new, previously unused world (w1) and this world would be accessible from w0.

If 'what would decide what was possible' is a question that you also want answered, I'd recommend a separate question. That question is really orthogonal to the main question about accessibility. To give a hint at one answer, David Lewis appeals to recombination principles (that you can sort of "cut and paste" parts of the world together and those amalgamations will be genuine possibilities) to explain a statement's possibility. So, for instance, you might argue that the existence of a unicorn is possible since there are horses and there are horned creatures and a unicorn is just a horned horse

  • +1 and accepted. I had been wondering if the axioms determined the relations, or if it was the other way around (it seems that both are possible). Fantastic answer, and thank you for answering so quickly as well. – Zzig May 24 '13 at 5:12
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It's true that, under one understanding of "accessibility" and "determining", your accessibility relation is determined by your axioms. While that perspective is a valuable one, there is a more direct one in which the accessibility relation is just some subset of the cartesian product of worlds.

Neither of these perspectives are, I think, helpful to understanding the meaning of an accessibility relation. It is more helpful, I think, to talk about the sorts of applications to which they are relevant. One application is in the sort of modality that philosophers like David Lewis tend to employ, where we are able to quantify over all possible worlds. Here we are essentially in the simplest accessibility relation, "determined" by S5, wherein every world can access every other world.

On the other hand we might use a modal operator to express something about moral obligation, wherein ◻️P means one is morally obligated to make P true, and ♢P means one is morally permitted to make P true. Here, ◻️P does not imply P, and therefore the accessibility relation that models this modal operator will not allow every world to access itself.

So hopefully this helps clarify the answer: You choose the accessibility relation you want, to model the phenomena you want. It's like the choice between using natural numbers or real numbers--you can use either, depending on the sort of thing that you are trying to model. If you are counting things, you use natural numbers. If you are measuring or describing smooth processes, you use real numbers. If it is not entirely clear (maybe you have a tank of water mixed with a chemical, and it's not obvious whether you want to model every discrete particle of water using a natural-number system, or maybe you just want to treat these quantities like continuous variables in a differential equation) then there is a debate to be had.

  • +1. Taken with Dennis's answer, yours paints a much clearer picture of the subject. Thank you. – Zzig May 24 '13 at 5:15

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