I have a question about the notion of possibility in modal logic. There are systems and worlds with this notion. They say that a world w1 is accessible to an other world w2 if and only if for any true proposition of w p be possible in w2. I also read about possible words of s5 system that have identical necessary an contingent propositions. Also these worlds are accessible worlds from each other. So i am confused. For example assume a world w3 that exist in it a 3-handed human being. A being that do not exist in the actual world w4. Assume q as a 3 handed human being exists. So we know by definition of contingency and possibility that q is contingent in both w3 and w4. And it seems that q is not possible in w4 actual world because q is false in it and possibility means to be consistent with the truths. It is not possible in w4 that q be true. We said q is contingent just because q is true in some equivalent accessible world w3. So knowing this come to return to the definition of accessibility or so-called relative possibility. We said a world is accessible to an other world if and only if any true p of the first is possible in latter. Who w3 and w4 will be accessible as they obviously claim in modal logic of s5 system and its possible worlds?

Does they mean contingency from possibility? If it is so does the quantifiers of the modal logic is necessity and contingency instead of the first and possibility??

https://en.wikipedia.org/wiki/Accessibility_relation to read about accessiblity realation

  • Hello again! You wrote: "And it seems that q is not possible in w4 actual world because q is false in it and possibility means to be consistent with the truths." But this isn't how (metaphysical) possibility is defined. For example, it's false that I ate breakfast this morning, but it's possible (i.e. there is a possible world) that I ate breakfast this morning. Also, could you clarify your last paragraph? I think there's a good question here but I'm not 100 percent sure I understand. – Adam Sharpe Mar 11 '20 at 1:10
  • Hello agian adam. – MHghasemi Mar 11 '20 at 1:18
  • I think i just mean what you said. I think that the (metaphysical) possibility just is synonymy with contingency? It is true? they say a proposition p is contingent in a world if there are some accessible worlds that it is true in it. So we have a unique notion with two different names: 1- contingency 2- (metaphysical) possibility. And when they say p is possible in modal logic they just mean it is contingent, For example One that claim 2 omnipotent being is possible. But if be there some contradiction to exists 2 such being together in fact it is not (metaphysical) possible. – MHghasemi Mar 11 '20 at 1:26
  • in the other words (metaphysical) possibility consists more and less than possibility at the same time:For example, it's false that I ate breakfast this morning, but it's (metaphysical) possible (i.e. there is a possible world) that I ate breakfast this morning. but it is not really possible because it is false. on the other hand I think may be there 2 omnipotent beings together but it is (metaphysically impossible because entails contradiction. Am I true? – MHghasemi Mar 11 '20 at 1:27
  • Is "possibility just is synonymy with contingency?" Not quite. Every contingent proposition is possibly true, but not every possibly true proposition is contingent (because some possibly true propositions are necessary). For example, 2+2=4 is possible, but not contingent. Here's a good explanation from Maverick Philosopher: maverickphilosopher.typepad.com/maverick_philosopher/2009/02/… – Adam Sharpe Mar 11 '20 at 1:34

This is only a partial answer - it's mostly meant to push back on the framing (hehe) of the question as currently posed. That said, I do think it meaningfully contributes to the topic.

Kripke frames and possible world semantics are one way to interpret modal logic. However, they're not the only way - and even if we commit ourselves to using Kripke frames, we still don't need to think in terms of possible worlds. There is no single "correct" meaning of accessibility, or of □. For example, in temporal logic when we use Kripke frames the accessibility relation corresponds to movement in time (maybe "is in the future of" or "is the next moment in time after").

Here's a list of the questions/ideas floating around here which each have many different answers/interpretations and are largely independent from each other:

  • How do we think about our modalities? What sort of non-truth-functional language are we trying to capture (is necessary, is permitted, might happen in the future, is consistent, ...)?

  • What axioms do we impose to reflect that motivation? This might not always be obvious - for example, in temporal logic do we want to imagine a single timely or multiple possible futures?

  • What mathematical structures do we want to use to model the formal system from the previous bulletpoint? Kripke frames are probably the best known, but they're not the only game in town - and frameworks other than Kripke frames don't necessarily even have an "accessibility relation" in any sense!

None of this is addressed at the outset by the initial idea of modal logic: that there are non-truth-functional concepts which are worth capturing - to some degree anyways - in a formal system.

  • +1 for the pun. – lemontree Aug 9 '20 at 13:21

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