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This is presumably an extremely basic question, but I didn't have any luck on my preliminary attempts to Google an answer or track one down here.

After hearing a number of debates and presentations in which something called "modal arguments" were referenced, I decided to look into modal logic by following this tutorial series. The author gives the following definitions:

  1. A proposition p is necessary if it is not possible that not p (□p/~◊~p)
  2. In what he stated was a loose definition, the author said that "for every way that things can be different, there is a corresponding possible world."
  3. Combining necessity with the possible worlds notion, we interpret □p to be "true" in some world w if and only if p is true in all worlds that are accessible from w.

This definition of necessity with respect to possible worlds makes sense in most cases. However, I'm still confused on an edge case (which may be impossible, but I'm not sure):

  1. What happens when p is false in "w" but true in all of the possible worlds it accesses?
  2. Also, along the same vein, is a world accessible from itself?

I figure that if a world is accessible from itself, then the problem is resolved, which is why I ask (2). If a world is not accessible from itself, though, I figure that this problem is impossible from the perspective of nomological possibility, but can't figure out a way to resolve it if we consider logical possibility.

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    Your intuited and familiar □p→p is only the "reflexivity axiom" of system T constructed upon K. An application could be some version of ontological argument (such as Pruss's WPSR) where a necessary first cause may not exist in this world. Similarly, it's a famous and surprising conclusion in K that □p⊬□□p, so modality is not idempotent unary operator like projection which we're all too familiar with. In other words, K is very general and can model some cases where one knows something, however, one doesn't know that one knows that same thing at all which seems not that an uncommon case... Jun 11 at 2:59

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Also, along the same vein, is a world accessible from itself?

No, not automatically. You will later get to different so-called frames aka modal systems where certain properties such as reflexivity of the accessibiliity relation do always hold, but for the so-called system K that you are currently working with, if wRw is not explicitly stated, then the world can not access itself.

This means that ...

What happens when p is false in "w" but true in all of the possible worlds it accesses?

... in this case, p is necessary in w and also false in w. This may sound counterintuitive, but is unproblematic as far as the laws of modal logic are concerned.

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  • Gotcha, thanks! Since it violates unspoken etiquette on some corners of StackExchange to accept an answer immediately, I'll hold off for a bit before accepting, but this gives me the confidence to continue on with the tutorial series. Thanks! Jun 10 at 23:17
  • Accepted after five hours; thanks again, lemontree! Jun 11 at 4:22

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