Am I right to assume that in no modal logic, whether in K or in a logic where the accessibility relation is specified as either reflexive, symmetrical or transitive, does ”p implies necessarily p” hold? In what (if any) ways does the accessibility relation between worlds have to be qualified in order for this inference to be provable?

  • Temporal modal logic, like Aristotle's ship-battle discussion uses, presumes p implies necessarily p as an axiom -- which means it is provable. Only the future (even if it is the immediate future, or the immediate past to be known only in the future) has possibilities and requires arguments about necessity. Later modal logics often take a more abstract version of necessary. But this one is not dead. – jobermark Jan 7 at 16:37
  • Very interesting! You mean that if we are speaking about the past, the inference could be provable? – simulacra Jan 11 at 15:49
  • The answer below makes sense, and renders this comment irrelevant, for this case. You aren't filtering p for temporal references. If you know the past and can totally predict the future, because it is all necessarily determined to be the same as the original world, you no longer have an interesting modality of necessity left to talk about. (So Scholastics who wanted primarily to talk about 'eternal' things then needed a different way of looking at necessity -- they pulled 'necessary' back to what could not have been otherwise rather than Aristotle's can no longer be otherwise) – jobermark Jan 11 at 18:17
  • For materialists, if we are talking about the past, this is not just provable, it is an axiom, right? We cannot change the facts from which we must start. Things are as they are and have the histories they already have. So that is then uninteresting again. – jobermark Jan 11 at 18:25

You are right.

If the equation p→☐p is provable, then the only world that is allowed to be accessible from world w is w itself. One could describe that as all worlds being isolated. More formally, this axiom forces the accessibility relation R to satisfy:

If wRv, then v=w.

The whole idea of having other worlds breaks down...

If R is also reflexive (i.e. if ☐p→p is an axiom), you can derive p↔☐p, and the logic behaves exactly as the usual propositional logic with an uninteresting additional symbol.

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