Possible worlds

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This is a chat I had on the internet with a couple of guys about the idea of "possible worlds". In the transcript below I use the name Mark.

Michael:Does the positing of possible worlds entail possible times?
Mark:I was thinking about this idea of "possible worlds" yesterday. And thought like this: Possible in the sense of "I am not able to see any fault in this idea", or "this world can actually work, it can actually exist, i know that"?
Then I thought: Maybe there is only one possible world (in the second sense), namely the one we live in.
Daniel:I like how possibility can have two (or maybe more layers).. there's the definitely possible, and then the possibly possible.
Mark:Yes, that is what i meant.
And the "possibly possible" is more like "with my limited knowledge, then i can not see any fault".
But possible in that sense is a rather arbitrary, so when we speak about "possible worlds" in a philosophical discussion it seems reasonable to assume we use the word "possible" in the other sense, the sense you describe as "definitely possible".
But then how do we know that there is any world which is "definitely possible" except the one we live in?
Michael:Logically possible worlds.
Mark:OK, but then we need to assume axioms, on which to apply our logic.
Michael:?
Mark:Axioms are not exactly logical, nor are they illogical.
Daniel:If you knew the generating principle behind worlds (and knew there are other worlds than what we live in) you could say what's definitely possible.
Mark:Daniel, right.
Michael:?
Michael:World equates to universe.
Mark:"I experience" is an axiom to me, not because it is logical, but because it is self-evident to me.
Sometimes I am not even sure our world is logical. It depends on what we mean by logical of course, but usually we have a rather certain idea of what is logical, but honestly speaking I am not so sure that the world which exists is of that kind of logic. Sometimes it feels as if the world is ending up in paradoxes no matter how we turn it around in the light of our "logic".
Michael:Mark, logic is grounded in 'reality' so is intrinsically 'logical'.
Mark:Yes, right, but with that view on logic, I am not so sure we know what "logic" is.
Michael:Logic is inductive, from the inductive we derive the deductive.
Mark:Say for example the discussion between monism and dualism. Both of these views end up in difficulties which seem impossible to resolve. Some philosophies therefore claim that the world is both, at the same time, however paradoxical or illogical that may sound.
It's not that I do not know what normal logic is, I am quite good at it. It's just that I see possible limits in our understanding of things here.