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Probabilty vs. Possibility

eijah said:
As I proposed earlier, and as someone else here said about either false or true, it seems to me rightly or wrongly that something is either not possible 0 or possible 1. So perhaps while probability is about statistical likeliness, a more rigorous than usual view of possibility is all about contrasting absolute non-existence vs. absolute existence?
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I don't really understand what it is you're after. Mathematically, probability is described by probability theory and possibility/necessity by modal logic. Those are very different things.
[latex]\[ \Diamond p \Leftrightarrow \lnot \Box \lnot p \][/latex]: "p is possible if, and only if, it is not necessary that not-p."
Possibility and necessity (let's call them M and L in text) act like quantifiers rather than values, and formally behave very analogously to existential and universal quantifiers:
[latex]\[ (\exists x)(Px) \Leftrightarrow \lnot(\forall x)(\lnot Px) \][/latex]: "there exists an x such that Px if, and only if, it is not the case that for all x, Px is false."

Jorghnassen said:
A better definition of a "possibility" would be any element (or even subset) of the probability space.
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There is no reason why a probability space can't include impossible events. If I roll a standard die, there is no possibility of getting a 10. That can be put in the probability space without any trouble. And technically, the empty set is an event that is in many situations not possible.
 
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Vorpal said:
I don't really understand what it is you're after. Mathematically, probability is described by probability theory and possibility/necessity by modal logic. Those are very different things.
[latex]\[ \Diamond p \Leftrightarrow \lnot \Box \lnot p \][/latex]: "p is possible if, and only if, it is not necessary that not-p."
Possibility and necessity (let's call them M and L in text) act like quantifiers rather than values, and formally behave very analogously to existential and universal quantifiers:
[latex]\[ (\exists x)(Px) \Leftrightarrow \lnot(\forall x)(\lnot Px) \][/latex]: "there exists an x such that Px if, and only if, it is not the case that for all x, Px is false."


There is no reason why a probability space can't include impossible events. If I roll a standard die, there is no possibility of getting a 10. That can be put in the probability space without any trouble. And technically, the empty set is an event that is in many situations not possible.
Click to expand...

Very well explained (on both points). Thank you.
 
If you think anything's possible, try winning a game of Chess with your first move. :)
 
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