Axiom S5

[Squawk]

Ahh, done a bit of reading. Modal logic seems to be pretty pointless since it rests on the definition of possible.

The best explanation I have seen thus far.

It is possible that 102394759087346987293875487692879870702347234234234234234234 is a prime number. I actually have no idea if it is indeed a prime number or not, so I must concede that it is possible that it is a prime number in some world somewhere.

Because it is possible that it is a prime number in some world somewhere I must conclude that it is necessarily true that it is a prime number somewhere. As such, I can conclude that indeed 102394759087346987293875487692879870702347234234234234234234 is a prime number.

Possible is the issue here.

 

[creativesoul]

Equivocating between possible and necessarily is a mistake. For if a claim is possibly true, then it also necessarily must be possibly false. That is what possibility implies. If everything which is possibly true were necessarily true, then there would be no distinction whiah can be made. That distinction between possible and necessarily is the existence of the possibility to be false. With possibly true comes possibly false, with necessarily true comes necessarily false.

 

[Zetetic]

According to Wikipedia and Standford: S5 states that if possibly p, then necessarily possibly p and that if possibly necessarily p, then necessarily p.

My analysis is this:

Necessarily possibly is quite distinct from Possibly necessary in the following way:

It is necessarily possible that an object exist if it is not contradictory by its definition.

If it is possible that Y necessarily follows from X, i.e. that X->Y could be a tautology, then S5 claims that ti is. I assume that a statement that is a contradiction by the other axioms of S5 could not possibly be necessarily true and so such statements are excluded.

Either way the two concepts are distinct, and so in conjunction do not give 'If possibly P then necessarily p'. That is, I believe, the root of the fallacy.

Also, it must be kept in mind that certain logics are useful for certain reasons, logicians may agree that that axiom in S5 is reasonable, but some still use adaptations of S4 and other variations. Stanford has a decent article on the topic: http://plato.stanford.edu/entries/logic-modal/

 

[GoodKat]

Whoops, turns out a minor misreading caused a massive misunderstanding. Possibly true does not imply possibly necessarily true, however possibly necessarily true does imply necessarily true:

That which is necessarily true cannot possibly be false
That which is possibly false cannot possibly be necessarily true
That which is possibly necessarily true cannot possibly be false
That which is possibly necessarily true is necessarily true

 

[ImprobableJoe]

Whoops, turns out a minor misreading caused a massive misunderstanding. Possibly true does not imply possibly necessarily true, however possibly necessarily true does imply necessarily true:

That which is necessarily true cannot possibly be false
That which is possibly false cannot possibly be necessarily true
That which is possibly necessarily true cannot possibly be false
That which is possibly necessarily true is necessarily true

I still don't get it... is this a special, limited rule of the sort you see in grammar? Because it sounds like you're speaking a foreign language, in a literal sense.

 

[creativesoul]

I find no practical use for 'possibly necessarily true'. If a claim is possibly true, then it is possibly necessarily true as well as being possibly false. If it is possible to be necessarily true then it means that it is not necessarily true, but may be. It adds only unnecessary confusion to say 'possibly necessarily true'. It is either necessarily true, possibly true, or necessarily false, or possibly false.

The 'possibility' depends upon the chance of being wrong. Therefore 'possibly neccessarily true' seems like a nonsensical statement to me. All things that are possibly true, may be necessarily true. That does not make them equal.

 

[ImprobableJoe]

I find no practical use for 'possibly necessarily true'. If a claim is possibly true, then it is possibly necessarily true as well as being possibly false. If it is possible to be necessarily true then it means that it is not necessarily true, but may be. It adds only unnecessary confusion to say 'possibly necessarily true'. It is either necessarily true, possibly true, or necessarily false, or possibly false.

The 'possibility' depends upon the chance of being wrong. Therefore 'possibly neccessarily true' seems like a nonsensical statement to me. All things that are possibly true, may be necessarily true. That does not make them equal.

So what you're saying is that "possibly necessarily true" does not equal "necessarily necessarily true" but "possibly possibly true" or " possibly possibly false" and/or "possibly necessarily possibly true/false"?

 

[GoodKat]

Whoops, turns out a minor misreading caused a massive misunderstanding. Possibly true does not imply possibly necessarily true, however possibly necessarily true does imply necessarily true:

That which is necessarily true cannot possibly be false
That which is possibly false cannot possibly be necessarily true
That which is possibly necessarily true cannot possibly be false
That which is possibly necessarily true is necessarily true

I still don't get it... is this a special, limited rule of the sort you see in grammar? Because it sounds like you're speaking a foreign language, in a literal sense.

S5 apparently only deals in absolutes. When you think about a statement X, it is ultimately either true or false, and if it is false, then it can't possibly be true, so if it can be true then it must be true. In S5, I would not be justified in saying "God possibly necessarily exists" without knowing that He does exist.

 

[ImprobableJoe]

S5 apparently only deals in absolutes. When you think about a statement X, it is ultimately either true or false, and if it is false, then it can't possibly be true, so if it can be true then it must be true. In S5, I would not be justified in saying "God possibly necessarily exists" without knowing that He does exist.

Yeah, which means that the language you are using is restricted by rules that in other uses do not apply. Really, the word "possibly" shouldn't be used in this case, because it muddies the waters? It seems to assume absolute knowledge, which means that there's no "possibly" involved.

 

[GoodKat]

S5 apparently only deals in absolutes. When you think about a statement X, it is ultimately either true or false, and if it is false, then it can't possibly be true, so if it can be true then it must be true. In S5, I would not be justified in saying "God possibly necessarily exists" without knowing that He does exist.

Yeah, which means that the language you are using is restricted by rules that in other uses do not apply. Really, the word "possibly" shouldn't be used in this case, because it muddies the waters? It seems to assume absolute knowledge, which means that there's no "possibly" involved.

Yeh that's basically it... I think

 

[Squawk]

It applies only in modal logic. The question is, can modal logic be used to verify existence postulates, and the clear answer is no.

 

[Zetetic]

I find no practical use for 'possibly necessarily true'. If a claim is possibly true, then it is possibly necessarily true as well as being possibly false. If it is possible to be necessarily true then it means that it is not necessarily true, but may be. It adds only unnecessary confusion to say 'possibly necessarily true'. It is either necessarily true, possibly true, or necessarily false, or possibly false.

The 'possibility' depends upon the chance of being wrong. Therefore 'possibly neccessarily true' seems like a nonsensical statement to me. All things that are possibly true, may be necessarily true. That does not make them equal.

I did some reading and here is what I have ascertained:

X is possible for a world W iff X is a possibility for a world that X can 'see', that is, if it is a possible state of affairs with respect to the conditions that form W.
X is necessary with respect to W if it is true for every world that X can 'see', that is; X is necessary for W if X holds for every possible state of affairs in W.

So if X is possibly true, X holds in a world that W can see, or equivalently X is a state of affairs that is consistent with W.

S5 makes the assumption that all worlds can 'see' one another and so any state of affairs is possibly true in any world W. So what 'Possibly necessarily P' says is that if X necessarily holds some world that is logically consistent with W, X necessarily holds for W. Thus if we are to note that every logically possible state of affairs is logically consistent with our world (if you take stock in Wittgenstein this might help), if we can form a world logically consistent with ours in which X is tautological, then X is tautological in our world, which seems reasonable to me.


As for " If a claim is possibly true, then it is possibly necessarily true as well as being possibly false" , this is false. By the definition of 'Possibly true', the only thing that 'X is possibly true' says is that X is a logically possible state of affairs. It may be that X is possibly necessarily true, and so it would be necessarily true, and thus X would be Possibly true, since we can choose any world V in which X is true (which would cover every world that W can see, including W itself). So given that X is possibly true, it does not follow that X is possibly false.

What does follow is this: If X is possibly true, then it is Either necessarily true or possibly false.


All things are made clear in well defined context.

 

[creativesoul]

Well it definitely cannot be both, necessarily true and possibly false. I implied that with 'possibly necessarily true and possibly false'.

If the term 'either' makes that much of a difference, then perhaps I should have used it. I definitely thought it.

 

[Master_Ghost_Knight]

The first thing I need here is to define spelicitly your terms. What do you mean With "possibly"? As far as i understand "possibly" is a judgement of character, if you say that it is possibly A it means that it can be A but it also mean that it doesn't not exclude it from being anything else and therefor the term possibly can not be dealt with absolutes because it doesn't give an absolut statment.