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If everything is possible, is it possible for something to be impossible?

Possibility and impossibility are modal notions; and are dual in the usual formulation; the SEP remarks:

It would seem to be a simple matter to fit a modal logic with quantifiers - 'all' and 'some' - however adding quantifiers involves a number of difficulties, some of them philosophical...Quine has argued that ... it is incoherent; ... his arguments don't carry the weight they once did.

How does one resolve the above paradox using quantified modal logic? A related question - which might be more directly the OPs concern - is are there dialethic modal notions?

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    Hi, welcome to philosophy SE. "Possible" occurs 4 times in your post, and I am not sure if any two of them mean the same thing. You'd have to explain what you mean in each case to turn this string of words into a question. – Conifold Sep 1 '16 at 21:04
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    @Conifold despite the awkward wording, I think the OP is on to something: If everything is contingent, then how can we declare that anything is necessarily impossible? – Alexander S King Sep 1 '16 at 22:43
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    So is this basically a question on the consistency of various formalisms of modal logic, when it's given that "necessarily possibly P" for any P? – commando Sep 1 '16 at 23:49
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    I don't think this question should be closed. Anyway, two things: (1) The premise that everything is possible is obviously false. (2) The short answer is "Yes", and there's no paradox. You're just confusing "It's possible that X is impossible" with "X is impossible and also possible". The first one is perfectly ok. Anything that is true is also possible, and so if X is indeed impossible, then it is also possible that it is impossible. – Eliran Sep 2 '16 at 6:01
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    @conifold: I said it was a good question - I didn't say I could answer it... – Mozibur Ullah Sep 5 '16 at 22:44
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There's no paradox here; the answer is simply yes, it is possible for something to be impossible. The appearance of paradox stems from confusing this:

It is possible that it is impossible that X

for this:

It is possible that X

which are obviously different.

Example

Consider the following statement:

There are married bachelors.

Call that 'X'. We know that X is impossible. Formally: ~◊X.

In standard accounts of modal logic, anything that is true is also possible. That is, if A, then ◊A.

By the above we get: ◊~◊X. That is, it is possible that it is impossible that there are married bachelors

  • Despite being accepted, this doesn't seem to match the original question --your comment on the OP seemed more on target, at least to me. – Chris Sunami Sep 8 '16 at 3:22
  • No, This is self-referential, "is it possible for something to be impossible" is like saying "This statement is false". – Charles Bretana Feb 17 '18 at 13:39
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Your question asks, "if everything is possible, is it possible for something to be impossible?" The antecedent of this conditional is "everything is possible" which is equivalent to "nothing is impossible". If the 'thing' that one is quantifying over here is taken to mean any proposition, then one cannot normally say that nothing is impossible, since contradictions are impossible, i.e. in classical logic, contradictions are not true in any possible world.

One could deal with this by restricting one's attention to logically contingent propositions only, that is, those that are neither logical truths nor contradictions. We could then ask, "if all contingent propositions are possible, is it possible for some contigent proposition to be impossible?" Note that this is not the same as asking, "if all contingent propositions are possible, is there some contingent proposition that is impossible?" The answer to the latter is clearly no, but the former is asking only, is it possible that there is an impossible contingent proposition? The answer to that depends on which modal logic one is using. In S5, which includes the axiom ◊P → □◊P this possibility is excluded. In weaker logics such as K and T, this is not excluded, so an impossibility may be possible.

It may help to think of this in terms of relationships between possible worlds, along the lines of Kripke-Hintikka semantics. Suppose that some proposition P is false in the actual world but it is contingent and we are willing to suppose that all contingent propositions are possible, so it is true in some possible world A, but false in some other possible world B, where A and B are both accessible to the actual world. Now we ask, could P be impossible in world B - i.e. is there a configuration of possible worlds under which P is false in all the worlds accessible to B? Under S5 there is not, because in S5 semantics, the relations between possible worlds are reflexive, symmetrical and transitive, so given that A is accessible to the actual world, and the actual world is accessible to B, it follows that A is accessible to B, and hence P is possibly true in B. But in weaker modal logics, transitivity does not hold and one can have iterated possibilities (and necessities) that do not collapse in this way.

So your question effectively reduces to: which is the appropriate modal logic for expressing the idea that anything is possible? Depending on what kind of possibility you have in mind, you may wish to use a weaker logic to allow for iterated possibilities.

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This is a paradox of a type that has historically been very significant in the field of logic. We can render the original premise as

FOR ALL X (X IS POSSIBLE)

If we make X be the statement "Y IS IMPOSSIBLE" then we yield "Y IS IMPOSSIBLE" IS POSSIBLE.

But if we make X be Y then the statement is Y IS POSSIBLE, which is incompatible with "Y IS IMPOSSIBLE."

The standard way to resolve this issue is forbid quantifying variables that range over sets, which is to say, we forbid replacing "X" with a statement like "Y IS POSSIBLE" that makes a higher order claim about Y. A logic with such a restriction, whether modal or not, is called "first order logic." Higher order logics exists, but they have to go through a lot of complex manipulations in order to avoid paradoxes of this type.

The more typical example of this type of paradox is FOR ALL X (X IS FALSE) where that sentence itself is allowed to be a possible substitution for X.

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Everything cannot be possible, because if 'everything is possible', then 'everything is impossible' would also be possible, which would negate the possibility for anything to be possible.

'Everything being an impossibility' is also impossible because that would require that it's possible that 'everything is an impossibility', negating the 'everything' part of the 'everything is an impossibility' (in other words, 'everything is impossible' except the possibility that 'everything is impossible').

This is all theoretically speaking, of course.

In reality, the idea that everything can be impossible is false, proven by the fact that you are reading this and are conscious and you exist, negating many things that would be part of the 'everything that is impossible'.

In reality, 'everything cannot be possible' remains impossible as well, for the reason stated both in paragraph one and in paragraph four.

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Everything is possible if and only if the rules stay the same in whatever universe we are in. E.g. if we say that 2 = 2, then this rule holds. If we change the rule to 2 = 3, then 2 = 2 no longer holds. Saying that 2 = 2 and at the same time 2 = 3 could be allowed in some notations but not in the functional notation [for every input value (domain), a function has exactly one output value (range)].

Assuming this notation, then saying that everything (x) = possibility means that only x = possibility is allowed (x = impossibility doesn't exist).

So to answer this, if everything is possible, then it is not possible for something to be impossible. This doesn't render the statement everything is possible invalid since that assumed something that could be impossible doesn't exist in the first place.

  • I am wondering if this addresses the modal notion in the question. If you have references of those taking a similar view they may help support the answer and give the reader a place to go for more information. Welcome! – Frank Hubeny May 10 at 18:26
  • I'm new to this so I can't really refer you to any group with similar views. Luckily, I'm working from first principles which should make it easier for you to seek more information e.g. I based on functional notation and any search engine should be able to define a function for you. Speaking about functions, there is a reason, they are defined the way they are for without that definition, chaos (or no chaos) would erupt. For this question, if we assume that everything is possible and no rules hold, then the first thing we could do is to make everything impossible (the self-referential issue). – Inno May 11 at 20:16
  • Other things we could do without rules would be to make all falsehoods true or to make truths = falsehoods or to say all truths or falsehoods mean 'a' (whatever is true or false or every/anything means 'a'). This is quite problematic as you may have guessed which means rules are necessary. Of course, we could ask whether the 'all things are possible' statement would include the possibility of no rules. Then, we would need to ask if we could be able to ask the above question in the first place for it is only rules (vocabulary rules) that facilitate its existence. – Inno May 11 at 20:28
  • And assuming that this question would exist without any vocabulary [rules] would take us to the old-age question of how the Biblical God came to exist in the first place - a field which doesn't permit asking questions (from my little experience). - Not that I believe these are impossible for if they were, then our coming out of nothingness/a singularity would also ask where that 'nothingness' originated from (if non-existence was really non-existent, then none of us would think about it). Of course, there is an argument that all that currently exists is nothing but let's not go there for now. – Inno May 11 at 21:09
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On the one hand it seems to me that if you live in a universe where "everything is possible" then that's your answer right there. Everything is possible. So you shouldn't have the option of im-possibility.

On the other hand, impossibility is a possibility covered by "everything being possible".

It's a paradox... unless you consider a Many Worlds scenario. Therein each separate world explores a different option. Some possible, others not. In that case I would imagine that you could choose which you'd want to be true(?)

Personally I like to believe that some things are impossible. If everything is possible then reality would make no sense. 2+2 should always = 4. But if it ALSO = 6 or 245 or Green then we'd live in a state chaos in which we could never reach a conclusion. We would always be wondering, ad nauseum, questioning, and never building upon anything.

I believe, and am glad, that some things are impossible.

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If everything is possible is it possible for something to be impossible?

Yes..

Everything is possible because it is possible for something to be impossible..

It is not a contradiction.. it is a loopback..

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