Over lunch yesterday, Peter Lupu questioned my assertion that possibility and contingency are not the same. What *chutzpah!* So let me now try to prove to him that they are indeed not the same, though they are of course related. To put the point as simply and directly as I can, possibility and contingency are not the same because every necessary proposition is possible, but no necessary proposition is contingent. Perhaps this requires a bit of explanation.

We first divide all propositions into two mutually exclusive and jointly exhaustive groups, the *noncontingent* and the *contingent*. The first group subdivides into two mutually exclusive and jointly exhaustive subgroups, the *necessary* and the *impossible*. A proposition is necessary (impossible) just in case it is true in every (no) possible world. A proposition is possible just in case it is true in some possible worlds. It follows that if proposition p is necessary, then p is possible, but not conversely.

Since we know that there are necessary propositions, and since we know that every necessary proposition is a possible proposition, we know that there are necessary propositions which are possible. But we also know that no necessary proposition is contingent. It follows that we know that there are possible propositions that are not contingent. It follows that the extension of 'possible proposition' is different from the extension of 'contingent proposition.' This suffices to show that possibility and contingency are not the same. Here are some definitions. I have included definitions not fomulated in the imagery of possible worlds for those who are 'spooked' by his imagery.

A proposition p is possible =_{df} there is a possible world in which p is true.

A proposition p is possible =_{df} it is not necessary that p be false.

A proposition p is contingent =_{df} there is a possible world in which p is true and there is a possible world in which p is false.

A proposition p is contingent =_{df} p is both possibly true and possibly false.

Example. *No color is a sound* is possible but not contingent. There is a possible world in which it is true, but no possible world in which it is false. *Tom's favorite shirt is red* is contingent. There is a possible world in which it is true and a possible world in which it is false.

UPDATE (21 February): David Brightly provides a very useful map of the modal terrain in the Comments.

Peter L. admits in an e-mail that I am right, "but only contingently." That's a good joke but it is also true. There are plenty of possible worlds in which I am wrong about the point under discussion.

Posted by: Bill Vallicella | Friday, February 20, 2009 at 02:31 PM

That's a point on which I was trying to get clear a while back - as a matter of technical terminology, does 'contingent' mean 'both possible and unnecessary'? But the alternative, it seemed to me, was that 'contingent' simply means 'unnecessary'. Now that alternate reading seems unlikely since (A) the impossible is a subclass of the unnecessary and (B) we would not call the impossible a subclass of the contingent. Yet I kept encountering statements that 'contingent' means 'unnecessary', so, worried that it was some sort of technical terminology quirk, I found myself weaseling around the issue when what I really wanted to do was make a hexagon of opposition (tildes to preserve spacing):

~ ~ uncontingent

necessary ~ impossible

possible ~ unnecessary

~ ~~ contingent

(Valid) disjunction and (inconsistent) conjunction of any two contradictories among them make for a total of 6+2 = 8 options. When you combine with 'true' and 'false', (e.g., 'contingently true' and 'contingently false'), then you get a 16-fold including a few odd combinations.

Anyway it would be nice if there were a dictionary or lexicon which plainly said that 'contingent' means 'possible and unnecessary', since that sure seems to be the meaning.

Posted by: Ben Udell | Friday, February 20, 2009 at 07:44 PM

Ben,

I think this follows Bill's usage. Does this help?

I think the problem arises because 'necessary' is often contrasted with 'contingent'. When used like this 'necessary' includes the necessarily false, ie, the impossible, which perhaps is a little counter-intuitive.

Posted by: David Brightly | Saturday, February 21, 2009 at 03:54 AM

"Peter L. admits in an e-mail that I am right, "but only contingently.""

Bill is right, about both: contingency and my admission of him being contingently right. (I blame the Indian cuisine!)

peter

Posted by: Account Deleted | Saturday, February 21, 2009 at 04:38 AM

David,

That's beautiful and exactly right. Thanks! I was thinking of making a 'modal map' but I'm 'graphically challenged.'

>>I think the problem arises because 'necessary' is often contrasted with 'contingent'. When used like this 'necessary' includes the necessarily false, ie, the impossible, which perhaps is a little counter-intuitive.<<

Right. Consider *There are round squares.* We could classify this as necessarily false, as not possibly true, or as impossible. But it would be misleading to classify it as necessary.

Posted by: Bill Vallicella | Saturday, February 21, 2009 at 08:31 AM

Ben writes, "Yet I kept encountering statements that 'contingent' means 'unnecessary'. . ." If a proposition p is contingent, then p is not necessary (i.e., not necessarily true). But if p is not necessary, it does not follow that p is contingent. For example, *There are round squares* is not necessary because impossible. But the proposition is not contingent. So 'contingent' cannot mean 'not necessary.'

>>Anyway it would be nice if there were a dictionary or lexicon which plainly said that 'contingent' means 'possible and unnecessary', since that sure seems to be the meaning.<< That is indeed what 'contingent' means. The discussion in Kenneth Konyndyk,

Introductory Modal Logicis good. He doesn't put it your way but he says things that entail it. He also has an interesting modal diagram on p. 17. But it is not as good as Brightly's!Posted by: Bill Vallicella | Saturday, February 21, 2009 at 08:57 AM

Thanks, Bill, for the reference, it may come in handy.

David, you hit it the nail on the head about 'necessary' sometimes including the necessarily false - in that sense it really means 'non-contingent' and confusion's seeds take root.

I did get Bill's sense originally. I meant that there are 16 non-redundant options, including 14 non-trivial options, which one can get for instance by combining 'True' and 'contingent', using the 16 binary connectives. So additionally there are the non-contingent, the not contingently-True, the not contingently-False, the contingent iff True, and the contingent iff False. The last two especially were what I meant by "odd combinations" (I haven't been used to thinking about them and have wondered in what connections they might be of practical concern).

I'll try to use David's graphic method, I really hope it works for me too, otherwise it will be a deletion-worthy mess. On the pattern of

Posted by: Ben Udell | Saturday, February 21, 2009 at 09:43 AM

Hello Bill, and thank you for the kind remarks. Some of us were doing tables this way long before MS Word and its like were invented!

But I've been puzzling over Peter's joke. You say you want to "try to prove to him [Peter] that they [contingency and possibility]" are indeed not the same. Both Peter and you agree that you are right, ie, that your proof is sound. But won't the proof work in any possible world and hence aren't you *necessarily* right? How could there be "possible worlds in which I[Bill] am wrong about the point under discussion"? After all, it's a matter of definition and the extent of possible worlds, hence 'lies outside all possible worlds'.

Posted by: David Brightly | Friday, February 27, 2009 at 02:03 AM

David,

You write, "Both Peter and you agree that you are right, ie, that your proof is sound. But won't the proof work in any possible world and hence aren't you *necessarily* right?"

There are two separate questions here. One concerns a psychological fact about me, the other concerns the soundness of a proof. Yes, the proof is sound in every possible world. But I am not right in every possible world because there are worlds in which I am not aware of the proof. So although the proof is necessarily sound, it does not follow that I am necessarily right.

Dou you buy this explanation?

Posted by: Bill Vallicella | Friday, February 27, 2009 at 12:45 PM