As I use them, 'imaginable' and 'conceivable' mean the following. Bear in mind that there is an element of stipulation and regimentation in what I am about to say. Bear in mind also that the following thoughts are tentative and exploratory, not to mention fragmentary. The topics are difficult and in any case this is only a weblog, a sort of online notebook.

To *imagine* X is to form a mental image of X. To imagine a two-headed cat is to form a mental image of (more cautiously: *as* of) a two-headed cat. To say that X is imaginable is to say that someone has the ability to imagine it. To *envisage* is to visually imagine. Not all imagining is visual.

To *conceive* X is to think X. To say that X is conceivable is to say that someone can think it, that is, has the ability to make it an object of thought. Trading Latin for good old Anglo-Saxon, conceivability is thinkability. Therefore, a round square is conceivable in that I now have it as an object of my thought, hence someone can have it as an object of his thought. If you balk at this, then you are probably confusing conceivability with conceivability without contradiction. Admittedly, round squares are contradictory objects. Still, one can think them. They are therefore thinkable or conceivable. If you weren't able to think of the round square you would not be able to judge that there cannot be a round square.

I grant that people often use 'conceivable' as elliptical for 'conceivable without contradiction.' But to be precise, one ought to add the qualifier.

It follows from the foregoing that the imaginable and the conceivable are distinct. A pentagon is a five-sided plane figure that encloses a space. A chiliagon is a thousand-sided plane figure that encloses a space. I can conceive of both without contradiction. But I can imagine only the pentagon. And I suspect the rest of you are in the same boat. Therefore, the imaginable is not to be confused with the conceivable. Not everything conceivable is imaginable. A second example is the round square lately mentioned. I can conceive it, although of course I cannot conceive it existing, but I certainly cannot imagine it. (If you are imagining a pulsating object that is first round then square then round again, you are not imagining a round square.)

To say as some do that the collapse of the World Trade Towers was unimaginable or inconceivable is an offensively loose way of talking.

Now for the hard part, which I can only touch upon briefly. Is imaginability a sure guide to real possibility? If one can imagine X, does it follow that X is really possible? Well, Ron Crumb of Zap Comix fame imagined and depicted all sorts of weird things that are not really possible, Tommy Toilet for example. (Do not click on the link if you are offended by scatological humor.) I conclude that imaginability does not entail real possibility. Perhaps it provides some evidence of real possibility.

A real possibility is one that has a mind-independent status. Real possibilities are not parasitic upon ignorance. Thus they contrast with epistemic possibilities. Since what is epistemically possible for a person might be really impossible (whether broadly-logically or nomologically), we should note that 'epistemic' in 'epistemically possible' is an *alienans* adjective: it functions like 'decoy' in 'decoy duck.'

As for conceivability, why should the fact that I can conceive something without contradiction show that the thing in question can exist in reality? Consider the FBI: the floating bar of iron. If my thought about the FBI is sufficiently abstract and indeterminate, then it will seem that there is no 'bar' to its possibility in reality. If I think the FBI as an object that has the phenomenal properties of iron but also floats, then those properties are combinable in my thought without contradiction. But if I know more about iron, including its specific gravity, and I import this information into my concept of iron, then the concept of the FBI will harbor a contradiction. The specific gravity of iron is 7850 kg/cu.m, which implies that it is 7.85 times more dense than water, which in turn means that it will sink in water.

The upshot is that neither imaginability nor conceivability without contradiction are sure guides to (real) possibility. Do they provide evidence of real possibility? I can both imagine and conceive my writing table being closer to the wall than it is. Is that evidence for the possbility of the table's being closer to the wall than it is? Uncontroversially, everything actual is possible. So if I move the table so that it touches the wall at time t, then I know that such a state of affairs is possible at t. The puzzle has to do with unrealized possibilities. Could the table yesterday have been closer to the wall than it was then? Of course! But how the hell do I KNOW that? The senses teach us what is the case, but not what could have been the case. Staring hard at the table and the wall, I see their actual relative position but not their merely possible relative positions. So how do I know that those merely possible relative positions are really possible as opposed to being merely excogitated *Denkmoeglichkeiten*?

(This epistemological puzzle is connected with the ontological problems of finding a place in reality for unrealized possibilities. My formulation is intentionally paradoxical in a manner to highlight the puzzle.)

So what do we say to the modal sceptic? I have beliefs about the merely possible. For example, I believe that Ted Kennedy's ill-treatment of Sam Alito at the latter's confirmation hearing was not inevitable, that the Massachusetts pol could have behaved in a more statesman-like manner. How do I justify my modal beliefs? Can I argue for them? Or are they just unarguable intuitions?

Is a round square actually contradictory, Bill? I hope you will forgive me if I stray from the topic a bit, but this thought has exercised me in the past.

First, to be clear about my definition, a contradiction is a property of propositions, it is the opposite of a tautology. That is, if P is a contradiction, then ~P is a tautology.

Second, since contradiction is a property of propositions, not of geometric figures, the figures themselves cannot be contradictory. What I suppose "a round square is contradictory" means is that the proposition schema

1. X is a round square

is a contradiction for all X. But 1 is not a contradiction. It has two distinct predicates, "round" and "square". It has the form

p(X) and q(X)

which is not a contradiction. Now 1 combined with

2. forall X . X is round implies that X is not square

is a contradiction. But 2 is not included in the mere concept of a round square; 2 comes from an extra-logical intuition. 2 is a priori, but it is not logical. So although 1 is false via a priori intuitions, it is not a contradiction.

I would propose the resurrection of a grand old term for such situations: "absurd". This used to be used in old proofs as something similar to "contradictory", but was used more broadly than mere logical contradiction. I suggest that a proposition is absurd if it is a priori false. Therefore, round squares are absurd, 1+1=9 is absurd, time travel that changes time is absurd, the idea that borrowing hundreds of billions of dollars out an economy suffering from lack of available credit will help the economy is absurd.

Posted by: Dave Gudeman | Friday, February 06, 2009 at 12:35 AM

Dave,

I'd say that

P) x is a round square

isn't equivalent to

Q) p(x) and q(x).

So proving that (Q) is non-contradictory doesn't prove (P) is.

You say "But 2 is not included in the mere concept of a round square; 2 comes from an extra-logical intuition." I rather think that your (2) is included in (1) - so long as you understand what you are really asserting - an object is round if it has exactly 1 side, and an object is square if it has exactly four. Bearing in mind that to understand a concept is to recognise what that concept extends to and what it doesn't, it seems obvious that the set that is the extension of ROUND and the set that is the extension of SQUARE will never have a common member.

Bill,

About the FBI: I think this example only works on the assumption that 2-dimensional semanticism (espoused by Chalmers' et al.) is false. They would say that your example confused what were really two distinct concepts.

"If I think the FBI as an object that has the phenomenal properties of iron but also floats, then those properties are combinable in my thought without contradiction."Agreed, I guess this concept of iron would be our pre-scientific one. Call it iron1.

"But if I know more about iron, including its specific gravity, and I import this information into my concept of iron, then the concept of the FBI will harbor a contradiction."

I'm not exactly sure what this "concept importation" would amount to. Surely all that is happening here is that the cluster of concepts which composed iron1 is having a few more added. So I'm tempted to think what you end up with is strictly a different concept, call it iron2. (I'm not sure what the criteria of identity for concepts could be.) The 2Dists say that to get concepts like iron2 all you are doing is rigidifying from the actual referents of a concept like iron1. (Or such was my understanding.)

Anyway, once we distinguish between iron1 and iron2 your claim that "necessarily, iron doesn't float in water" can be seen to be self-evidently false if iron = iron1, and seen to be self-evidently true if iron=iron2. But then you no longer have any reason to doubt that conceivability is a good guide to possibility.

Matt.

Posted by: Matt Hart | Friday, February 06, 2009 at 03:19 AM

Matt, given that a circle has one side, that a square has four sides, that 1=/=4, and that if a figure has n sides then it cannot also have m sides where m=/=n, then you can find a contradiction. But all of those facts come from non-logical intuitions of mathematics and space --they are not deducible.

Just to clarify, I'm using "intuition" in the Kantian sense. In Kantian terms, it is common to conflate logic with the analytic. If you view it that way, then what I'm saying is that the absurdity of a round square is not analytic but that it is a priori.

Posted by: Dave Gudeman | Friday, February 06, 2009 at 09:39 AM

Dave,

I doubt that we have any substantive disagreement once we adjust for differences in terminology.

Matt,

I don't have the time to be rigorous, but concepts can be more or less determinate. If my concept of X is not very determinate, then the impossibility of X might not show up within my concept. Suppose someone has the concept of a red point and infers that red points are possible because 'x is both red and a point' is not formally self-contradictory. But then this person comes to realize that points are unextended and that necessarily colors are extended. Importing this new information into his original concept and thus making it more determinate he comes to realize that nothing in reality can instantiate the concept

red point.Posted by: Bill Vallicella | Friday, February 06, 2009 at 10:34 AM

Greetings Bill,

I've run into this argument on several occasions and while the author(s) insist theists will accept the premises it's more the validity I'd appreciate your take on.

1) If God is possible, then God is a necessary being.

2) If God is a necessary being, then unjustified evil is impossible.

3) Unjustified evil is possible.

Therefore, God is not possible.

Posted by: modally challenged | Thursday, February 12, 2009 at 08:47 PM

Greetings and thanks. Response in a separate post on front page near the top.

Posted by: Bill Vallicella | Friday, February 13, 2009 at 05:31 PM

Too much logic.....theology is not logic, faith....a mis understood concept of rational beings. Augustine was the man....flawed yes but on the right track.

Posted by: Mike moran | Monday, February 16, 2009 at 11:49 AM