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?
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