Here is a four-limbed aporetic polyad:
1. The merely possible is not actual.
2. To be actual is to exist.
3. To exist is to be.
4. The merely possible is not nothing.
Each limb is plausible, but they cannot all be true. The first three limbs, taken together, entail the negation of the fourth. Indeed, any three, taken together, entail the negation of the remaining limb, as you may verify for yourself.
Now which limb ought we reject in order to avoid logical inconsistency? (1) is non-negotiable because purely definitional. Everything actual is possible, but not everything possible is actual. 'Merely possible,' by definition, refers to that which is possible but not actual. This leaves us three options.
(2)-Rejection. One might reject the equivalence of the actual and the existent analogously as one might reject the equivalence of the temporally present and the existent. Just as one might maintain that past events exist just as robustly as present events despite their pastness, one might maintain that merely possible items exist just as robustly as actual items. David Lewis' extreme ('mad dog') modal realism is an example of (2)-rejection. On his modally egalitarian scheme there is a plurality of possible worlds all on an ontological par. Each is a maximal mereological sum of concreta. Each of these worlds is actual at itself, but no one of these worlds is actual simpliciter. For each world w, w is actual-at-w, but no world is actual, period. Thus there is no such property as absolute actuality. It is not the case that one of the worlds is privileged over all the others in point of being actual simpliciter. What is true of a world is true of its occupants: I enjoy no ontological privilege over that counterpart of me who is bald now and living in Boston. Actuality is world-relative and 'actual' is accordingly an indexical term like 'now.' When I utter a token of 'now' I refer to the time of my utterance; likewise, on Lewis' theory, when I utter a token of 'actual,' I refer to the world I am in.
Having rejected (2), a Lewis-type philosopher could gloss the other limbs of the tetrad as follows. To say that the merely possible is not actual is to say that merely possible objects (e.g. bald Bill the Bostonian) are denizens of worlds other than this one. To say that to exist is to be is to say that there is no distinction between the existence of an object and its being in some world or other. To say that the merely possible is not nothing is to say that objects which are not denizens of this world are denizens of some other world or worlds.
I am tempted to say that this solution, via rejection of (2), is worse than the problem. For one has to swallow an infinity of equally real possible worlds. Further, my possibly being bald is not some counterpart of mine's being bald in another possible world. (This critique of course needs to be spelled out in detail.)
(3)-Rejection. A second theoretical option is to reject the equivalence of being and existence, of that which is and that which exists. Accordingly, there are things that are but do not exist. They have Being but not Existence. Everything is, but only some things exist. The early Russell, in the Principles of Mathematics from 1903, toyed with this view although he rejected it later in his career. If existents are a proper subset of beings, then one could locate merely possible items in among the beings that do not exist. The merely possible would then have Being but not Existence or Actuality.
This solution leads to an ontological population explosion much as the Lewis theory does.
(4)-Rejection. A third option is to deny (4) by affirming that the merely possible is nothing in reality, that it has no ontological status. One might construe the merely possible as merely epistemic, as being merely parasitic on our ignorance, or as having no status outside our thought. A view along these lines can be found in Spinoza.
Intuitively, though, it seems mistaken to say that there are no genuine, mind-independent possibilities. My writing desk, for example, is one inch from the wall, but it could have been two inches from the wall. It is not just that I can imagine or conceive it being two inches from the wall; it really could be two inches from the wall even though this possible state of affairs was never actual and never will be actual. (Moreover, what I CAN imagine or conceive refers to real but unactual possibilities of imagination and conception; or will you say that these possibilities are themselves derivative from acts of imagining or conceiving? If you do, then a vicious infinite regress is in the offing.))
Now suppose I had provided more rigorous and more convicing rejections of each of the three theoretical options. Suppose that a strong case can be made that all four propositions must be accepted. Then we would have four propositions each of which has a very strong claim on our acceptance, but which are collectively inconsistent. (Assume that the inconsistency is demonstrable.) What might one conclude from that? (A) One possibility is that we ought to abandon the Law of Non-Contradiction. (B) A second is that one of the solutions must be right even though we have good reason to think that every solution is mistaken. (C) A third is that the aporetic tetrad is an insoluble problem, a genuine intellectual knot that cannot be untied.
Note that (A), (B), and (C) form a meta-aporia. Each of them has a claim on our acceptance, but they cannot all be true.
Suppose there are genuine but absolutely insoluble philosophical problems. What would that show, if anything?
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