The asymmetry of logical possibility and impossibility


L = Logically possible.

O = Something an omnipotent being can cause.

Now there is a very old tradition that if ~L then ~O, since there is no such being to be caused. Said more poorly, “omnipotence does not extend to logical contradictions.” It follows logically from this that if O then L.

What I want to deny, but what has a long tradition of being argued in modern philosophy, is if L then O. We get a famous suggestion of such an argument by Descartes, who is clearly borrowing on a long line of voluntarist thought when he says:

[T]he belief that there is a God who is all powerful, and who created me, such as I am, has, for a long time, obtained steady possession of my mind. How, then, do I know that he has not arranged that there should be neither earth, nor sky, nor any extended thing, nor figure, nor magnitude, nor place, providing at the same time, however, for [the rise in me of the perceptions of all these objects, and] the persuasion that these do not exist otherwise than as I perceive them ? And further, as I sometimes think that others are in error respecting matters of which they believe themselves to possess a perfect knowledge, how do I know that I am not also deceived each time I add together two and three, or number the sides of a square, or form some judgment still more simple, if more simple indeed can be imagined.

Descartes immediately suggests that he agrees that if ~L then ~O, and so this argument seems to be an attempt to move from L to O. The “How, then, do I not know”* claim seems to be a claim of logical possibility, which he then claims falls immediately under O.

Voltaire, in clarifying an argument from Locke, makes the L–>O move unmistakably:

The point is not whether we know if matter can think by itself…The point is not whether we know if our soul is spiritual or not. The point is simply whether we know enough about matter and thought to be bold enough to assert that…God cannot communicate thought to the being we call matter

Locke, of course, could see no reason why this could not be the case, and so judged that it did fall under omnipotence.

But that’s where the magic happens. As soon as we say that logical possibility indicates falling under divine omnipotence, what is logically possible becomes really possible. The problem with this is that logical possibility can follow from ignorance, forgetfulness, or imperfect knowledge. As far as a blind man is concerned, it’s logically possible that blue and green are the same color. Maybe the whole sighted world is confused on the point (they’ve been confused about colors before), maybe they’re all involved in some grand conspiracy to trick him, whatever. But omnipotence is not some philosopher’s stone that empowers us to magically transform ignorance into existence, which is what the if L then O move requires. 

The same move rules out the even more dubious move to go directly from logical to real possibility, which seems to be involved in any talk of “possible worlds”. At any rate, it’s certainly involved in the attempt to move from the possibility of God to his existence. Such a move might work if we posit from the outset an ideal knower that is in no way deceived and has all possible knowledge, but then we’ve just posited God from the outset.

There is, in other words, an asymmetry between logical possibility and impossibility. A thing is logically impossible precisely because we see a contradiction in it; and so L is a matter of what we see, and failure to see something is not the same thing as to see it’s not there.


*The distinctive mark of the L—>O move is some variant on “but how do we not know that” or “for all we know it might be that”.


  1. whitefrozen said,

    January 31, 2014 at 8:40 pm

    L is in the eye of the beholder?

  2. Timotheos said,

    February 1, 2014 at 1:37 am

    And this is exactly why I think the term “logically possible” needs to be abolished, or at least severely reprimanded. I mean how “logical” is the “logically possible” suggestion that a prime minister is a prime number?

    Rather, we need to say that what’s “logically possible” is conditional to what we’re talking about; that, dare I say it, what’s ”logically possible” depends on our domain of discourse, and thus we need to define our domain of discourse before we go analyzing what is and is not logically possible.

    As for the “how do we know that it’s not possible” claim, couldn’t it equally be wondered “how do we know that it’s possible”, and thus the two claims effectively cancel each other out?

    • February 1, 2014 at 10:07 am

      Logical possibility does have a role to play in explaining omnipotence, and so in describing the extent of being as such. But it is not the same thing to notice a contradiction and not to notice one.

      It also has value as an idealization: the conceivable might be identified with the really possible for an ideal knower, though not in the sense that the conceivable would be a contingent thing. If we could eliminate ignorance, forgetfulness, or imperfect knowledge as influencing judgement, there might be a more robust power to logical possibility. But this sense of logical possibility could only be, at best, a subset of theology as a discourse about the nature of the divine mind. But taken in this way it probably collapses as soon as we try to get it off the ground: God does not need sets of propositions to think, nor does he need modalities that characterize our thinking, like the contingency of the future.

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