Thursday, January 13, 2011

Hunting Contradictions

In the context of time (temporality), change may be characterized by a fact or state of affairs holding at some time t which does not hold at some other time t'. That is to say, at t=0, proposition P is true; but at t+1, P is not true. From this, we may say time "allows" a contradiction (P&~P) to exist by spreading it out: the conflicting natures of P and ~P may coexist as long as they are "side by side" and not in the "same place", temporally.

With the above scare-quotes, I meant to emphasize the use of spatial metaphor in characterizing time; and this leads directly to the other pathway toward contradiction. A proposition and its contrary may exist simultaneously if they are separated from one another spatially; and this is such a natural part of existence that we hardly ever think of it. Expressed more intuitively, this simply means that things are different, when you look around yourself. The universe is not 100% homogeneous. Expressed a bit more formally, we say that (P&~P) can obtain at one and the same time t, if P and ~P occur respectively in different spatial regions.

In short, I am saying that contradictions are apparently possible when they are separated by space or time; that is, when they do not share identical space-time coordinates.

A counter question, however, is whether these are really "contradictions" if specified precisely enough. Suppose one says, "The space-time point (x,y,z,t) is [tense-lessly] filled (by some entity)".* If one pose against it the statement, "The space-time point (x,y,z,t) is empty", there is nowhere left "to go" in order to escape contradiction. To allow contradictions, according to my foregoing claims, one proposition must vary in time or space from another; but since we have exactly specified identical space-time coordinates, the contradiction is impossible. (This follows, at least in spirit, Quine's comments on temporal logic.)

* I seem to recall that some philosopher or other popularized the use of several tense-less terms and syntaxes, but I have no idea who it was or what the details of it are, now. I would be much obliged if anyone could point me in the right direction.

So, in response to the above question, I actually must agree: if we prevent them from "colliding", contradictions aren't really contradictions after all. This is trivially true, and hardly new information, but I like to think I have presented a slightly different framework for thinking about the matter.

A more interesting question might be, "Does this apply to all propositions?" I think the answer must be an assured "No", because not all propositions have to do with space-time per se, and thus this avenue of approach is not available. For example, "1=1" seems to be a general claim, without reference to any particulars, and it is hard to see how we could situate such a claim in the above schemes.

I think we can fairly safely conclude that a "robust" or fully specified contradiction is impossible, simply by definition; but we may be led to wonder whether there aren't other "channels" beyond the four of space-time which one might slide along to allow apparent contradictions. It strikes me that, should we ever somehow encounter a "real life contradiction" or paradox (whatever that would be like), we could, and likely would, posit a new channel of metaphorical space in which the contradictory elements could be separated. (We may speak of these as "dimensions", but I am cautious of using that word thanks to its abuse by overly enthusiastic New Age sorts and science fiction writers.)