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.)

Link Dump

So here are some blog entries (or whatever) I have recently found interesting.

At Theoretical Atlas, Jeffrey Morton recounts a talk given by Gregory Chaitin where Chaitin urges us to create a more rigorous "theoretical biology". Something where we could prove interesting theorems about evolutionary models, for example.

At (Blog&~Blog), Ben Burgis writes a bit about Graham Priest's Sylvan's Box story and the (im)plausibility of drawing conclusions about "real life" logic from the logic that exists in fiction.

Alan Rhoda discusses the Law of Excluded Middle in the context of propositions about the future, responding to an example supplied by David Hunt.

This is a somewhat older paper (1997) by Allen F. Randall entitled "Quantum Phenomenology". He attempts something of an updated version of Descartes' cogito (maybe mixed with Kant's transcendental project), attempting to logically derive the principles of quantum mechanics from the basic awareness of experiential existence (my phrase, not Randall's). Randall claims, "Far from being "bizarre" and "weird", as is usually thought, the strangest paradoxes of quantum theory turn out to be just what one ought to expect of a rational universe." I admit, I've only read the first part of this paper, but it's still a very fascinating idea.

At Mixing Memory, Chris reports on several studies where unpleasant stimuli (such as foul odors, messy/sloppy workspaces, disgusting video footage) harshen the severity of people's moral judgments.

Vacuous Truth

From my post in a topic in the xkcd forums:

"Every woman currently living on Pluto is male."

"Every true falsity is false."
"Every false truth is true."

"Every circular square is neither circular nor square." (This one almost makes an intuitive sense... but then, it really doesn't, when we consider that every circular square is also circular and square.)

"Every present King of France is simultaneously bald and not bald."

By the mathematical/logical principle that propositions are vacuously true when their subject does not exist, no matter what is predicated of that subject, the above statements should be considered true.

This relates to the problem of negative existentials in the philosophy of language, but I don't have more to say about it at the moment. May investigate further.

Hegelian Precursor to Derrida

If we cannot indeed make true statements about the whole (because they inevitably lead to an incomplete thesis-antithesis-synthesis triad, which in turn becomes a leg of a new triad...), then yes, I think Derrida's theory of deconstruction--wherein all statements undermine themselves--follows more naturally.

Note, by the way, that I know next to nothing about both Hegel and Derrida here. But anyway.

My objection to deconstructionism hitherto has been similar to my complaints about extreme skepticism generally: by attempting to demolish all foundations, one must necessarily presuppose a new foundation from which to do that demolishing. In other words, the statement "all propositions undermine themselves" necessarily undermines itself, rendering it false. Discussing that with a lit-theory friend of mine, he laughed and called that a beautiful part of the theory; to him, it "shows" the theory in operation on itself. It bothers me, however, since it leaves us left with a paradox and/or a jumble of contradiction.


Parallels

We can view the deconstruction of "All propositions undermine themselves" as analogous to the eternal war between a skeptic and a dogmatist, or an unraveling of the Liar's paradox. To wit, each valid step may be succeeded by a contradictory, equally valid step in the argument. E.g.,

(1) All propositions undermine themselves.

(2) "All propositions undermine themselves" undermines itself, by (1). Ergo, (1) is false.

(3) If (1) is false, then there can be propositions which do not undermine themselves after all. In which case, (2)'s reasoning is incorrect, because (1) might be one of those propositions, and so (1) might be true. Since (1) was given as an initial premise, we should then consider (1) true.

(4) (3)'s conclusion is a proposition which, by (1), undermines itself. Hence (3) must be false, and if (3) is false, then (1) is not true after all.

...etc, etc. This may be argued back and forth as long as we like without resolution.

[Note again: as said before, I know crap-all about Derrida. I have no idea if genuine deconstructions follow the form I just gave, and at the moment am too lazy to verify. Hooray, I'm a bad scholar. You caught me, want a prize?]

Just as with the Liar's Paradox,

(5) This statement (5) is false.

(6) Because (5) is false, (5)'s negation, "This statement (5) is true", must be true. Hence (5) is true.

(7) Since (5) is true, we know that the proposition "(5) is false" is true. Thus, (6)'s conclusion is false, because (5) is not true after all.

(8) Yet, if (5) is false, then its proposition "(5) is false" is false itself, meaning that (5) is really true. That means (7)'s conclusion is falses.

... etc. Clearly, an infinite succession of licitly derived contradictions. I'm trying to make a point about steps directly contradicting the directly previous step, but that's probably confusing, and under an ordinary analysis it is not necessary, so let me show a more intuitive route which is equivalent:

(9) (9) is false.

(10) Because (9) is false, "(9) is false" is false. Thus (9) is true.

(11) But if (9) is true, then "(9) is false" is true. And that means (9) is really false.

(12) If (9) is false, the same reasoning as from (10) shows that (9) is true.

(13) But if (9) is true, then the same reasoning as from (11) shows that (9) is false.

(14) The same reasoning from (10) and (12) shows that (9) is true.

(15) The same reasoning from (11) and (13) shows that...

The main difference is that I'm not spelling out the contradiction of the last step so much as just reasserting either (10) or (11) to refute the last conclusion. Which is really the same thing, so why am I making a fuss about it? The Lord only knows. Really, the smart thing to do is to stop as soon as you've found a contradiction in the argument (since otherwise we run into problems with explosion), but I'm trying to make the analogy to Hegel more explicit.


And speaking of whom, back to Hegel.

Suppose we call the assertion that "The Liar's Paradox statement is true" our thesis, and "The Liar's Paradox statement is false" our antithesis. Clearly, we can always reason toward thesis or antithesis, successfully proving or disproving each conclusion however many times we like, without ever reaching a final resolution. To Hegel, I believe, we should then realize the futility of this exercise, at which point we need to step outside of the system and create a synthesis between thesis and antithesis. Being a Hegelian neophyte, I don't know what the synthesis should be in this case, but it might be something like, "The Liar's paradox is both true and false" or it is"partially true, partially false," or "true at one time, false at another," or some other means of effecting reconciliation.

Now, the fun part about Hegel is that he says the new synthesis, whatever it is, now becomes the thesis or antithesis of a new thesis-antithesis-synthesis triad, which will need its own extra-dichotomous resolution. And we approach (but never reach?) truth through an infinite chain of these triads--reminiscent of Kant's moral progression, where practical reason must postulate an infinity of time (or lifespans?) through which we imperfect beings aspire toward perfection.

Now, where this is relevant more specifically to my thoughts, is that an infinite chain of dialectical syntheses reminds me very strongly of the warring double-helix Ouruborus I mentioned earlier. It seems to me wishful thinking on Hegel's part to claim that the addition of a synthesis makes a new kind of "progression" or development. Rather, the chain of triads fighting with each other is precisely ismorphic to the chain of contradicting (10)s and (11)s I outlined above, the simply unending contradiction. So, either the Liar's Paradox and Derrida's deconstruction already exemplify Hegel's described growth, or there exists simply no progression to speak of either way.

Hegel's syntheses are attempts to establish a new "groundless ground" or self-supporting justification in each controversy. By my thinking, however, this is not progression, since the new synthesis remains just as much a part of the very system it attempted to escape. This is just like trying to defuse Gödel's incompleteness theorem by adding axioms to a formal system; it doesn't matter how many or what axioms you introduce: by the very nature of the system at hand, you will leave yourself open to a new version of the incompleteness theorem. (Unless you reduce your system's axioms to a point where they express less than you originally wanted.)

Now, we might be able to argue that there's a kind of progression/development/growth/whatever here anyway, but I'm not going to investigate further at this point. It may explain (in part) the impossibility of halting philosophical inquiry.

By now, it's beginning to seem to me, naively, that the history of thought is little more than a horribly convoluted deception, an intriciate illusion, contrived to hide the fact that all we've been doing is saying "Nuh-uh!" and "Yeah-huh!" to each other for the last two thousand years.

Silly children.

Conflicting Sensations

In my last post, I wondered what a world which allowed simultaneous contradictions would look like.

Earlier this morning I had a very fascinating experience – shortly before I woke up, I became aware that I was simultaneously dreaming and yet in my bed at the same time. That is to say, I received two mutually exclusive sets of sensations, one informing me that I was lying on my back underneath sheets and covers with my eyes closed, the other that I was walking through some dreamscape with my eyes open.

I can hardly remember the details of the dream now, but still, the memory of that dual experience is striking. It makes me wonder – perhaps the brain is better equipped to deal with simultaneous contradictions than I'd thought, since I was apparently making sense of feeling that I was in two places at once. Both sensations felt as though they were centered on me, my brain, my consciousness/awareness/etc. It was as though I temporarily had two perspectives on "life" (even though, of course, only one of these was a veridical perspective), and I was able to feel both simultaneously.

So perhaps experiencing a contradiction would be a little like that, somehow.