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.
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.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.
(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...
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.