In exploring the dialectic, I plan to examine conceptions of it one by one, and consider whether they can be sustained. The goal is to move more or less from the broadest to the narrowest, until we find a core that is true.
So, let's start with a very broad conception of dialectics: as a successor to mathematical formal logic, which "arose out of the criticism of formal logic, overthrew and replaced it as its revolutionary opponent, successor and superior," in the words of George Novack. The idea is that dialectics has a relationship to conventional logic which is analogous to the relationship between the theories of Einstein and of Newton, or between an obsolescent tool and a state-of-the-art one.
This is so immediately problematic that I do not even know how to present it more fully before abandoning neutrality.
Mathematical logic allows the mechanical deduction of results - one can start with an expression like "(A OR B) AND (NOT A OR C) AND (B OR NOT C)" in terms of Boolean (true or false) variables A, B, and C, and then use a computer to determine, given values for the variables, whether the expression is true or false. Or one can even use a computer to check whether the expression is valid - whether it is true for all possible values of A, B, and C - or satisfiable - whether it is ever true given any values of A, B, and C - using an algorithm like resolution which does not require plugging in any specific values for the variables at all.
One cannot do anything like this for dialectics. As Rees writes, dialectics is "not a calculator into which it is possible to punch the problem and allow it to compute the solution." But calculability is precisely what distinguishes logic as logic. It is not an incidental benefit. The reason we can use mathematical logic in a calculator is that mathematical logic, given whatever premises, can tell us unambiguous conclusions which follow necessarily. If the "laws" of dialectics were laws in the same sense that commutativity is a law of Boolean logic, we could tell computers how to obey them, and so use them in programs. We can't.
Logic enables us to go from one or more true statements to another equivalent true statement without the need for additional empirical verification. If I see, say, one protest result in a victory, I know that it is not the case that all protests are defeated, without having to check any additional protests, thanks to a basic equivalence in first-order logic. And when we go beyond single-step deductions, very often logic will take us to non-obvious conclusions.
But deduction of this kind is not available with dialectics. Rees concedes that "once we are sure we have made an accurate abstraction", that does not mean "we can also be sure that any further categories that emerge as a result of contradictions which we find in our concept will necessarily be matched by contradictions in the real capitalist world. This... is only a safe assumption on the basis of constant empirical verification."
So dialectics' non-calculability is not merely a question of a missing formalization. Logic is a mathematical way of examining a process that is essential to reasoning, where we place our knowledge and beliefs into relationship with one another in order to draw conclusions and form a coherent whole, without any necessary mediation by new or additional experience. Dialectics, whatever it is, is a different kind of beast.
Engels writes, "Even formal logic is primarily a method of arriving at new results, of advancing from the known to the unknown — and dialectics is the same, only much more eminently so." But if we are to set dialectics up against formal logic, as an alternate method of deduction, then if it is more "eminent", that is only in the sense that the Pope is more eminent than a subway conductor. The latter does practical work herself, while his eminence has subordinates for that.
Of course, why should one set dialectics up as a rival to logic as studied by mathematicians and computer scientists? Why disparage formal logic with dishonest word games - like Novack's attempt to slide from "A = A", to "A always equals A", to "A cannot be non-A", to "a man cannot be inhuman, democracy cannot be undemocratic", without the reader noticing a non-equivalence, in order to cast doubt on the "law of identity"?
This kind of rhetorical trick has nothing in common with science, but is necessary if dialectics needs to conquer logic's territory in order to have living space. So why not abandon the attempt?
We should. The only problem is that if you do not view dialectics as a logic like Boolean logic or first-order logic, you do not get universality for free. Engels describes dialectics as "the science of the general laws of motion and development of nature, human society and thought." But if dialectics cannot take over logic's universal applicability, and if it cannot be formalized and studied in isolation independent of topical content, there is no clear reason why we should assume it is so general. We will have to reconsider how universally its principles really apply.
 Logic of Marxism, p. 28.
 The Einstein/Newton analogy is used by John Rees, Algebra of Revolution, p. 271. Of course, he says in the same place that dialectics is not an "alternative" to formal logic, but this seems to be just a confused retreat from the full implications of the original idea. Einstein's theory was, of course, an alternative to Newton's, which is a mere low-energy approximation.
 Of course, for general expressions checking satisfiability is NP-complete, and so infeasible in practice, but that's off topic.
 Algebra of Revolution, p. 271.
 ibid., p. 110.
 Anti-Duhring, chapter 11.
 Logic of Marxism, p. 18. This is a truly awful passage, which stands out even in a very bad book. On top of the idiotic hackery of the claim that "man cannot be inhuman" follows from the law of non-contradiction, Novack asserts that this law follows "logically and inevitably" from the law of identity, and in turn "flows logically" to the law of the excluded middle. Of course, not only are these three axioms not derivable from one another (or they would not be axioms), there are in fact consistent logical systems which deny the law of the excluded middle, used in constructivist math. And then there's the fact that the three laws Novack discusses are archaic and do not constitute the complete axioms of any modern logical system...
 Anti-Duhring, chapter 11.