The Laws of the Laws of Thought

“History continually effects totalisations of totalisations” — Sartre, Critique of Dialectical Reason

One of the blogs I’m currently very interested in is Metalogic is Ethics, run by a graduate student in Philadelphia. John and I agree about the importance of formal concerns to “Continental” issues, and we are both thankful for the liberalizing influence of Badiouianism on that interface without quite having the grateful consciousness of disciples. Something we’ve discussed is the significance of second-order logic for considering dialectics: although I doubt anyone ever completely agrees with what I say, hopefully this work-up of my position on that topic will mark out an area broad enough to be occupied by a group larger than a party of one.

To put it mildly, formal logicians are not Hegel fans; going back to Russell’s turn away from British Idealism, formal logic has been informally defined as everything Hegel’s “logic” was not. The closest any formal thinkers have gotten to appropriating Hegelian themes is “dialetheism”, the Australasian philosophical movement which holds that paraconsistent logics (which have rules for reasoning with contradictions that are more sophisticated than the traditional “principle of explosion”) demonstrate that it’s coherent to believe there are real contradictions, “contradictions in the object” as a traditional dialectician might say. People like Graham Priest have mentioned Hegel in connection with this project, as well they might; but I think the real story of Hegel and logic is a little bit more complicated than simply accepting dialetheism. The story begins, as well it might, with Plato.

I’m no Plato scholar, but I imagine it’d be an uncontroversial observation that Platonic dialogues operate in this fashion: Socrates gets one of his interlocutors to produce a description of an Idea, and then they collectively reason about the consequences of that Idea for reasoning with Ideas generally, and the consequences of reasoning with Ideas generally for the employment of that Idea. This is clearly a “second-order” process of reasoning, but those less familiar with formal logic may not know there’s no need to leave “second-order” as an inexact descriptor: there is “second-order” logic. First-order logic allows the reasoner to quantify over objects in the universe of discourse, which produces universal and existential statements about the application of predicates to those objects: second-order logic allows one to quantify over those predicates, producing universal and existential statements about predication in general.

Sounds great, huh? In fact, using second-order logic one can describe all mathematical concepts without resorting to set-theoretic axioms, as Frege did with his second-order logic, his “laws of the laws of nature”. Or at least you could, if that didn’t produce paradoxes like Russell’s “set of all barbers that shave themselves”. Some people have recently tried to salvage Frege’s logicism from the paradoxes (by restricting his Basic Law V), but that’s not quite what I want to talk about here — although his mathematical “platonism” may shed some light on the original article, he was certainly no dialectician. No, what I aim to talk about is the relationship between Platonic and Hegelian dialectics in light of second-order considerations.

Between Plato and Hegel, we have Kant’s “Transcendental Dialectic”, his logic of metaphysical illusion. Unlike the understanding, which operates by subsuming intuitions under concepts (much as constants are included in the extension of predicates), Kant’s Reason works with Ideas (concepts involving totality, the unconditioned, and the perfect) and gets entangled in antinomies and contradictions on account of their character. I guess you could anachronistically characterize Kant as a Quinean of sorts, interested in restricting theoretical cognition to “first-order” concepts of the understanding, and I think that would not be an unreasonable way to gloss the influence of modern science on modern philosophy which culminated in his work.

Hegel accepts the results of Newtonian physics, and the constraints of experimental method on philosophy of nature: but unlike Kant he held no truck with skepticism, and wanted a modern version of Plato’s productive dialectic. Consequently, Hegel returned to the second-order, and his dialectic is much more nearly a process of moving back and forth between orders of abstraction than cookie-cutter application of a “thesis-antithesis-synthesis” schema. That contradictions seem to occur in this process is perhaps an indication of the fact that this second-order reasoning falls prey to the affliction of any formal system powerful enough to generate the principles of mathematics, incompleteness: the interrelation between dialectically demonstrated truths cannot be systematized axiomatically.

Maybe that’s tendentious — but there’s an interesting feature of second-order logic which I think is quite illuminating for considering Hegel and Hegelianism. “Standard semantics” for second-order logic is incomplete, but the logician Leon Henkin (who developed the proof of first-order completeness which is commonly taught today) devised a “Henkin semantics” for second-order logic which is complete. Henkin semantics only allows second-order statements over defined first-order totalities: this is similar to the restriction in the Zermelo-Frankel “axiom of comprehension” which replaced Frege’s unrestricted law of comprehension. If we consider Hegel as second-order, perhaps Henkin semantics is a good “model” of Left Hegelianism (where dialectic reasoning is preserved but only in reference to “real abstractions”, concepts that are incarnated by material realities); Right Hegelianism preserves the full expressive power of second-order dialectic (Henkin second-order logic is no more expressive than first-order logic), but at the cost of rational cogency and lack of “mystification”.

I’m definitely floating with the universe by the end of this line of thought, but there is a more recent “dialectical” concept that reminds me very much of another thought I independently had about two approaches to logic and geo-linguistic correspondences to them. One of Sartre’s major concepts in his Critique of Dialectical Reason is the “practico-inert”, elements of social organization which are resistant to the subjective projects of praxis and form the ground of social struggle. Now, in 1970s logic a distinction was made between “Western model theory” and “Eastern model theory”; the former being exemplified by the work of Alfred Tarski and his students at Berkeley, the latter being exemplified by the work of Abraham Robinson and his students at Yale.

In pure logic there’s not much heavy weather to be made over this distinction; both Tarski and Robinson were from Central Europe (by way of Palestine and Britain in Robinson’s case), and Robinson had quite happily taught at UCLA. However, I think the distinction is not without its geographical aptness. Western model theory was more heavily “logical”, and focused on the significance of models for definitions of logical consequence and other “abstract” features of logic. Eastern model theory was more heavily “mathematical”, and focused on the significance of models for proving things about mathematical theories and other “concrete” formal phenomena. This parallels a distinction in discursive styles between the western and eastern US. In the West, people have traditionally been quite fond of solecism and sophistry as devices for getting points across indirectly, whereas in the East there’s more of a focus on ineliminable realities bound up with noble sentiment: a Western raconteur might be “temporarily disabled” by a stunning woman, whereas an Easterner might be discomfited by the plight of personally aggressive people with “disabilities”.

It seems to me that this “Eastern model theory” seems to capture the presence in language of the practico-inert which Sartre touches on at one point, the crude and tasteless plays on words you just can’t get away from, low “interpretations” of signifiers which distract us from drawing the appropriate conclusions. But lest this seem to be mere provincial boastfulness, I will mention that the reason people don’t use this distinction in logic anymore is that all new work in model theory today is “Eastern” model theory, leading to Wilfrid Hodges’ redefinition of the subject as “universal algebra minus fields” rather than Chang and Keisler’s “universal algebra plus logic”. And, like Sartre says, as undesirable as many aspects of the practico-inert are from the standpoint of revolutionary subjectivity it’s a fundamental existentiale of sociality which you can’t get away from.