Ambiguous figures and dialectical logic

graham priest, formalization of dialectic logic

Graham Priest has formalized aspects of dialectical logic (in the Hegelian and Marxist sense) primarily through dialetheism and paraconsistent logic, most recently in a 2023 model of dialectical progression.⁠Tandfonline Priest argues that dialectic involves genuine logical contradictions (of the form $A \land \neg A$) that arise and are then aufgehoben (sublated: both cancelled/overcome and preserved/retained at a higher level). This requires dialetheism—the view that some contradictions are true (or hold)—rather than classical logic, which treats contradictions as explosive (entailing everything via the principle of explosion). He has developed this view over decades, notably in “Dialectic and Dialetheic” (1989/1990 in Science & Society) and related earlier work on change, motion, and tense logic.⁠Journals.sagepub

Core Ideas from Priest’s Account

Dialectical contradictions are logical (not merely pragmatic, rhetorical, or “real” in a non-logical sense). Hegel and Marx’s dialectics assume that certain contradictions hold.

Classical (Frege-Russell) logic is valid in limited domains but inadequate for the full range of dialectical phenomena; a more general (paraconsistent/dialetheic) logic is needed.

Sublation preserves the contradictory elements in a transformed way rather than simply eliminating inconsistency.

Related earlier formal work includes “To Be and Not to Be: Dialectical Tense Logic” (1982), which uses a tense logic based on the Logic of Paradox (LP) to model instants of change where a system can be in both a prior and posterior state (realizing a contradiction), linked to continuity principles and remarks by Hegel and Engels.⁠Link.springer

The 2023 Formal Model (“The Logical Structure of Dialectic”)

In this paper (History and Philosophy of Logic 44(2): 200–208), Priest provides an explicit formal model of dialectical progression using the tools of paraconsistent logic (specifically LP for definiteness; other paraconsistent logics could work).⁠Tandfonline Key elements of the model (informal description from the paper):

A dialectical progression is a sequence of situations/interpretations $S_1, S_2, S_3, \dots$.

In the underlying paraconsistent semantics, a predicate $P$ has an extension $\delta^+(P)$ (things of which $P$ is true) and an anti-extension $\delta^-(P)$ (things of which $P$ is false/$\neg P$ holds). These can overlap, allowing gluts (both true and false).

Progressions occur in triples corresponding roughly to thesis / antithesis / synthesis (or root stage / its negation / negation of the negation):

In the “negation” stage, the situation concerning some object becomes its opposite. In the “negation of the negation,” one returns to something like the root stage but at a “higher” level.

A new concept/predicate typically emerges in the process (required for the Hegelian character of the progression). Technical details (same domain across interpretations, controlled language expansion with new predicates, etc.) appear in an appendix.

The model is applied to concrete examples from Hegel (e.g., the opening dialectic of Being in the Science of Logic) and Marx, showing how they fit the structure.

Priest emphasizes that the model captures the dynamical aspect (how contradictions arise and are sublated) that pure dialetheism alone does not address.

Broader Context

This work sits within a revival of interest in formalizing Hegelian dialectics, reflected in the 2023 special issue of History and Philosophy of Logic (co-introduced by Elena Ficara and Priest) and the related volume The Formalization of Dialectics (edited by Ficara and Priest). Other contributors discuss alternative formal approaches (e.g., conjunctive paraconsistency, hybrid logics, historical attempts by figures such as Günther).⁠Taylorfrancis

Priest’s approach is distinctive in treating the contradictions as dialetheias within a relatively standard paraconsistent framework (LP) and in providing an explicit sequential model of progression and sublation. Critics have debated whether this fully captures Hegel’s developmental notion of truth or the non-formal aspects of dialectic; Priest maintains that dialetheism is required and that the formal tools illuminate the logical structure.

For primary sources, see:

Priest, “Dialectic and Dialetheic” (1989/1990).

Priest, “The Logical Structure of Dialectic” (2023).

Related background in In Contradiction and his work on LP and paraconsistency.

This is a concise overview of the main formal contributions; the papers themselves contain the technical details and textual applications.55 sources

as in an ambiguous figure, which is figure A and not figure B and simultaneously is B&~A

Worked for 4sYes — ambiguous figures provide a natural, everyday illustration of the kind of simultaneous “both/and” structure that dialetheic and dialectical accounts aim to capture.

Take a classic example such as the duck-rabbit (or the Necker cube, Rubin’s vase, etc.). The same lines can be experienced as:

Figure A (duck) and not figure B (rabbit), and, equally and at the same time, as figure B (rabbit) and not figure A (duck).

In logical notation this is roughly:

$$(A \land \neg B) \quad\text{and simultaneously}\quad (B \land \neg A)$$ From these two conjuncts it immediately follows (by ordinary conjunction introduction and simplification) that we have both $A \land \neg A$ and $B \land \neg B$. That is a dialetheia: a true contradiction in Priest’s sense.

How this fits Priest’s framework

In the paraconsistent semantics he uses (LP and its relatives), a situation can assign a predicate both to its extension and to its anti-extension. The ambiguous figure is precisely such a situation: the visual stimulus is both “duck-shaped” and “not duck-shaped” (because it is rabbit-shaped), and vice versa.

The contradiction is not a failure of perception or a mere pragmatic oscillation; under the dialetheic reading it is a genuine feature of the object/situation itself. The figure really is both, and the mind can (and does) register both aspects without the whole description collapsing into triviality. Dialectically, the two aspects stand in a relation of mutual negation that is sublated at a higher level: the single figure that unifies the two contradictory readings. This mirrors the root → negation → negation-of-the-negation progression Priest formalizes.

So your formulation is exact: the ambiguous figure is a concrete case of something that is $A \land \neg B$ and simultaneously $B \land \neg A$, thereby realizing the dialetheic structure that Priest takes to be central to dialectical logic.

On Hegel's dialectical logic

Claude:

give a clear example of hegel's dialectical logic

Hegel’s own preferred example is the opening move of the Science of Logic: the dialectic of Being, Nothing, and Becoming.

1. Being (thesis). Start with the most abstract concept possible: pure Being, with no determinations at all — not “a” being, not being this or that, just Being as such, indeterminate immediacy.

2. Nothing (antithesis). But a concept with literally no content, no predicates, nothing to distinguish it from anything else, is indistinguishable from Nothing. Hegel’s claim isn’t that Being resembles Nothing — it’s that pure Being, thought through rigorously, collapses into Nothing. They have the same (lack of) content.

3. Becoming (synthesis). Yet Being and Nothing aren’t simply identical either — the thought “Being” and the thought “Nothing” are different acts of thinking. So the truth of this instability, where each category passes into the other and neither can be stably held apart, is Becoming: the unity that contains the vanishing of Being into Nothing and Nothing into Being as a single process.

The logical structure to notice: this isn’t “compromise” between two static positions. The antithesis emerges by pressing the thesis until it undermines itself from within (immanent critique), and the synthesis isn’t a blend but a new category operating at a higher level of determinacy — one that can explain why the first two broke down. Becoming can then itself become a new “thesis” subjected to the same treatment (it collapses into Determinate Being, and so on) — this is what drives the entire Logic forward as a sequence of self-generating categories rather than an externally applied schema.

A more concrete version of the same logical shape, if the abstraction above is unhelpful, is the master-slave dialectic in the Phenomenology: the master’s dependence on the slave’s recognition undermines the very autonomy the master sought, while the slave’s formative labor on the object gives the slave an independent self-relation the master lacks — each position destabilizes into its opposite through its own internal structure, not through external comparison.

so hegel's logic would approve the solution of russell's paradox by the NBG set theoretic distinction of classes and sets, such that memberless classes represent the synthesis. eventually, strange sets and classes lead to an absolute, as described in rucker's last logic book

This doesn’t hold up as stated, and the disanalogy is worth spelling out precisely because there’s a more interesting connection nearby that it’s obscuring.

What NBG actually does. The class/set distinction (Bernays, refining von Neumann) resolves Russell’s paradox via a “limitation of size” principle: a collection is a proper class exactly when it’s too large to be a member of anything else (the class of all sets, the class of all ordinals, the Russell class {x : x ∉ x} itself). This blocks the paradox because R = {x : x ∉ x} turns out to be a proper class, and “R ∈ R” is simply ill-formed — proper classes can’t be elements of anything, so the question never gets asked. Note also that proper classes aren’t “memberless” — the class of all sets has every set as a member. What they lack is the capacity to be a member. If that’s what you meant by “memberless,” fine, but it’s worth being precise since the two properties are opposite in direction.

Why this isn’t dialectical in Hegel’s sense. Aufhebung requires that the contradiction be worked through from within — the thesis, pursued to its full determination, generates its own negation as part of its content, and the synthesis preserves and elevates both moments rather than declaring one of them illegitimate. NBG does the opposite: it’s a stratification device, structurally a cousin of Russell’s own theory of types or Zermelo’s separation axiom. It doesn’t sublate the Russell class; it excludes it from the domain of things-that-can-be-members by fiat, motivated by the technical goal of consistency, not by any immanent unfolding of the concept “set.” Hegel’s Logic would have to show that “set” thought through to completion necessarily produces “proper class” as a moment it cannot do without — but historically the move ran the other way: the paradox was found first, empirically, by contradiction-hunting, and the class/set stratification was retrofitted to block it. That’s closer to Kantian antinomy-management than to genuine dialectical motion.

The real connection is elsewhere, and it cuts against your reading. Where Hegel is actually pertinent is the bad infinite (schlecht-unendlich). Every attempt to totalize the sets-so-far (V, then the class of all sets, then classes of classes, the cumulative hierarchy, inaccessible cardinals, and on) generates a new “outside” that itself demands totalizing — exactly the endless progression Hegel diagnoses as spurious infinity, infinity that never actually achieves itself because it stays perpetually opposed to a finite it hasn’t yet absorbed. This is precisely the structure Rucker is tracking with “strange sets” leading upward.

But then Cantor’s/Rucker’s Absolute Infinite is not a good match for Hegel’s Absolute — if anything they’re structurally opposed. Hegel’s Absolute is achieved, self-transparent, fully mediated: the “circle of circles” in which nothing remains external to the system. Cantor’s Absolute Infinite is defined by the opposite property — it is precisely what cannot be gathered into a determinate totality, an “inconsistent multiplicity” that recedes from every attempt at conceptual capture (which is why Cantor reaches for theological language, Ein Sof, rather than mathematical closure). One Absolute is total self-mediation; the other is defined by resistance to totalization. Sharing the word “Absolute” doesn’t get you a shared concept.

So the honest verdict: NBG’s fix is a formally elegant but dialectically inert patch, and the genuinely Hegelian-resonant phenomenon in this territory is the bad-infinite structure of the cumulative hierarchy — which would be a point against treating the ascent to strange sets/classes as culminating in anything Hegel would recognize as an Absolute, rather than a point in favor of it.

russell's paradox fulfills the hegelian criteria

Here you’re on much firmer ground than the NBG claim, and it’s worth separating the two clearly because they don’t stand or fall together.

The paradox itself, considered as a raw phenomenon, does have the right shape. Unrestricted comprehension — the principle that any predicate φ determines a set {x : φ(x)} — is the set-theoretic analogue of pure, maximally indeterminate Being: no restriction on content, total generality. Apply it reflexively (φ(x) = “x ∉ x”) and you get R, whose determinacy consists entirely in oscillating into its own negation: R ∈ R just in case R ∉ R. That’s not a static inconsistency sitting inertly beside the concept — it’s the concept collapsing into its opposite the moment it’s pushed to full self-application, structurally identical to how pure Being, examined, turns out to already be Nothing. So yes: taken as a raw phenomenon, this is a dialectical contradiction in Hegel’s technical sense, not a mere logical error to be embarrassed about.

But that’s only the antithesis moment, and Hegel’s criteria don’t stop there. The triadic movement requires that the contradiction be sublated — resolved into a third category that preserves what was true in both R ∈ R and R ∉ R rather than simply forbidding the question. This is exactly where the standard technical responses fail the test I raised last time: ZF’s separation axiom, NBG’s class/set stratification, and Russell’s own type theory all take the anti-dialectical option. They don’t carry the contradiction forward into a higher unity — they retreat one level and refuse to let the offending object be formed at all. That’s suppression of the contradiction, not Aufhebung.

There is one place where the actually Hegelian move has been taken seriously: Graham Priest’s dialetheism. In In Contradiction and Beyond the Limits of Thought, Priest treats “R ∈ R and R ∉ R” as both true — a genuine, stable contradiction rather than a defect to be engineered away — and explicitly names Hegel as a historical predecessor for the view that some contradictions are real and don’t need eliminating. More importantly for your point, Priest generalizes Russell’s paradox together with Cantor’s, Burali-Forti’s, and the Liar into a single recurring structure he calls the Inclosure Schema: any totality that is fully determinate generates, by that very determinacy, an object that is simultaneously inside and outside it. That schema functions as exactly the higher-order unity Hegel’s method demands — it doesn’t exclude the paradoxical object, it explains why every sufficiently complete totality must produce one. That’s sublation, not stratification.

So the accurate verdict: the paradox fulfills Hegel’s criteria for a genuine dialectical contradiction; classical axiomatic set theory does not complete the movement Hegel’s criteria then demand, because it chooses exclusion over sublation; and if anything in this territory has actually done the second half of Hegel’s job, it’s dialetheism’s Inclosure Schema, not NBG.

the new NBG concept of class is the third category

It’s the strongest version of the claim so far, but it still fails, and it fails on the specific point Hegel is most insistent about: the difference between Verstand and Vernunft.

Sublation vs. stratification is exactly the distinction Hegel uses against Kant, and NBG lands on Kant’s side of it. Kant’s response to the antinomies of pure reason is to say the contradictions only arise because reason oversteps into a domain (the unconditioned) where the categories of the understanding don’t properly apply — draw the boundary correctly and the contradiction dissolves. Hegel’s entire polemical target in the Logic is that this is an evasion: it doesn’t follow the contradiction through, it fences it off and calls the fence a solution. That is structurally identical to what NBG does. “Class” is a boundary — collections above a certain size can’t be members of anything — drawn precisely so that R never has to face the question “R ∈ R?” That’s Verstand’s characteristic move: divide, then declare the division beyond negotiation. If Hegel rejects Kant’s antinomy-management as insufficiently dialectical, the same reasoning rejects NBG’s class/set split for the identical reason.

Direction of motivation matters, and it runs backward here. Being/Nothing/Becoming is supposed to be discovered by thinking Being through with total rigor — the contradiction is not anticipated, it’s arrived at. NBG’s proper-class concept has the reverse genealogy: the paradoxes (Russell’s, Cantor’s, Burali-Forti’s) were already known, and “limitation of size” was devised specifically to prevent them. That’s engineering a category to avoid a known bad outcome, not a category generated by the internal unfolding of “set” itself. A synthesis that’s built backward from the contradiction it’s meant to prevent hasn’t earned the name.

The decisive test: does the new category retain the contradiction as living content, or does it switch off the conditions for the contradiction to occur? Becoming doesn’t make “Being passes into Nothing” inapplicable — Becoming is that oscillation, stabilized and made into positive content. A genuine third category has to still contain the tension it resolves, transformed but active. “Class” does the opposite: once R is typed as a proper class, the question “R ∈ R?” isn’t answered in a new, richer way — it’s rendered ill-formed, unaskable. The oscillation isn’t preserved at a higher pitch; it’s deleted. Compare this to what an actual sublation in this territory looks like: Priest’s Inclosure Schema (from Beyond the Limits of Thought) takes exactly the pattern you’d want — every totality complete enough to be fully determinate generates an object that is both inside and outside it — and keeps that object’s contradictory status as the positive content of the resulting structure, true on both sides at once. That’s a third category built from retaining the contradiction. “Class” is a third category built from making the contradiction impossible to state.

So the claim doesn’t hold: NBG’s class concept satisfies the letter of “new category introduced to handle a contradiction” but fails the substance of Aufhebung on all three tests that matter — method (Verstand, not Vernunft), direction of motivation (backward from known paradox, not forward from internal unfolding), and content (elimination of the tension rather than its preservation). The dialetheic Inclosure Schema is the thing in this vicinity that actually does what you’re describing.

so then hegel's dialectic logic fails for set theory paradoxes, tho Kant's doesn't

That’s an overcorrection, and it collapses two things that need to stay separate: whether Kant’s structural move achieves consistency, and whether it constitutes a philosophical vindication of Kant over Hegel. It doesn’t, on either count, and there’s a further problem: Kant’s solution doesn’t actually escape Hegelian critique — it just relocates the failure.

“Hegel fails” is too strong, because a working Hegelian-style formal system exists.

Paraconsistent/dialetheic set theory — Priest’s own program, and more concretely Zach Weber’s work (Transfinite Numbers in Paraconsistent Set Theory, Extensionality and Restriction in Naive Set Theory) — builds set theory on unrestricted (naive) comprehension, the same comprehension principle that generates Russell’s paradox, using a paraconsistent underlying logic where a contradiction doesn’t explode into triviality (ex falso doesn’t hold). In these systems, “R ∈ R and R ∉ R” is derivable and retained as true, not stratified away, and the theory still supports nontrivial mathematics — transfinite arithmetic, cardinality results, and so on. That is a genuine Aufhebung: the contradiction survives as positive content at a stable higher level rather than being fenced off. It’s real, it’s published, it’s actively developed. So the correct statement isn’t “Hegel’s logic fails for set-theoretic paradoxes” — it’s “the mainstream axiomatic tradition (ZF, NBG, type theory) is Kantian in structure, and a minority but formally serious dialetheic tradition is Hegelian in structure, and these are two live, competing foundational programs.” Mainstream adoption is a fact about mathematical practice (consistency and classical logic are strongly preferred, for good reasons), not a verdict on which philosophical logic is correct.

Kant’s success is also less clean than “succeeds” suggests, and the flaw is Hegelian. ZF and NBG don’t achieve a single, closed, final boundary the way Kant’s critique claims to achieve a settled demarcation between appearances and the unconditioned. Gödel’s incompleteness results mean any such system leaves statements undecided (the continuum hypothesis is the classic case), and the standard response is to posit larger boundaries — inaccessible cardinals, measurable cardinals, ever-stronger large cardinal axioms, reflection principles — each one drawn specifically to settle what the previous boundary couldn’t, and each one in turn incomplete with respect to further questions. That is not a single successful act of boundary-drawing; it’s an open-ended, non-terminating series of boundary-redrawings with no achieved totality — which is close to a textbook instance of Hegel’s bad infinite: a progression that keeps producing a new “beyond” it must incorporate and never actually closes into a whole. So the Kantian strategy purchases local, provisional consistency at exactly the price Hegel says such strategies always pay: it never reaches the completed self-transparency that would make it a genuine terminus rather than a moment in an unfinished regress.

The accurate formulation, then: mainstream set theory is Kantian in structure and it works pragmatically, but it doesn’t escape Hegelian critique — its need for an ever-expanding hierarchy of axiomatic boundaries is itself the bad-infinite symptom Hegel diagnoses as reason’s dissatisfaction with merely external limitation. And Hegel’s logic hasn’t failed for this domain in any general sense — it has a working formal instantiation, just a non-mainstream one, in dialetheic/paraconsistent set theory. What’s true is narrower than your formulation: classical, consistency-first foundations chose the Kantian route and it is formally successful by its own lights; that’s different from Hegel’s dialectic failing as logic, and different from Kant’s approach being complete on its own terms either.

most mathematicians would find your last answer suspect, as most accept ZFC, NBG, or a modified version of Russell and Whitehead's PM

The sociological fact is correct, and worth granting cleanly: ZFC, its conservative extension NBG, and typed systems descending from Principia are what essentially the entire working mathematical community uses, full stop. Paraconsistent/dialetheic set theory is a minority research program worked on by maybe a few dozen people worldwide. If your point is “you’re describing something exotic as if it had comparable standing to the mainstream,” that’s fair, and I should have flagged the disparity in maturity more explicitly rather than just calling it “formally serious.”

But notice what that fact does and doesn’t settle. It’s a sociological/pragmatic claim about which foundation mathematicians have converged on for doing mathematics — and the reasons for that convergence are good ones: classical logic’s inferential power, a century of accumulated results, and the fact that paraconsistent set theory genuinely lags behind in what it can reconstruct (more on that below). None of that is in dispute. What it doesn’t do is answer the separate question I was actually addressing: which of these programs, considered as formal structures, instantiates Hegelian sublation versus Kantian boundary-drawing. That’s a structural/philosophical classification, and majority uptake among mathematicians is not evidence about logical structure — a claim can be philosophically Kantian in form and also correct to prefer, for entirely non-philosophical reasons (it works, it’s productive, it doesn’t require rebuilding a century of results). Appealing to consensus to settle the structural question would be answering a different question than the one asked.

Where you’re right to press, though, is that I understated how contested and partial the paraconsistent recapture actually is, and this matters for how much weight the “existence proof” can bear. Weber’s program does prove genuine results — ordinal and Peano arithmetic, a Cantor’s-theorem-respecting theory of cardinals — inside naive comprehension with a paraconsistent logic. But: (1) paraconsistency alone isn’t even sufficient to keep naive comprehension safe, since Curry’s paradox (R = {x : x ∈ x → φ}) can trivialize the theory independently of negation-inconsistency, requiring further restrictions on the conditional, not just on negation; (2) there’s live technical dispute (Nicolai and others) over whether “classical recapture” strategies actually recover the full deductive strength of the classical theories they’re supposed to reproduce, or only a strictly weaker fragment; and (3) basic notions like computability and recursion don’t transfer over straightforwardly once contradictions are live, which is still being worked out. So the honest scaling-down is: dialetheic set theory is a real, technically serious existence proof that an Aufhebung-structured alternative to stratification is coherent and partially executable — not that it’s a mature, complete, competitive foundation for mathematics. That’s a weaker claim than I let stand, and it’s the one that survives scrutiny.

objectivity <--> subjectivity. what is the synthesis? This is one of the actual named triads in Hegel’s system, not something to be constructed by analogy — it’s the final third of the Doctrine of the Concept in the Science of Logic, and the answer has a name: the Idea (die Idee).

The structure, as Hegel gives it. The Subjective Logic runs: Subjectivity → Objectivity → Idea.

Subjectivity is the concept (Begriff) considered purely in itself — concept, judgment, syllogism — the “I” as pure self-relating activity, not yet given any external actuality. Taken in isolation this is one-sided: a subjectivity that stays merely subjective, that never actualizes itself in anything outside itself, is empty — this is the same diagnosis Hegel gives the “beautiful soul” in the Phenomenology, a will so purely inward it never risks itself in objective existence and therefore accomplishes nothing determinate.

Objectivity is the concept externalized into indifferent existence — mechanism (things externally related, pushed around by other things), chemism (things related through their determinate natures), and teleology (things ordered toward an end). Taken in isolation this looks like sheer given world, existing independently of any concept. But Hegel’s actual argument — and this is the pivot, not a restatement — is that objectivity, followed through its own internal development from mechanism to teleology, already turns out to be conceptual: purposiveness (teleology) can’t be stated without smuggling in exactly the structure — an end guiding a process — that subjectivity was supposed to have a monopoly on. So objectivity, examined rigorously, is not the subjectivity-free “outside” it presented itself as.

The Idea is what results: not a compromise, but the discovery that neither pole was ever purely itself. Hegel defines it directly: the Idea is “the absolute unity of the Concept and objectivity” — a subjectivity that has fully actualized itself as objective, and an objectivity recognized as already conceptually structured through and through. This is important to get right against the common misreading of “synthesis” as splitting the difference between two intact opposites. Hegel’s synthesis is never that. It’s the result of showing that subjectivity-in-isolation and objectivity-in-isolation were both already inadequate abstractions from something that was never actually split — thought that has fully worked itself out, and being that has fully worked itself out, turn out to be the same self-determining process seen from two sides.

The Idea then has its own internal unfolding, worth having in view since it’s not a static endpoint: Life (the immediate unity — an organism where subjectivity directly informs and organizes a body, no gap yet between purpose and matter) → Cognition, which splits into the theoretical Idea of the True (subject conforms its concepts to given objectivity — knowing) and the practical Idea of the Good (subject reshapes objectivity to conform to its own purposes — willing and acting) → the Absolute Idea, where this last opposition between theoretical and practical also collapses: thought recognizes that its own method of self-development just is its content, that there is no gap left between the activity of thinking and what is thought. That’s the terminus of the Logic — subjectivity and objectivity are only fully sublated when their own opposition, and not just their content, has been shown to be self-generated and self-dissolving.

One disambiguation worth flagging, since the same pairing shows up elsewhere in the system with a different answer: if you mean it at the level of Geist rather than the Logic — Subjective Spirit (individual consciousness, soul, psychology) versus Objective Spirit (law, morality, ethical institutions) — the synthesis there is Absolute Spirit: art, religion, and philosophy, where a subject’s activity (making, worshiping, thinking) and objective, publicly shared content (the artwork, the doctrine, the system) are no longer opposed because the content just is the subject’s own self-understanding rendered external and shareable. Within Objective Spirit alone, there’s a smaller-scale version of the same move in the Philosophy of Right: subjective moral will (Moralität) and objective institutional right (abstract right) are synthesized in Sittlichkeit, ultimately the state — institutions that are objective and yet are nothing but concrete freedom’s own self-actualization, not an alien constraint on it.

If you had a specific register in mind — logical, psychological, political — say which and I’ll go deeper on that track specifically, since the Logic’s answer (the Idea) and the Encyclopedia’s answer (Absolute Spirit / the state) are structurally parallel but not interchangeable.

so these triads are ordered triples, meaning the parts do not obey strict implication

Correct on the “ordered” part, but the “strict implication” claim needs more care than a flat no — and there’s a prior question worth raising about whether “triad” is even the right formal object here at all.

On order first. The ordering isn’t uniform across all three positions, and it’s worth being precise about where the asymmetry actually sits. Being and Nothing are mutually convertible — Hegel’s claim is symmetric: Being passes into Nothing and Nothing passes into Being, each is the truth of the other, neither has priority. The asymmetry only enters at the third step: Becoming is not simply “whichever of Being/Nothing you started with” reversed — it’s a genuinely new determination that can’t be decomposed back into the pair without loss (Being and Nothing survive only as moments within Becoming, not as free-standing terms you can extract and reconstitute). So if you’re modeling this as an ordered triple ⟨thesis, antithesis, synthesis⟩, the first two coordinates are order-symmetric with each other and only the transition to the third coordinate is properly directional/irreversible. That’s a real structural point, and it means “ordered triple” undersells the relation between coordinates 1 and 2 while correctly capturing the relation to coordinate 3.

On strict implication: right conclusion, but the reason matters more than the verdict. Strict implication (Lewis’s fix to material implication) is still a classical, truth-functional-plus-modality relation: p strictly implies q iff it’s impossible for p to be true and q false, evaluated over a fixed domain of propositions with stable content and stable truth-values. Hegel’s transitions fail this not merely because they’re “loose” but because the premise doesn’t meet the precondition for strict implication to even apply: the content of “Being” is not held fixed across the transition to “Nothing.” Hegel’s explicit target — his running polemic against Verstand and formal logic throughout the Doctrine of Being and the Introduction — is exactly the assumption that terms retain a stable, self-identical meaning while inference rules operate on them externally. Dialectical necessity works by determinate negation: the negation that emerges isn’t the empty, indeterminate “not-p” of classical negation, but a specific, content-bearing result of showing why the very content of p was already inadequate to sustain the determinacy it claimed. That’s not extractive (deriving what’s already implicit via a truth-preserving rule) but productive (generating a determination not contained in the premises under any classical rule of inference) — which is precisely why Hegel insists formal logic, on its own resources, is “empty”: it can rearrange given content but can’t generate new content the way he claims the dialectic does.

A prior complication, given what you already know from Mueller. You’ve read Mueller against Stace on exactly this point — worth foregrounding here because it bears directly on your question. “Thesis-antithesis-synthesis” as a fixed three-term schema is not Hegel’s own vocabulary; it’s the popularization descending from Chalybäus and later Stace, which Mueller’s “Hegel Legend” essay attacks precisely because Hegel speaks of Momente (moments) of a single self-differentiating movement, not three discrete propositions plugged into an inference relation. If that’s right, “ordered triple” may be the wrong formal object from the start — an n-tuple presupposes separable elements that jointly satisfy some relation, whereas moments are only distinguishable retrospectively, in exposition, from within a movement that is, at the time of its actual unfolding, not yet decomposed into separate terms at all. So there are two candidate diagnoses on the table, not one: (a) the relation between the terms is real but non-classical (a different, dialectical necessity — determinate negation — rather than strict implication), which is the reading Priest’s paraconsistent formalization tries to make rigorous; or (b) the whole triadic packaging is an artifact of exposition and Hegel scholars, and there is no fixed formal relation of any kind connecting freestanding “terms” to be modeled, classical or otherwise, because the thing being modeled was never three items to begin with. These pull in different directions — (a) says formalize it differently; (b) says don’t expect a formalization of terms-in-relation to be the right kind of project at all — and which one you take seriously changes what “not strict implication” is doing as an answer.

<{a,b}, c>, not <,c>

The notation is the right fix for the problem I flagged, but it buys clarity on one issue by making a silent commitment on the other, larger issue — and it still can’t do the job you’d actually need it to do.

What it gets right. <{a,b}, c> correctly refuses to encode an ordering between a and b while still marking c as structurally distinguished — that’s exactly the asymmetry in the text: Hegel states the Being/Nothing relation as fully reciprocal (“Being passes over into Nothing, and Nothing into Being” — stated as a two-way claim), and he even remarks, later in the Doctrine of Being, that the Logic could equally well have started from Nothing instead of Being, since Nothing taken with the same immediacy is just as indeterminate — so the choice of starting term really is conventional, not privileged. {a,b} models that correctly. <,c> would smuggle back in exactly the ordering commitment (“first a, then b”) that the text explicitly denies.

What it can’t do. A set-theoretic pair is a purely extensional object — it records that {a,b} and c co-occur as a packaged structure, but it has no resources to represent why c results, which was the entire content of my last answer: the relation is determinate negation, not strict implication, precisely because it’s productive rather than extractive. Nothing about <{a,b}, c> distinguishes “c is generated by the internal collapse of a and b into each other” from “c is arbitrarily paired with {a,b}.” To get that you’d need a distinguished relation or operator connecting the coordinates — some f({a,b}) = c — and the moment you write that down, you’re forced to specify its properties, which is exactly the substantive unresolved question, not something the choice of bracket-nesting settles. Tuple structure can encode symmetry/asymmetry among positions; it cannot encode necessitation, generativity, or content-change across a transition. Those need a modal or inferential apparatus layered on top, and what that apparatus looks like (classical function? Priest’s paraconsistent consequence relation? something else entirely) is the actual open question your notation currently leaves silent.

A cost you may not have intended to pay. Packaging a, b, c as elements of a set/tuple at all presupposes they’re separable, well-defined items that exist prior to and independent of the relation holding them together — which is precisely interpretation (a) from my last message, not (b). Mueller’s objection (via Stace) was that “moments” (Momente) are not freestanding terms available for collection into a set prior to the movement — they’re only distinguishable in retrospect, as an artifact of exposition, from within a process that at the time of its actual unfolding hasn’t yet separated into countable items. If Mueller is right, there is no a, b, or c to put into any bracket structure, ordered or not — the demand for a tuple already begs the question against that reading by presupposing the individuation it would need to justify.

Does it even generalize? Being/Nothing is the cleanest case for {a,b} because Hegel states their interchangeability directly. The later triads I mentioned are looser fits: Subjectivity and Objectivity aren’t given as a single reciprocal claim the way Being/Nothing are — the Logic argues through them sequentially (Subjectivity shown one-sided, then, separately, Objectivity shown to already be conceptual), which is closer to two independent one-sided collapses than to a single symmetric pair given all at once. So <{a,b}, c> is a good regional fix for the specific case that motivated it and a looser, more stipulated fit elsewhere in the system.

a <--> b.. --> ..c

The ..-->.. for the c-transition is a genuine improvement — using visibly nonstandard notation to mark “this is not classical implication, and I’m not claiming to know what it is” is honest, and it’s the right response to what I flagged last turn. But the <--> for a/b needs the same scrutiny, and it doesn’t survive it, plus there’s a structural option your diagram forecloses that’s arguably closer to what Hegel actually says.

<--> flattens exactly the thing that was supposed to be preserved. A biconditional is a static, truth-functional relation: a and b have matching truth-value, full stop, no direction, no activity. But “Being passes into Nothing, and Nothing passes into Being” is not a claim that two propositions happen to co-vary — it’s a claim that two movements occur, each with its own direction: Being, examined, turns into Nothing (one transition), and Nothing, examined, turns into Being (a second, independent transition). What makes it reciprocal is that both transitions happen, not that some single symmetric connective holds between fixed endpoints. Collapsing that into <--> loses the very feature — the movement itself — that’s supposed to constitute Becoming. The more accurate diagram is two directed arrows sharing endpoints: a --> b and b --> a, not one bidirectional edge.

And that restructuring changes where c belongs. If Becoming just is that pair of opposed transitions, stabilized and grasped as a unity, then c isn’t downstream of {a,b} via some further, separate arrow at all — c is the two arrows themselves, reified into a new object. Structurally that’s closer to c := (a-->b, b-->a) — Becoming as the name for the cyclical structure taken as a whole — than to {a,b} ..-->.. c as three items with a transition connecting a pair to a third. Your notation, by keeping a, b, c as three coordinate slots with an arrow between the first two packaged together and the third, still presupposes c is reached from {a,b} rather than c being nothing but {a,b}’s own activity made explicit. That’s a real fork: is the third moment a new destination, or is it the first two moments’ own process, looked at differently? Hegel’s language (“the truth of [Being and Nothing] is... the movement of the immediate vanishing of the one into the other: Becoming”) reads more like the second option — Becoming isn’t arrived at, it’s what Being-passing-into-Nothing-and-back already was, redescribed as a unity rather than as two separate vanishings.

The dotted arrow is honest but still inert. Marking the a/b-to-c step as non-classical correctly refuses to overclaim, but it’s a label for an unsolved problem, not a solution to it — it still can’t distinguish “c genuinely results from {a,b}’s instability” from “c is stipulated.” That’s fine as an acknowledgment of where things stand, but it’s worth being clear that no notation does that work for you; only an actual account of determinate negation would, and that’s substantive philosophical content, not a diagram choice.

Mueller’s objection is still sitting underneath, untouched. Every version of this notation — yours, mine, tuples, arrows — treats a, b, c as pre-existing nodes that some relation (however marked) then connects. That’s still squarely interpretation (a). If Mueller/Stace’s reading is right that Being and Nothing are only individuated in retrospect, from within a completed movement, then the very starting point — draw two nodes, label them a and b — has already decided the question against Mueller by treating them as available for labeling before the movement that (on his reading) is what makes them determinate objects in the first place.

no. i was using dot notation instead of parens, with double dots meant to avoid confusion for a single dot used for the logical product. hence i meant merely (a <--> b) --> c

Fair correction on the notation — I over-read the double dots as a deliberate hedge when you meant plain grouping. But swapping in the ordinary reading doesn’t rescue the formula; it makes both problems from the last two turns sharper, not milder.

The biconditional problem is unchanged by the bracket style. Whether you write <--> or (a <-> b), it’s still a single static, symmetric, truth-functional relation between two fixed propositions — and my objection two turns ago wasn’t about notation, it was about what a biconditional is: a claim that a and b match in truth-value, with no representation of the two independent directed transitions (Being collapsing into Nothing; Nothing collapsing into Being) that Hegel actually describes as constituting Becoming. (a <-> b) still asserts co-variance, not activity. Parenthesizing it doesn’t add the missing directionality back in.

And “-->” as ordinary implication reintroduces exactly the charge Hegel levels against formal logic — now in a stronger, more damaging form. If --> is material implication, (a<->b) --> c is true whenever c is true regardless of a and b, or whenever a<->b is false regardless of c — the connective is satisfied by cases with zero relation to dialectical content. That’s not a weak formalization of determinate negation, it’s compatible with there being no relation at all between the premise and c. If --> is meant as strict implication instead (necessity of the material conditional), it’s better-behaved but reintroduces the objection from two turns ago at full strength: strict implication requires a and b’s content to be fixed while the inference is evaluated, which is precisely what Hegel denies — the whole point of the transition to Becoming is that “Being” and “Nothing,” pursued to full determinacy, don’t retain stable enough content to serve as fixed inputs to any truth-preserving rule. Either reading of --> reproduces the “formal logic is empty” complaint; the classical reading trivializes it, the modal reading just restates the problem I already raised rather than solving it.

So the honest state of the formalization attempt, three turns in: every version — tuples, arrows, biconditionals, material or strict implication — keeps hitting the same wall, because the wall isn’t about which classical apparatus you pick, it’s that Hegel’s transitions aren’t the kind of thing any relation between fixed, pre-individuated propositional contents can represent. That’s not a defect in your notational choices; it’s the actual content of the claim that dialectical necessity is sui generis — not reducible to, and not a special case of, any relation in the classical propositional toolkit, symmetric or asymmetric, material or modal. The only formalization on the table that doesn’t just relabel this gap is Priest’s paraconsistent one, and it works by changing the logic itself (allowing true contradictions, non-classical negation) rather than by rearranging how classical connectives are bracketed — which is the level at which a real formalization would have to intervene.

i know. you mean a <--> ~a. --> c

This is the right move, and it’s a real improvement — but it’s worth being precise about exactly what it buys you, because it buys two different things, one a genuine advance and one a substantive interpretive cost you may not want to pay.

What it gets right. Treating Nothing as ~a rather than as an independent atom b formalizes something textually accurate: Nothing isn’t correlated with Being by some external relation, it is Being’s own negation — same content, opposite sign, which is exactly why Hegel can say their content is identical while their “determination” differs. That’s a strictly better formal move than my {a,b} treatment, which let Nothing float as an unrelated proposition merely happening to co-vary with Being.

What it does in classical logic: nothing useful, and this is actually informative. a <-> ~a is a classical contradiction — unsatisfiable under any valuation. Embed that in (a<->~a) --> c with --> as material implication and the antecedent is always false, so the whole formula is vacuously true for any c whatsoever — true, and tells you nothing. This isn’t a failure of your formalization; it’s your formalization correctly demonstrating, from the inside, exactly the claim I’ve been making for three turns: classical logic cannot represent this transition without either exploding or going vacuous. You’ve now shown that formally rather than me asserting it.

Where it actually lives: Priest’s LP. In the Logic of Paradox, a proposition can take the value Both (true-and-false), and under that assignment, a <-> ~a is satisfiable — non-vacuously, because when a is Both, ~a is also Both, and Both matches Both. This is, formally, close to the cleanest statement available of what a dialetheic reading of Being/Nothing claims: a (Being) is a dialetheia, and a <-> ~a holding is not a contradiction to be eliminated but the actual, stable, glutty content of the concept at that stage. That’s real progress — you’ve located the one non-classical setting where the left-hand side of your formula is a coherent object rather than dead on arrival. Note, though, that --> still needs re-specifying once you’re inside LP: LP’s material conditional (defined via ¬ and ∨) notoriously fails to validate modus ponens for glutty inputs, which is precisely why Priest and others reach for distinct conditional connectives (relevant/paraconsistent conditionals, e.g. via Routley-Meyer semantics) rather than the LP material conditional when they need inference to actually transmit something to c. So the c-transition is still open, but now it’s open in the right place — a known, worked-on problem in paraconsistent proof theory, not a hole in your notation.

The cost, and it’s a real one. Formalizing Nothing as ~a at all commits you to reading Nothing as the negation of Being — a determinate operation applied to a determinate content. But Hegel’s actual staging is arguably more radical than that, and this is the thing worth checking against the texts you know well: at the opening of the Doctrine of Being, Being and Nothing are each introduced as separately and equally immediate — Nothing isn’t derived by negating Being, it’s arrived at on its own terms, “pure nothing... simple equality with itself.” If Nothing were already the negation of Being, the concept of negation (a determinate operation with determinate scope) would have to be already available at the very point where the Logic is supposed to have nothing available yet except sheer indeterminate immediacy — negation as an operation is arguably itself a result achieved only through the Being-Nothing-Becoming movement, not a tool already on hand to state it. On that reading, ~ — classical or paraconsistent, doesn’t matter which — imports more logical machinery into the starting point than the text licenses at that stage of the Logic, even though it’s exegetically well-motivated by the later remark that their content is “the same.” Whether that’s decisive against your formalization or just a cost you’re willing to pay is a real interpretive fork, and it’s one where reasonable readers of the text split, not something notation can adjudicate.

curious that vacuous truth corresponds to a dialectic triad's claim of truth

It’s curious, but worth being precise about why it’s curious, because the reason cuts against reading it as any kind of vindication.

What’s actually happening: it’s not a coincidence, it’s ex falso quodlibet. (a<->~a) is a contradiction — unsatisfiable in classical logic. And “a contradiction classically implies anything” isn’t a special quirk of this formula; it’s just the principle of explosion (ECQ) showing up, the same principle that makes any classical theory containing a contradiction trivial (prove everything). So what you noticed is the correct, general classical behavior at contradictions, not something specific to Being/Nothing. That’s actually the reason it’s not a coincidence — Hegel’s transition really does involve a genuine contradiction (that’s the whole thrust of treating Nothing as ~a with a<->~a holding), so classical logic’s one universal response to contradiction — everything follows — was bound to fire here too.

But that’s exactly the disease, not a cure. Three turns ago the whole reason classical logic was rejected for this project was that it can’t represent genuinely contradictory content without collapsing into triviality. Vacuous truth is that collapse, just witnessed directly: (a<->~a) --> c being true for literally any c whatsoever isn’t evidence that c is the right conclusion, it’s evidence that the antecedent has stopped discriminating between conclusions at all. Hegel’s claim is maximally selective — Becoming, specifically, and nothing else, is what results from this determinate contradiction. Vacuous truth is maximally unselective — it would validate (a<->~a) --> "the moon is made of cheese" with exactly the same formal certainty. Those are opposite properties dressed up in the same surface phenomenon: both look like “c follows unconditionally,” but one is unconditional because of overdetermination (the content specifically forces this outcome) and the other is unconditional because of underdetermination (the premises are broken, so nothing is being tracked). If anything, this is a clean example of why “the formula comes out true” is a bad proxy for “the formula says something” — precisely the complaint against material implication that motivates relevance and paraconsistent logics in the first place.

And it evaporates exactly where the formalization is supposed to actually live. Move to Priest’s LP, where a is a glut (Both true and false) rather than classically inconsistent: a<->~a is now satisfiable, not vacuously false-always. That means the antecedent is no longer trivially unsatisfiable, ECQ doesn’t fire, and (a<->~a) --> c is no longer automatically true for every c — you’d actually have to do work, with a real conditional connective, to show Becoming specifically follows rather than something else. So the “curious correspondence” is a purely classical artifact that disappears the moment you take the dialectical content seriously enough to formalize it in the logic that can actually hold it without exploding. That’s worth sitting with: the appearance of validation was itself a symptom of using the wrong logic, and it’s gone precisely in the logic we already established is the right one to be using.

well, within the triad, ANYTHING is true

Two very different claims are being run together here, and they turn out to point in opposite directions rather than converging.

Sense one: “anything is true” via ECQ. This is exactly last turn’s diagnosis — a contradictory antecedent classically implies anything. But that reading is specifically what paraconsistent logic exists to block. LP is defined by rejecting ECQ: a and ~a both being true does not license deriving arbitrary c. That’s not an incidental feature of Priest’s system, it’s the entire motivation for building it — to have a formal setting where you can hold a<->~a without the collapse into “anything follows.” So if “within the triad” means within the logic we established actually houses this content (LP, or a relevant/paraconsistent conditional), the claim is false, not trivially validated: paraconsistency’s whole point is that not everything is true once you’re inside it.

Sense two: Hegel’s holism, “das Wahre ist das Ganze.” If instead you mean something more Hegelian — every determination eventually finds its place and its moment of truth within the completed System — that’s a real doctrine, but it’s structurally the inverse of trivialism, not a version of it. Trivialism is indiscriminate: every proposition is asserted, flatly, with no ordering or qualification, because the logic has stopped tracking distinctions. Hegel’s doctrine is maximally discriminating: his explicit claim (Encyclopedia Logic, the doctrine that “the finite is untrue,” das Endliche ist unwahr) is that every partial, isolated determination is false taken on its own — not true. What a finite category gets is not unconditional truth but a specific, located, and only partial validity, contingent on exactly how it’s aufgehoben into what follows. Nothing gets truth for free; each moment has to earn its place through the specific work of being shown both necessary and insufficient. “The whole is the true” describes the terminus of an immense amount of differentiation, not the absence of it — it’s closer to “only the fully-worked-out totality is true, and everything short of it is false-but-necessary” than to “everything is true.”

So the apparent overlap — both look like “everything, without restriction” — is actually surface resemblance between opposite mechanisms: explosion produces indiscriminate assertion by the logic breaking (losing the ability to discriminate at all), while Hegel’s holism produces comprehensive inclusion by the system working, differentiating every category from every other and assigning each its exact, earned, partial share of truth. Conflating them would mean mistaking the failure mode of classical logic under contradiction for the success condition of Hegel’s system — and the paraconsistent formalization we landed on last turn is specifically the apparatus built to keep those two apart, which is the strongest evidence that they’re not secretly the same thing.

who is priest

Graham Priest (b. 1948) — British/Australian logician and philosopher, best known as the leading defender of dialetheism: the view that some contradictions are literally true, formalized through paraconsistent logics like LP (the Logic of Paradox) that block explosion (ECQ) so a true contradiction doesn’t trivialize the whole system. His main technical/philosophical case is that the Liar, Russell’s paradox, and structurally similar antinomies all fit a single recurring pattern he calls the Inclosure Schema — the one I invoked several turns back as the actual sublation Hegel’s method would demand, where NBG’s class/set stratification only excludes.

He’s trained in mathematical logic (PhD, LSE; thesis on type theory, under John Lane Bell) but has worked across metaphysics, philosophy of mathematics, Buddhist philosophy, and the history of philosophy — including explicit engagement with Hegel as a historical dialetheist. Career-wise: long stretches in Australia (Western Australia, Queensland, Melbourne), St Andrews, and since 2009 Distinguished Professor at the CUNY Graduate Center — as of this past summer he’s reportedly moving to Tohoku University in Japan this October, so that may be shifting.

His central relevant books, for direct citation: In Contradiction (1987/2006) lays out dialetheism and LP; Beyond the Limits of Thought (1995/2002) develops the Inclosure Schema across the paradoxes; An Introduction to Non-Classical Logic is the standard textbook covering the technical apparatus.

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