Can Hegel’s Logic be formalised?

Introduction

I argue there is a formal logical operator corresponding to three moments of Hegel’s Logic. The central question “Can Hegel’s Logic be formalised?” is answered in the affirmative. The framework in question is category theory. Throughout the essay I use “Logic” to refer to Hegel’s method, and “logic” to refer to modern formal systems composed of axioms, syntax and inference rules. I do not seek to reinvent formal logic qua Hegel, nor to confine his system to a formula. My argument is restricted to three moments.

Hegel’s dialectic is a triadic movement between three moments of determination whose formalisation depends on the questions

Can the moment of the:

1. understanding be captured as an abstraction operator in logic?

2. dialectical be captured as an operator of determinate negation in logic?

3. speculative be captured as an ordering operator on levels of concepts?

This essay answers all three in the affirmative.

The first (abstract) “moment of the understanding” appears trivial in two-valued logics but has generalised application in the richer framework. Its formal property is identified as idempotency in section 1.2, and the formal definition is given in section 3.

The second (dialectical) “moment of determinate negation” is prehended in the disambiguation of senses in which an operator “negates”. This leads to two formal constraints: content-preservation and non-involutivity. Both are met by adjoint pairs of functors.

The third (speculative) “moment of sublation” is the movement upward in ‘Aufhebung’ levels of concepts, which is defined as an objectively organisational operator, where a determination is a “unity-and-identity-of-opposites” (UIO).

In Section 1, Hegel’s Logic is disambiguated from formal systems of mathematical logic and a working example of the ‘dialectic’ is introduced.

In Section 2, I examine why analytic logic dismissed Hegel’s method. Russell’s objection that Hegel equivocates between different uses of the verb “to be” in the being-nothing-becoming dialectic is examined. I argue that Russell’s general convictions were a case against Bradley more than against Hegel directly.

In Section 3, I introduce category theory as the continuation to the project of objective logic. Lawvere’s work at the intersection of category theory and Hegelian Logic is shown as supporting formal mathematical structure in Logic. Further work by Lambek et al. motivates the reconciliation of Hegelian and formal systems of logic. A translation of the general dialectic, and its treatment of the copula, into mathematical form is offered. Section 3 closes by recreating the being-nothing-becoming dialectic inside Lawvere’s example.

The conclusion returns to Russell’s claim that Hegel leaves the nature of self-consistency undiscussed, showing it is answered by Lawvere.

1. Hegel’s Logic and Method

To understand Hegel’s Logic and method we need to clarify Hegel’s use of the term Science.

Science (Wissenschaft). A body of knowledge brought into self-grounding order, where each determination is necessitated by its connection to all others – i.e. an organisational structure for knowledge, not an empirical methodology.

Science in this sense is what can be articulated completely. It will be key that Hegel’s criterion for science is in this sense organisational.

Equally important is Hegel’s use of “Logic” (Logik), as not only different from contemporary formal logic, but different from the classical logic of his time.

logic (formal logic). Mathematical formal systems of logic, constituted by a set of axioms with rules of inference. There are many such systems in modern mathematics. The closest analogue to such a system in Hegel’s time is what he designates as Subjective Logic (cf. Objective Logic). The defining property it shares with today’s systems of logic is abstracting propositional form from content.

To Hegel, Logic ought to be a Science of cognition which unifies thought content and form. Hegel’s Science of Logic could thus be read as proposing a Logic which is a self-development of the concepts themselves (Hegel 2010, Introduction).

Logic (Logik). A Science of cognition that does not presuppose a separation of cognitive content from cognitive form (cf. logic or formal logic). I use (capitalised) ‘Logic’ as shorthand for Hegel’s Objective Logic, which later merges with Lawvere’s category theoretic objective logic.

Hegel sought to achieve such a Logic by a method of ‘determinate negation’, which in the secondary literature is used synonymously with ‘the dialectic’.

1.1 Determinations and the Three Moments of Hegel’s Method

I rely on an interpretation of Hegel’s Logic as operating on ‘determinations’. The term ‘determination’ should be understood as any cognitive content, be it the content of propositions, truth evaluations, concepts or categories. Anything one may introspect about can be taken as determinate and therefore may qualify a determination.

Determination (Bestimmung). Any given determination is the particular cognitive content specifying: first (moment of abstract negation), the isolation of a specific concept, object or category from its relations to others such that it is reduced to a pure form; second (moment of determinate negation), the boundary of this form’s limit as its sole cognitive content.

While Hegel takes this use of determinations as given, much of the secondary literature is plagued by an emphasis on propositional determinations as opposed to conceptual or categorical ones. In my view, this emphasis on propositions stems from the popular misattribution of the Fichtean “thesis-antithesis-synthesis” formula to Hegel (Chalybäus 1837; Mueller 1958). The inability to translate some Hegelian determinate categories into propositions is a common pitfall in formal analytic logicist dismissals of Hegel’s Logic. Rather, I claim, formalising Hegel’s logic requires a emphasis on determinations as specifying concepts or categories as cognitive content. This allows us to work with determinations as variables, and with negation as function on said variables.

The Logic works on determinations through a structure of three moments, each named by its function. Stated analytically, the essay uses the same names for the corresponding logical operators. The moments in order are:

1. The moment of the understanding / moment of abstraction (Abstract Negation)

2. The dialectical moment (Determinate Negation)

3. The speculative moment (Aufhebung/sublation)

1.2 Four Ways to Negate: Abrupt, Boolean, Abstract, Determinate

Hegel employed a method which negates a determination to produce a new determination containing a higher level of ‘truth’. Hegel’s negation sought to extract the rational kernel of Spinoza’s omnis determinatio est negatio¹ (trans. “every determination is negation” ; Spinoza 1995, Letter 50). Hegel, like Spinoza, repeats the claim, but it should be read with an caveat: every determination is a result of applying different negations. A determination is first pinpointed through an abstract negation which isolates it from its relations to other determinations. For Hegel, this is not the end of the story: the determination should be developed through determinate negation.

What does it mean for there to be different negations? I present four types.

The four negations are differentiated by how they split on two axes: content-indifference (which bifurcates), and what an operation does under iteration (repeated application).

Content-indifference.

(i) indifference of rule function – the operator performs the same rule on any input; (ii) indifference of output – the only rule is the uniform output.

Involutivity. An operation is involutive if doing it twice (on a binary type) returns you to the start.

Idempotency. An operation is idempotent if doing it twice does no more than doing it once – for example, rounding a number to the nearest integer.

Abrupt Abstract Negation

Hegel attacks the traditional notion of negation by designating it “abstract”. Its function is content-indifferent. It is a way to cancel a determination such that one is left with nothing. For Hegel this was unproductive. The paradigmatic case of abstract negation is the skeptic's negation, which “sees only pure nothing” (Hegel 1977, Introduction).

Abrupt Abstract Negation (Constant Map). Content-indifferent cancellation resulting in nothing (no information).

To Hegel, content-indifferent negation defeats the endeavor to organise Science. There is a nuance in content-indifference which explains why a specific type of abstract negation is nonetheless indispensable in Logic. Hegel’s hostility, stated analytically, has two targets: uniform indifference of output, and finality – the mistaking of an act of fixing a determination for the terminus of its internal logic. He never objects to holding a determination fixed (abstraction) as such; only to declaring its fixity its completed understanding.

Boolean Negation

Modern formal negation was first introduced as Boolean truth-functional negation in Boole’s Laws of Thought, after Hegel’s death (Boole 1854). This type of negation cannot have been a direct target of Hegel’s critique. This means that any attempt to align Hegel’s disapproval of abstract negation with a critique of symbolic logic is a retrojection.

Boolean Negation. A unary logical operator that takes a truth value and flips it to its opposite. In classical binary systems of logic this means sending true to false and false to true. Also called the “NOT” operator. This is the most common type of negation in modern systems of logic.

The temptation to equate Boolean negation with abstract negation perfected was one which late British idealists succumbed to. At first, it seems that truth-functional negation’s content-indifference is identical to that of abstract negation. However, content-indifference can refer to two things:

1) indifference of rule function;

2) indifference in output.

Boolean negation is indifferent in rule function because it does the same thing in every case – flip a binary truth value.

In symbolic logic there are four unary operations on a variable: the identity, Boolean negation and the two constant maps (Enderton 2001). The constant maps – everything to the False, or everything to the True – are the two operations which are content-indifferent in output. Thus the constant maps are exactly the operations capable of destroying determinacy (the information content of the input). Boolean negation is neither of these, as it is involutive. It does not leave you with nothing; it is content-preserving.

Abstract Negation Proper

Abstract Negation (moment of the understanding). An operation isolating a content from its relations and holding it at a limit, such that applying it again adds nothing more (AA = A). Abstract Negation proper pinpoints a determinate fixed point, one pole of an opposition; formalised as the idempotent endofunctors L and R (see section 3).

While the constant maps are idempotent, not all idempotent operators leave us with nothing.

Elusivity of the first moment. There is a formal reason the first moment remained indiscernible for a long time. The understanding’s operation (fixing a determination so that doing it again adds nothing) is idempotent, yet inside two-valued logics the only unary idempotents erase all information. In binary logic abstract negation is indistinguishable from annihilation. The first moment only becomes formally perceptible in a setting rich enough to contain non-trivial content-preserving idempotents. Category theory, as section 3 shows, is exactly such a setting, where a negation operator which reaches idempotency identifies a limit object.

Determinate Negation

Hegel’s case was directed at the methodic doubt which annihilates any content into a uniform “unknown”. Take Cartesian doubt’s result as an example of this uniformity of output – the blank slate. Descartes recovers the negated knowledge through an insight which on the Hegelian view is uncomfortably accidental (the cogito). Hegel sought a negation which could, by itself, guarantee the first principle in any category of thought, instead of suspending judgement. Determinate negation must be productive in this sense: it preserves what was true of the starting content while developing it into a truer determination.

We thus place the following constraints on a formal definition of determinate negation:

Determinate Negation (bestimmte Negation). both content-preserving and productive (non-involutive) – Formalised in section 3 as the adjunction L⊣R between two idempotent fixed poles of a level.

In summary:

1. Abstract negation, in its abrupt (trivial/degenerate) form, neither preserves content nor produces new content;

2. Boolean negation preserves but does not produce;

3. Abstract Negation proper preserves a limit of content but does not produce;

4. Determinate negation ought to do both.

1.3 The Being-Nothing-Becoming Dialectic: Working Example

This method runs through the entire Logic: each determination, abstracted from its relations and developed without presupposition, reveals simultaneously its one-sided limitation and the immanent necessity of the higher determination which resolves its contradiction. More precisely, the determinate negation of the double abstract negation sublates two opposed determinations into a single new determination at a higher conceptual level.

The moment of resolution is termed Aufhebung (sublation), which (from German) has three simultaneous meanings: to cancel, to preserve, and to elevate. (Hegel 2010, Doctrine of Being, ch. 1, Remark).

Thus Determinate negation names the Method’s dialectical operation (negating yields a definite new cognitive content); Aufhebung names the culmination of the triadic structure of the dialectic.

The Science of Logic begins by taking the concept of being as abstracted from all its determinate content – the first determination is to entertain the thought of being, not of any particular entity, but of anything whatsoever: that which is common to all entities given that they exist (Hegel 2010, Doctrine of Being, ch. 1).

(i) This first stage of the method is the moment of abstraction, during which the Logical operation is understanding. In Hegel’s idiom, to understand a thing, as it is in-itself, one must abstract it from all determinations of the thing as it is for-other things.

But this pure being, as lacking any differentiating relation to any other, taken only as it is in-itself, is indistinguishable from nothing. The only thing common to all existing entities is not itself an existing entity; namely, it is nothing. Being has “passed over” into nothing.

(ii) This is the dialectical moment, during which the Logical operation of determinate negation takes place.

At the same time as a concept is stripped of all determinations which differentiate it from everything else it is not, it becomes indistinguishable from its opposite – that which it determinately is not.

And yet being precisely is not nothing: both determinations are experienced as distinct. This contradiction is what the third stage deals with – there is an immediate third determination which acts simultaneously as the unity and identity of these opposites, namely becoming.

Becoming is the immediate determination capable of simultaneously holding the “being passed into nothing” and the “nothing passed into being”.

(iii) This is the speculative moment, which concludes with the Logical operation here referred to as sublation (Aufhebung).

What Hegel seeks to show in the dialectical development of the categories of being is that no consistent determination can be a self-contained immediacy.

To be determined as something always requires a relation to some other, and at the highest level of abstraction cognition is capable of, the other is the opposite of the determination.

In other words, the attempt to think pure being-in-itself fails by collapsing into thinking pure nothing, and this failure to think the determination alone tells you something. Something occurs at the moment the determination negates itself.

In Hegelese: “the unity of being and nothing is not a state; … rather, this middle and unity, the vanishing, or equally the becoming, is alone the truth of being and nothing.” (Hegel 2010, Doctrine of Being, ch. 1).

What first appears in cognition as immediate moments, when introspected upon, now gives three determinations:

• “When trying to think pure being (abstraction), I had a feeling that being passed into nothing (negation)” [Abstract Negation 1 (AN1)]

• “When trying to think pure nothing (abstraction), I had a feeling that nothing passed into being (negation)” [Abstract Negation 2 (AN2)]

• What validates the next determination as true – “the two earlier feelings prehend a singular new determination (determinate negation)” – can be determined as “becoming” (sublation). [Cf. section 3.5]

I have shown that the first moment consists of two abstract negations: the fixing of being and the fixing of nothing run in opposite directions, and it is their relation that the second moment names.

The real form of dialectic is thus:

(double) Abstraction – Determinate Negation – Sublation. It is not “thesis – antithesis – synthesis”.

2. The Rift Between Traditions

By the beginning of the twentieth century a new conception of logic began to take shape. The verdict among the founders of this new logic, as applicable to philosophy, was that Hegel’s Logic resisted formalisation.

Moore and Russell led “the revolt against idealism”, ending the generational dominance of British Idealism in academia (Moore 1903). At the same time, Frege’s function-argument logic with quantification (Frege 1879) displaced the subject-predicate form on which logic had considered itself a complete science since Aristotle (Kant 1781/1787, Bviii). The Vienna Circle, influenced by Wittgenstein, began to treat post-Hegelian metaphysics as linguistic nonsense (Carnap 1932). In the Tractatus, Wittgenstein made truth-functionality canonical; his conception of negation maps exactly onto Boolean negation (Wittgenstein 1922, §5.5).

Hegelian Logic claimed to incorporate contradiction; the new programme required consistency – therefore the two were taken as incompatible. It is worth flagging that part of the above verdict rests on an equivocation between contradiction as defined internally in formal logic and the contradictions internal to the determinations Hegel worked with (cf. Hegel 2010, Doctrine of Essence).

Contradiction. (i) In a formal system, where one can derive a proposition simultaneously with its Boolean negation, every other proposition trivially follows – rendering the whole system inconsistent. Consistency is the absence of any such pair of propositions. (ii) Internal to a determination (Hegel), where the fixed content of a determination passes into its opposite while remaining conceptually distinct – the opposition is sustained and the contradiction confined to the two determinations’ mutually limiting relationship.

2.1 Russell’s Objection

The revolt against idealism was aimed at Russell and Moore’s contemporary academics representing the Hegelian tradition: Bradley, Bosanquet, McTaggart and others (Hylton 1990). Russell was especially successful in putting forth arguments against Bradley, and he interposes such arguments in sections addressed against Hegel in general:

“The way in which, as it seems to me, Hegel’s system assumes the ordinary logic which it subsequently criticizes, is exemplified by the general conception of ‘categories’ with which he operates throughout... Mr. Bradley has worked out a theory according to which, in all judgment, we are ascribing a predicate to Reality as a whole; and this theory is derived from Hegel.” (Russell 1914/2009, Lecture II, p. 31).

Bradley argued that external relations, when fixed as determinate, need further attachment to relata in an infinite regress, which he identified as an internal contradiction in the notion of external relation – thus declaring all such relations universally impossible (Bradley 1893). Russell was adamant about the indispensability of external relations within mathematics (“greater than”, “equal to”) (Russell 1903), and opposed the tendency of British idealists to promote rational abstractions to universal judgements about reality. He attributed this tendency to Hegel:

“Hegel believed that, by means of a priori reasoning, it could be shown that the world must have various important and interesting characteristics, since any world without these characteristics would be impossible and self-contradictory. Thus what he calls ‘logic’ is an investigation of the nature of the universe, in so far as this can be inferred merely from the principle that the universe must be logically self-consistent... Logic itself would be concerned rather with such questions as what self-consistency is, which Hegel, so far as I know, does not discuss.” (Russell 1914/2009, Lecture II, p. 30)

Russell’s closing clause (on self-consistency) should be kept in mind, as it is answered in section 3.3 and the conclusion. Russell does not there support his claim that Hegel held such judgements; the example given thereafter is Bradley’s. Bradley’s exercise in fixing the limit of a concept – “external relation” – is an instance of what Hegel calls the standpoint of the understanding. From the standpoint of Hegel, who criticises the halt at the moment of understanding, there is agreement with Russell in being skeptical of rational abstractions being universally fixed. Russell’s success against Bradley cannot be translated into a direct attack on Hegel; rather, what emerges is functional agreement in critique of the end-all and be-all of subject-predicate logic.

2.2 Where Russell Directly Addresses Hegel

Russell’s adapted Cambridge fellowship dissertation An Essay on the Foundations of Geometry (1897) was conceived as the first instalment of a scientific Hegelian model: geometry's inherent contradiction in empty space being pure relativity forces an immanent transition to Physics (Russell 1897; Griffin 1991). Russell abandoned the project due to advances by Cantor, Dedekind and others which dissolved the contradictions he had identified. His disillusion with the Hegelian method merited never revisiting the project of formalising the dialectic (Russell 1959). Despite this anecdote, Russell scarcely offered direct formal rebuttals of Hegel’s Logic.

One objection considered by contemporary Hegelians, especially in Morris (2008), is an argument in the margins of Russell’s Our Knowledge of the External World accusing Hegel’s dialectic of an equivocation in the verb “to be”. The verb “is”, as used in natural language, can take on four jobs when held to the formal standard of logic (Russell 1903; Russell 1919):

1. Predication → φ(a)

Socrates is wise

2. Existence (Quantification) → ∃x φ(x)

There is at least one number between 7 and 12

3. Class inclusion → A ⊆ B

The rose is a plant

4. Identity → a = b

Hesperus is Phosphorus

In logical atomism, each sense of “is” is an axiomatic primitive presupposed in foundation, with no implicit connection. Russell’s (1945) diagnosis is that a dialectical passage can look necessary because a claim asserted in one sense of “is” is then taken up in another; when these senses are distinguished, the apparent dialectical transition dissolves.

On Hegel’s account, the four senses of ‘is’ are arranged as subsequent developments rather than presupposed. In the chapter “Judgement” in Development of the Concept, one finds the dialectical development of the copula – the ‘is’ joining subject to predicate. The four movements are (Hegel 2010, Doctrine of the Concept, “The Judgement”):

(i) Predication (inherence). The positive judgement asserts the immediate inherence of a predicate in a subject. Its copula does no more than hold the predicate to the subject – the first and most abstract work of ‘is’.

(ii) Quantification (subsumption). The subject is neither exhausted by nor identical with its predicate – and this determinate negation yields the judgments of reflection, whose forms are the quantified ones: ‘this’, ‘some’, ‘all’. The copula now expresses subsumption under a universal, and the existential ‘is’ appears as the judgement’s ‘some’.

(iii) Inclusion (the genus). The subsumption practised by the judgements of reflection completes itself in the judgement of necessity: the universal judgement’s ‘allness’ collapses into the genus, and the categorical judgement “the rose is a plant” states kind-membership as an objective relation rather than an external comparison. The order through which subsumption moved: singular under particular under universal is posited as content: inclusion.

(iv) Identity (the copula’s real sense). The disjunctive judgement already posits the genus as identical with the totality of its species; the judgement of the concept therefore measures the singular against its own concept. The copula becomes the posited identity of subject and predicate. That this identity cannot be primitive is Hegel’s older result: in the Doctrine of Essence, identity is posited by reflection as essentially containing difference. The bare ‘A = A’ of the understanding is an empty tautology (Hegel 2010, Doctrine of Essence, ‘Identity’, Remark). The copula’s last sense is the judgement’s first presupposition made explicit: the original unity whose division (Ur-teil) the judgement is.

What formal logic stipulates as four distinct axiomatic symbols, the Logic generates as successive determinations of the Judgement, each forced by the determinate negation of its predecessor.

Russell argues against the British Hegelians in Our Knowledge of the External World (1914) that the ability to abstract the form of a determination from its content is the paradigmatic requirement for a formal system of logic. He states:

“In all inference, form alone is essential: the particular subject-matter is irrelevant except as securing the truth of the premises... In logic, it is a waste of time to deal with inferences concerning particular cases: we deal throughout with completely general and purely formal implications, leaving it to other sciences to discover when the hypotheses are verified and when they are not.” (Russell 1914/2009, Lecture II, p. 35)

This leads onto a new reading of the analytic verdict. Russell’s logical atomism sought to dissolve complex and unclear metaphysical questions through a philosophical logic in joint foundation with mathematics, where complex proofs can be reduced to fixed and self-evident axiomatic constituents which hold universally (Russell 1908). In this project each true fact can be independently intelligible through its fixity – independent of any particular content, but as a universal form derived from logical atoms. In the Preface to the Phenomenology, Hegel calls the analytical, abstractive activity (form from content) of the understanding “the most astonishing and mightiest of powers, or rather the absolute power” (Hegel 1977, Preface). The founders of formal logic built a powerful instrument for developing the pure form of determinations abstracted from any content, and declared that anything not expressible in that instrument was not logic. Section 3 examines whether category theory has since moved this instrumental boundary – and if so, by how far

3. Category Theory for Hegelian Logic

3.1 The Heir of Abstraction

The Historical Context

The project for grounding Mathematics in the twentieth century underwent groundbreaking developments – this I mean literally (Russell 1902; Gödel 1931).

In the period from 1945 to the 1960s, category theory took a structural turn in the foundations of mathematics (Eilenberg and Mac Lane 1945), moving from axiomatising everything in terms of sets to focusing on relations (morphisms) between objects.

Before we declare category theory the logical successor of logical atomism, a word in the claim must be disambiguated, for “abstraction” now names two operations.

There is abstraction as omission: drop the content and keep a placeholder – the variable, a type parameter, a constant letter that stands for anything (cf. section 1.2, rule-indifference).

There is also abstraction as isolation: hold a determinate form apart from its relations to fix it in its own content (cf. section 1.2, idempotency). The former yields an empty slot; the latter a specific content. While the abstractive power of logical atomism is confined to the former sense of omission, the Hegelian understanding abstracts mainly in the sense of isolation: fixing being as being reads as isolating the determination through suspending its relations to all others.

Category theory is in a formal sense a synthesis of these two senses. Where logical atomism still posits intrinsic content in self-evident primitive logical truth, category theory declares that the form of such atoms may change from one category to another – some may be admitted across categories, others may not – it is maximally content-indifferent. What the Hegelian understanding declares negatively – the determination’s content contradicts itself when abstracted from all relations – category theory positively formalises: what an object “is”, in a category, is substantively exhausted by its relational mapping to other objects. What is referred to as an object in category theory is isomorphic to the conception of a Hegelian determination. Furthermore, category theory derives internally a proof that identity is defined up to isomorphism by an object’s role in a web of relations, including relations to itself (Yoneda lemma; Mac Lane 1998, §III.2).

The Curry–Howard–Lambek Correspondence

The form of the claim that category theory supersedes logic finds its content in the Curry–Howard–Lambek correspondence.

Curry and Howard first observed that propositions correspond to types, and proofs to programs. To prove a propositional inference is the same as constructing a function which transforms a proof of the antecedent into a proof of the consequent in typed lambda calculus (Curry and Feys 1958; Howard 1980). Lambek’s contribution was to show that, in category theory, typed lambda calculus is the internal language of cartesian closed categories: propositions and types are the same objects, and logical proofs and computer programs the morphisms between them (Lambek and Scott 1986). Thus I argue that whatever the moment of the understanding achieved in logical form, it generalises in categorical form.

Category theory thus inherits and generalises the abstractive power of the first moment; whether it can also express what the understanding cannot – a productive negation – is the question of the next section.

3.2 Adjointness in Foundations (determinate negation)

Category theory, through a higher level of abstraction, achieves what last century’s formal systems of logic could not: it gives a precise formal meaning to exhibiting what initially seem distinct, fixed, atomic logical operations as instances of a single relation. In “Adjointness in Foundations”, Lawvere (1969) showed that the basic operators of logic are “adjoint functors”. An adjunction is best approached through category theory's preference for a hierarchy of equivalence relations over equality, each proceeding by weakening a constraint of the previous – expressing more.

Equality sits at the bottom of this hierarchy due to its rigidity. In category theoretic practice one never needs two objects to be the same (echoing Hegel’s critique of equality). Next is isomorphism – an invertible map between two objects as freely interchangeable functionally – equality being the special isomorphism whose inverse is the identity. Next we consider maps between object relations – functors. The map between two functors is called a natural transformation, encoding a family of relations, one at each object. When a natural transformation from category F to category G can be reversed by a natural transformation from G to F, the two natural transformations are in adjoint equivalence. "An adjunction is the general case in which this fails: the round trip from F to G and back falls short of the identity (the two maps oppose each other without being inverses) – that composition is called the “adjoint functor”.

Adjoint functors (L ⊣ R). Two functors in opposite directions, L : B → A and R : A → B, whose maps correspond: to each L b → a, precisely one b → R a. They are not inverses: a round trip need not return to the start – from b you come back not to b but to RL b, from a to LR a. The irreducible excess produced in the round trip is measured by the unit b → RL b and the counit LR a → a (cf. Determinate Negation, section 1.2).

Lawvere proves that all logical connectives stand in adjoint relations (Lawvere 1969). Conjunction and implication stand in adjoint relation. Existential and universal quantification are the left and right adjoints, respectively, of substitution. Category theory organises all of logic into such adjoint relations.

This organisation parallels section 2.2: the four senses of ‘is’, which formal logic must stipulate as separate primitives, follow in category theory from a single operation.

(i) Substitution (predication). The base operation of categorical logic is substitution. Predication is its minimal act: to say Socrates is wise is to substitute Socrates into wise.

(ii) Quantification (existence). Substituting along the projection which forgets the subject loses information: distinct predicates collapse into one proposition. To ask what a predicate amounts to once its subject is forgotten is to ask for a left adjoint. That adjoint is the existential quantifier ‘∃’. φ becomes ∃x φ(x) when no particular subject (S) is determined for it (Lawvere 1969). Universal quantification ‘∀’ is right adjoint: when the predicate is true of all unspecified content ∀ x φ(x).

(iii) Inclusion. Predicates stand in an order of inclusion. The rose is a plant is ∀x (Rose(x) → Plant(x)). Inclusion is the substitution expressing this order between two right adjoints.

(iv) Equality (identity). The movement of (ii), applied not to forgetting a subject but to repeating one, generates identity. Substitution along the diagonal maps each element with itself turning φ(x, y) into φ(x, x). Equality is the left adjoint of this diagonal substitution. Leibniz's law follows from the same adjointness (Lawvere 1970, §1). Hesperus is Phosphorus asserts the everywhere-legitimacy of substituting one for the other.

Each transition is an adjunction pointing to the unique minimal answer to a problem the previous determination poses to itself. The senses of 'is' succeed one another by determinate negation, with the immanent necessity Hegel claims for the dialectic.

Hegel’s development mirrors the formal adjoint relations down to the order of derivation. Russell's distinction of the verb is the understanding's standstill of this derivation: it succeeded in holding the determinations fixed, yet blind to the unified movement which gives rise to all four. His most direct argument against the dialectic thereby describes, in the formal framework of category theory, the Judgement chapter's development of the copula.

3.3 Unity and Identity of Opposites

Having isolated the formal properties an operator needs to qualify as determinate negation , I now present the categorical development of using it, as applied to the being-nothing-becoming dialectic and culminating in the formal definition of Aufhebung – the third stage of Hegel’s method. I reproduce Lawvere’s theorems as relevant to my argument and direct the reader to his original paper (Lawvere 1989) for the mathematical proofs.

Picture categories A and B with a functor A → B. We show under what conditions two opposite sub-categories of A are, at the level of B, one and the same.

Lawvere defines the unity-and-identity-of-opposites (UIO):

Unity-and-identity-of-opposites (UIO). A functor A → B is a UIO if it possesses a left adjoint and a right adjoint – two functors running back from B to A – and these adjoints are full and faithful, meaning that each projects B into A as a subcategory (due to the properties of adjoint functors, if one is full and faithful, then so is the other).

First, the left and right adjoints present B inside A twice, as two subcategories. These two copies of B are opposed in A, where they appear as a bare (void) pole and a saturated (pure) pole. They are united because both sit inside A, and identical because both are full and faithful representations of B.

To understand these adjoint functors we take the simplest case. In category theory an object is terminal if exactly one morphism (relation) runs towards it from every object. In sets, any singleton is terminal: from any set exactly one function maps onto it; dually, the empty set ∅ is initial. The terminal category is named “1”, as the smallest category is: an object and its identity arrow. From any category A there is exactly one functor A→1, and this functor is a UIO whenever A has its initial and terminal objects as the two internally opposing poles, which Lawvere terms non-being and pure being respectively. The left adjoint from B to A selects non-being; the right adjoint, pure being. In general, L reduces an object to the void pole of whatever opposition the level concerns, and R completes it to the pure pole. Any category with an initial and a terminal object carries the adjoint string 0 ⊣ ! ⊣ 1, and hence this simplest UIO . Categories of logic (CCCs, a fortiori any topos) guarantee that both objects exist, hence the UIO. In a CCC, 0 ≅ 1 forces its internal logic to be inconsistent. A category’s logic is consistent, then, precisely when 0 ≇ 1 (void and pure being remain distinct poles of a real opposition).

In most cases A is a topos: a category structured like a universe of sets in which one can define a sheaf. A sheaf can be intuitively taken as a saturated object: it contains everything required for the level to be complete, so that completing the level adds nothing. In topoi, R is sheafification. Sheafification R projects any object to the nearest sheaf on the level above, are the sheaves – those already saturated, the fixed points of R – are the sheaves. L is the opposite operation: it reduces an object to its skeleton, the bare residue in the level below – the fixed points of L – we call skeletal. Most objects in categories are neither sheaves nor skeleta but sit in between; the sheaves and skeleta are the limit-cases which the L and R functors reach.

In Lawvere’s framework, A is the category of applications, whose objects are finite sets carrying a right action of a monoid M.

A monoid is a set which can combine any pair of its elements under associativity, while having a neutral element that, when paired with any other, leaves it unchanged (the identity element).

Here an application is some finite set which can be transformed by each element of M through M’s law of combination. The monoids we will be considering are graphic: obeying the identity relation xyx = xy for all elements x and y. Setting y to the neutral element gives xx = x; thus each element of a graphic monoid is idempotent.

A partially ordered set, or poset, is a set where each element can be ordered as below or equal to another element, subject to the following conditions: if two are each below or equal to the other, they are the same; and the relation is transitive (if a ≤ b and b ≤ c then a ≤ c).

Lawvere takes the poset of left ideals of M, where

a left ideal is a subset closed under multiplication on the left, so that multiplying any element of the subset on the left by any element of the monoid yields a result inside the original subset.

In graphic monoids, where multiplication need not commute, every left ideal is two-sided: substituting s for x and x for y in the graphic identity gives sxs = sx, so right multiples of ideal elements remain inside. This makes their ideals enumerable as levels. The intersection of any collection of left ideals is again a left ideal, so that among any collection there is always a smallest one: their intersection. This leads to the following:

Proposition (Lawvere 1989, Proposition 16 and Corollary). For A, the category of all applications of a graphic monoid M, the poset of all UIOs with domain A is parametrised by the poset of left ideals of M.

So the UIOs of A (its conceptual levels) form a poset, with the simplest case being a finite, distributive lattice as the canonical order of conceptual levels for the determinate M. Every level that could be determined in M is a point of this lattice, as part of a chain of unity and identity of opposites. I have put in place the formal setting for levels and their two poles. The third moment can now be defined as a relation between levels.

3.4 The Aufhebung

We can now define the two poles introduced in the previous section. Recall the left ideals as the conceptual levels of M. Determine a level: take a left ideal and call it S. The S-skeleton of an application X is the set of elements of X determined by at least one member of S. For an element s of S and an element x of X, the right action is written x·s. The element x belongs to the skeleton when x·s = x for some s in S. X is S-skeletal when this action exhausts it – when every element of X is determined by some member of S. The dual of this is called the sheaf condition. Recall that every left ideal is also a right ideal, hence an actionable set; thus we can take S as an application, and X is an S-sheaf when every map from S into X is given by the action of a single element of X. Skeletal objects are thus assembled entirely out of fixed points; sheaves are objects entirely determined by their elements. These are, respectively, the void and pure poles introduced earlier.

The constants of M – elements for which c·x = c for any x – form the smallest non-trivial left ideal; call this level 0. The skeletal objects at level 0 (the void pole) are the discrete applications: those whose elements are all constants, where every element is disconnected from every other. The sheaves at level 0 (the pure pole) are the codiscrete applications, with everything that can be joined being joined.

The discrete and codiscrete applications are identified as the Hegelian “non-becoming” and “pure becoming”.

Definition (Aufhebung) (Lawvere 1989). For left ideals S ⊆ T. T sublates the opposites of S, written S << T, if every S-skeletal application is a T-sheaf – that is, if every object that is void at the lower level S has become complete at the higher level T. Because the intersection of left ideals is a left ideal, for every S there is a minimum T with S << T. Lawvere calls this smallest T the Aufhebung of S, and writes S’.

Therefore the Aufhebung of the opposition pure being versus non-being is the opposition codiscrete versus discrete, or pure becoming versus non-becoming. Iterating this method computes a chain of levels, each the Aufhebung of the previous, with the length of this chain specifying the conceptual dimension of M. Within each level, two opposed poles (skeletal via the left adjoint, sheaf via the right) are unified by the functor to that level’s B category. Each level’s void pole is carried to the pure pole of the smallest level above by an Aufhebung relation S << T.

Lawvere held that Hegel’s project of an Objective Logic “is in a way the program which category theory has been carrying out within mathematics for the past fifty years” (Lawvere 1989, Note 2).

3.5 Being-Nothing-Becoming: Categorically

Recreated herein is the being-nothing-becoming dialectic in Lawvere’s framework. Take a graphic monoid M and its category of applications A, with the lattice of left ideals as its conceptual levels, and consider the lowest level: the UIO given by the unique functor to the terminal category, whose poles in A are the initial object (non-being) and the terminal object (pure being). A determination begins as an opposing pole within a conceptual level (skeletal or sheaf), and the level is the UIO category (B) that holds both poles within itself.

(i) The moment of the understanding is the function of the level’s two idempotent operators. R fixes a determination toward the pure pole, the attempt to hold being as pure being, or AN1 (see section 1.3). L fixes toward the void pole, as the attempt to hold pure nothing – AN2. Each “fixes” determinate content idempotently, yielding one pole, or half of a full determination.

(ii) The dialectical moment is the adjunction L ⊣ R. The “passing over” of being into nothing and of nothing into being corresponds to the unit and counit of this adjunction. Where the relation between the two abstract negations falls short of the identity morphism (the non-trivial case), this relation preserves the distinct natures of AN1 and AN2. We call this relation the determinate negation, as it is content-preserving and non-involutive.

(iii) The speculative moment is the Aufhebung relation. The opposition of the lowest level is sublated by the unique minimal level in the level above it. In this example, the opposition of the void and pure poles is sublated by the codiscrete and discrete applications – pure-becoming against non-becoming. “Becoming” is the first determinate UIO in the lattice of levels. This resolution is forced, as S’ is determined by a universal property as the unique minimal resolving level. The Aufhebung formally traverses the canonical lattice of levels, reflecting Hegel's claim each subsequent determination in the dialectic is immanent and necessary.

This movement’s only presupposition is that the poles of the lowest level are not identical – in other words, that the universe is consistent (0 ≇ 1). Where this condition collapses, there is no formal logic.

4. Conclusion

“Can Hegel’s Logic be formalised?” – It already has been.The argument confirms three counts, one for each moment. The moment of the understanding is formalised as idempotent fixing, with the operators L and R holding a determination at a pole (fixed point), whose non-degenerate form is imperceptible to two-valued logic except in the meta-language of category theory (sections 1.2, 3.1). The dialectical moment is formalised as the adjunction L ⊣ R: the productive, non-involutive relation between a level’s two opposed fixed points (section 3.2). The speculative moment is formalised as the Aufhebung operator S → S’ on the lattice of levels: the minimum jump operator carrying the void pole of one level to the pure pole of the next (section 3.4). On Lawvere’s terms, these three operators recreate the being-nothing-becoming dialectic (section 3.5), where being and nothing are the fixed points of the lowest level, their passage is the adjunction between determinate poles, and becoming is the level forced in sublation.

Thus we can return to Russell’s direct demand of Hegel. The clause remaining from section 2 – that logic “would be concerned rather with such questions as what self-consistency is, which Hegel … does not discuss” (emphasis added) – is addressed in the proposed framework. In any cartesian closed category the first opposition, nothing to pure being, either holds distinct or collapses into unity; its collapse is inconsistency, under which every determination can become any other. Self-consistency is the sustained opposition (cf. Contradiction, definitions (i)–(ii)) internal to the category’s poles. Category theory's Objective Logic presupposes no axiom of non-identity; it concerns what self-consistency is in any formal logic (all CCCs, as opposed to solely binary systems). Section 2.1 showed that Russell’s other arguments struck Bradley’s idealist arguments, which amount to the abrupt abstract understanding. Russell’s programme never sought a positive implementation of formal Hegelian notions.

This essay’s answer sides with Lawvere’s (1989, Note 2) remark that Hegel’s Objective Logic “is in a way the program which category theory has been carrying out within mathematics for the past fifty years”. The conclusion is that the remark can be extended by thirty-seven years.

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