I. Main questions and main results

It is a heroic enterprise to develop a systematic reading of Hegel's Doctrine of Being, the first of three books in the Science of Logic. The basic idea of Houlgate's grand interpretation of Hegel's three parts of this Doctrine, which I call the Logic of Quality, Quantity and Measure, is that Hegel demands a ‘presuppositionless derivation of categories’ (II: ix). Hegel on Being (in short: HoB) presents a thoroughgoing and rigorous interpretation in a kind of dialogue (as I would characterize what HoB marks as ‘Excursus’) with Kant (I: 19–46, 307–73; II: 171–79) and especially Frege's formal logic and his attempts of a predicative definition of numbers (II: 25–138). In any case, HoB makes us aware of the really problematic presuppositions in post-Fregean Analytic Philosophy.

The 880 pages in the two volumes provide an indispensable starting point for asking the following very general and perhaps crucial questions for a second thought on Hegel's text:

1. 1. What are logical categories and what is a ‘scientific’ or presupposition-free ‘derivation’ of say, nothing or non-being, becoming and being-here-and-now (Dasein) from pure being?

2. 2. How far can HoB's linear approach to the text convince us? It somehow forbids us to use later texts for understanding earlier passages. However, Hegel himself famously speaks at the very end of the third book in his Doctrine on Method of a circle of circles (WdL: 252; cf. also Stekeler Reference Stekeler2022: 1150)Footnote ¹ and explains (Cf. WdL: 251; Stekeler Reference Stekeler2022: 1146–49) that we have to start again to read the first two books in view of the results at the end. Only in hindsight can we fully understand what it means to begin with what Hegel calls Objective Logic, followed by Subjective Logic, i.e., the Doctrine of the Concept. This is so because we presuppose the concepts of Subjective Logic already in all categories from the very beginning.

3. 3. This leads us back to the questions, which HoB wants to answer. What can it mean to say that we must start logic with the category of pure being and then must by some necessity go over to ‘nothing’, and from there via further steps to ‘essence’ and ‘concept’?

II. Elucidations of historical context and preconditions

There are preconditions in understanding any text that we cannot formulate as axiomatic assumptions in schematic deductions in the modern sense of the word. Such apagogic proofs use the modus ponens as the (only) meta-rule in deriving conclusions from a set of premises.Footnote ² Making the preconditions of understanding explicit, presupposes much more knowledge about contexts and facts, about cultural history and language.

Clearly, Kant's Transcendental Analytic is the main text to which Hegel's logic refers almost all the time; its most contested and least understood part is the so-called deduction of the categories. In reference to these texts, Hegel ridicules the idea that there could be a fixed number of categories, namely precisely 3 × 4 in the four groups of quantity, quality, relation and modality. Hegel agrees with Kant, however, that in these contexts a deduction of categories is, like in the jurisprudence of the time, no apagogic proof at all, but explication of presuppositions in the generic forms of judgements together with justifications of their proper application. We must, therefore, distinguish the different meanings of ‘deduction’. Kant's transcendental deduction of the (use of) categories shows how to check presuppositions in attempts to refer to concrete, objective things in the world—such that the conditions for possible objective experience are virtually the same as the conditions for being a possible object of joint experience.

In Kant's context, the word ‘quantity’ is a headline for the different contents of noun-phrases N, ‘quality’ for verb-phrases P.Footnote ³ In the case of a noun-phrase like ‘the lion’, for example, N can refer to all or many objects; to a generic, to just one, or to no object of some sort. In the case of quality, the ‘normal’ question is if in a sentence N is P the verb phrase P or its negation or ‘complement’ P^C applies to N. Formal logic totally underestimates the third, most important, case. It is the case of in(de)finite negation, which is always true and false, i.e., inconsistent, contradictory in itself. Famous examples are ‘the rose is no elephant’ (cf. WdL: 70; Stekeler Reference Stekeler2022: 420), ‘Caesar is no number’ (cf. II: 113) and ‘I am hereby lying’. Infinitely negative sentences or assertions like the famous paradox of the liar in the last case are ‘true’ because the negated assertion is also ‘false’. Some such sentences are mere ‘category mistakes’, others are speculative oracles as, for example, ‘God does not exist’ in negative theology, not only in philosophical enlightenment. The noun phrases do not fulfil the presuppositions of semantically well-formed designation. Infinite negation thus does not express truth function, but asks us to remove the sentences or assertions somehow from the class of propositions and thoughts. In fact, we silently assume a verbal rule according to which talking about a thought presupposes that what is said is consistent. All this shows why the ‘principle of non-contradiction’ only ‘seems to be obvious’ (I: 10). However, Hegel does not at all ‘accept’ contradictions or even claim that there are contradictions in the world. He rather makes the implicit and idealizing presupposition in any formal principle of bivalence explicit. These presuppositions are by no means trivial. They hold in the end only in purely mathematized contexts and domains. In the real world, there are always intermediary and accidental cases that can—even unwittingly—undermine the ideal presupposition of the formal principle Tertium non datur (i.e., ‘there is no third’).

In any case, Hegel sees that we have to generalize Kant's insight. We have to ask for all noun-phrases (names) and all (de)finite predicates: to which relational and modal domain of objects or entities do they belong?Footnote ⁴ Zeus, for example, belongs to the Greek mythos, but does not exist in the real world. The number 12 belongs to the pure numbers, the amount of legs of my cat Emily does not. In Frege's Humean definition of the ‘cardinality’ of a set, there is no distinction between sets of empirical and of time-general things (see II: 112ff.). Even though amounts and sets of temporal things are abstract, they are not pure. We arrive at pure sets and time-general numbers only via arbitrary reproductions of representing terms. Frege grossly underestimated the role of the numerals as children's numbers (Kleinkinderzahlen).

Houlgate's idea of a presuppositionless logic thus seems to stand in tension with what Hegel really does, namely explicating implicit pre-conditions in practical knowledge (1), in formal presuppositions of using categorical schemes (2) and hidden contradictions (3) that arise if we neglect the (de)finiteness of all sortal domains: There is no universe of all (pure) being. Only in this reading, Hegel's categorical analysis is, at the same time, logic, critical ontology and ipso facto a critique of any merely metaphysical belief-philosophy.

III. The beginning of logical analysis

HoB's interpretation of the core text begins in Chapter 6 (I: 135) with the following remark:

I think we must disagree with at least the last sentence: Hegel does not think that we can discover by logical analysis anything that we do not already know. However, after making practical knowledge and intuitional differentiations by logical commentaries explicit, we know things differently, more (self-)consciously.

Houlgate also says that pure being is absolutely indeterminate; even though neither he nor Hegel tells us ‘what the reflections are that have to be set aside’ (I: 125). Hegel's criticism of Kant's Ding an sich even shows why it is impossible to abstract from all determinations. It is, therefore, unclear what it could mean to ‘set aside all determinate assumptions about being and begin from the least that it can be: namely pure and simple being’ (I: 135).

It is a mere appeal to try to focus on the category of formal existence, as I would like to say in short, when Hegel asks us to consider pure being without any further predicative determination. The very failure of the attempt leads us immediately to a presupposed differentiation of being and nothing or between being and not-being. Moreover, the word ‘pure’ always signals that we talk about an ideal form (eidos, concept). If we just focus on the form of distinguishing between a P and a complement P^C, we abstract from the specific P. As a variable, P is equivalent to P^C, even though the latter is its formal negation. This is the reason why pure being is the same as pure not-being. Hegel develops his idiosyncratic language in order to articulate such logical facts, which we today can express more perspicuously, namely by using letters as variables and negation signs like C. All this shows why, and how, we have to replace the talk about being by the form ‘is P’, about non-being by ‘is not P’, about becoming by ‘ceasing to be P^C’ and ‘beginning to be P’ and about Dasein by ‘being here and now P’.

In a sense, pure being relates to the distinction between being and not-being just as man to woman and man. Already Parmenides, Plato's ‘father’ of all logic, urged us accordingly to say that there are no non-existing entities. Hegel is just following Aristotle when he sees that the real problem of the obvious ambiguity of formal existence lies in the wrong presumption of a universal domain of all beings as opposed to an absolute nothing. In other words, all well-determined domains for the variable ‘something’ (cf. I: 38, 54; II: 97ff.) are already conceptually limited.

Moreover, interpretation gets lost in translation if we do not account for the fact that Hegel's word ‘Beweis’ refers to a demonstrative showing (Aufweis), ‘Notwendigkeit’ to a resolution of some need or problem—whereas necessitas means: ‘it cannot be otherwise’. All this shows that the developmental steps in Hegel's generalized version of transcendental logical analysis consist in elucidating implicit presuppositions in using categories and talking about abstract entities or concrete objects.

IV. On Frege's unquestioned presuppositions

Houlgate is right that Fregean logic presupposes too many things. However, it is not only ‘the logical distinction between a concept and an object’ (II: 136) and ‘the law of identity’ (II: 137) that pose problems. Hegel and Frege even agree that there are always different representations of the same object or entity. They do not agree, however, about the very constitution of the domains for interpreting the category ‘something’. Hegel calls such a (sortal) domain ‘Begriff’, ‘concept’, whereas a Kantian or Fregean concept just is the content of a sortal predicate in an already presupposed domain of entities with equalities, inequalities and some other relations. Even though nobody would call a prime number a concept, the Fregean schemes of formal logic define it as a predicate in the natural numbers in the following way ‘for all x and y larger than 1 and smaller than z the inequality x⋅y ≠ z holds’. But any attempt to define the numbers in this way fails because there is no predicative definition of a Hegelian or primordial concept like the relational domain of the pure numbers or Cantor's pure sets. Defining a Hegelian concept like being an animal or a middle-sized physical object or body is, in fact, logically entirely different than defining a predicate by Fregean formal means in an already presupposed domain.

Even though some of the arguments have to be sharpened or even turned upside down (since we cannot define numbers starting with pure being—cf. II: 137), the topic is immensely important. Indeed, any sufficiently educated logician and philosopher of science and nature has, from now on, to take at least some of Hegel's considerations on logic and being into account.