Foreshadowing the Concept

This will conclude my walk-through of the Preface to Hegel’s Phenomenology. Here he gives an explicit anticipation of what he calls “the concept”, which will be a key term in the Phenomenology‘s successor work, the Science of Logic. Along with “idea”, “concept” for Hegel represents something that is already beyond the naive opposition of subject and object.

If we imagine the naive view of subject and object as a pair of distinct points, a preliminary analogy for Hegelian concept and idea would be a line between the subject and object points. This can be understood as including all the meaningful content of experience, and can be taken as close as you like to either or both endpoints, but does not include either endpoint. Then the inversion of point of view that Hegel speaks of between ordinary consciousness and the standpoint of his logic would correspond to the relation between seeing experience in terms of the static duality of subject and object, and seeing it in terms of “living” concept and idea.

He begins this part with some remarks about mathematics that are overall very negative-sounding. This is setting up for a contrast between mathematics as the paradigm of static “formal” reasoning, and the meaning-oriented, becoming-oriented “dialectical” reasoning he wants to promote. To put the negative remarks in proper perspective, it is worth knowing that he will devote nearly 200 pages in the early part of the Science of Logic to a serious and sympathetic discussion of mathematics that shows good familiarity with the leading mathematical writers of his day.

“As for mathematical truths, one would hardly count as a geometer if one only knew Euclid’s theorems by heart without knowing the proofs” (p. 25).

Here he repeats the important point that knowledge does not consist in bare conclusions or propositions.

However, I think he goes astray when adds that really, “The movement of mathematical proof does not belong to the object but is a doing that is external to the item at hand” (ibid). I would say almost the opposite: the appearance of externality between theorem and proof — the idea that theorems have a status of simple truth independent of their proofs — reflects the very same kind of error that he pointed out before in the separation of results from the development that produced them.

It is true that a mathematical proof viewed as an object does not consist in the kind of becoming of knowing that Hegel attributes to good philosophical thinking. To mix terminology from computer science and Aristotle, mathematical proofs are in principle “statically” evaluable; this means they do not depend on any runtime accidents. In Platonic terms, mathematical objects are “eternal”, and proof is a kind of strict unfolding of their essence that we can imagine after the fact to have been predetermined, even though we don’t see the full predetermination in advance.

Earlier in the Preface, Hegel has argued that in the genuine becoming of knowing, “accidents” play an essential role, just as I would say they do in any actual working out of Aristotelian teleology. The means is not irrelevant to the end to the extent that we care about the end’s actualization. Like Aristotle, Hegel treats the process of actualization as primary.

Thus he is right that the becoming of knowing that philosophy ought to aim at does not — and ought not to — follow the canons of mathematical proof. In philosophy, we learn as much from our mistakes as from our successes, but errors in mathematics do not present the same kind of opportunities for improving our wisdom. Mathematics is not philosophy but something else. It is not “conceptual” in Hegel’s sense that involves a kind of “life” and “self-movement” of the concept.

However, he goes on to say that “In mathematical cognition, insight is an external doing vis-à-vis the item at issue” (p. 26, emphasis added). I don’t find this to be true today, and think it was, if anything, further from true in Hegel’s day.

Surely the maximal externalization of human insight from proof would be today’s computer-based proofs. While it is now possible to produce purely symbolic proofs whose validity depends only on the syntactic rules of a functional programming language, and sometimes even to produce proofs in a fully automated way, the really big successes of computer-based mathematical proof in recent decades have involved automated proof checkers that eschew fully automated proof development in favor of “dialogue” with an insightful human. At least in the current and foreseeable state of the art, human insight is not at all external to the development of mathematical proofs, even though the checking of completed or partial proofs for errors can be fully automated.

I say that mathematics is not philosophy, but its practice is far from being the mindlessly formal “defective cognition” he makes it out to be here in the Preface. Mathematical objects including completed proofs are static, but I say that the doing of mathematics essentially involves the activity of human intelligence.

“[W]hat is formal in mathematical convincingness consists in this — that knowing advances along the line of equality. Precisely because it does not move itself, what is lifeless does not make it all the way to the differences of essence…. For it is magnitude alone, the inessential difference, that mathematics deals with” (p. 27).

Mathematics only deals with things that are in principle strictly univocal. Strictly univocal things lack “life” for Hegel, and are therefore inessential.

“In contrast, philosophy does not study inessential determinations but only those that are essential. The abstract or the non-actual is not its element and content; rather, its element and content is the actual, what is self-positing, what is alive within itself, or existence in its concept. It is the process which creates its own moments and passes through them all; it is the whole movement that constitutes the positive and its truth. This movement just as much includes within itself the negative ” (p. 28).

Philosophy for Hegel is especially concerned with actuality, and as with Aristotle, what is actual is not simply to be identified with what is factual.

“Appearance is both an emergence and a passing away which does not itself emerge and pass away… which constitutes the actuality and the living moment of truth…. Judged in the court of that movement, the individual shapes of spirit do not stably exist any more than do determinate thoughts, but they are also equally positive, necessary moments just as much as they are negative, disappearing moments” (pp. 28-29).

Here he is using “appearance” in a very different way from what Plato called mere appearance. It seems to be something like the concrete manifestation that is necessarily implicit in actuality.

“In the whole of the movement… what distinguishes itself in it and what gives itself existence is preserved as the kind that remembers, as that whose existence is its knowing of itself” (p. 29).

Previously, he said that the true is the whole. In this movement of self-knowing, which is quite different from being an object for oneself, the subject and object that are quite distinct for ordinary consciousness become interwoven.

“It might seem necessary to state at the outset the principal points concerning the method of this movement…. However, its concept lies in what has already been said, and its genuine exposition belongs to logic, or is instead even logic itself, for the method is nothing but the structure of the whole in its pure essentiality” (ibid).

The entry point for what Hegel calls “logic” is what I have glossed as being at home in otherness. For Hegel, logic is not about formal manipulations. It is a very non-ordinary way of looking at things that leaves distinctions of subject and object behind. The Phenomenology is supposed to provide a way into this perspective, starting out from what Aristotle would call the way things (ordinarily) are “for us” (see Otherness; At Home in Otherness).

“In everyday life, consciousness has for its content little bits of knowledge, experiences, sensuous concretions, as well as thoughts, principles, and, in general, it it has its content in whatever is present, or in what counts as a fixed, stable entity or essence…. [I]t conducts itself as if it were an external determining and manipulation of that content” (p. 30).

Ordinary consciousness regards things in the world as fixed, pre-known, and manipulable. It regards itself as somehow standing off to the side from the order of the world, and implicitly as able to act in complete independence from that order. It is “Cartesian”. The weakness of this point of view is progressively exhibited in the Phenomenology.

“Science may organize itself only through the proper life of the concept…. [D]eterminateness… is in science the self-moving soul of the content which has been brought to fulfillment. On the one hand, the movement of ‘what is’ consists in becoming an other to itself and thus in coming to be its own immanent content; on the other hand, it takes this unfolding back into itself, or it takes its existence back into itself, which is to say, it makes itself into a moment, and it simplifies itself into determinateness” (p. 33).

Hegelian rational “science”, sustained in otherness, examines a movement of “logical” unfolding and return that (unlike the unfolding and return in neoplatonism) occurs not in eternity but in worldly coming-to-be. The fact that the return occurs in becoming and in time gives it the form not of a simple circle but of an open-ended spiral that never literally returns to its origin.

“[S]cientific cognition requires… that it give itself over to the life of the object” (ibid, emphasis added).

In the main body of the Phenomenology, the Consciousness chapter shows the limitations of the ordinary view that we are wholly separate from the object, and the Self-Consciousness chapter develops a sharp critique of the attitude of the master who attempts to claim unilateral control over both objects and other people.

“[T]he stable being of existence… is itself its own inequality with itself and its own dissolution — its own inwardness and withdrawal into itself — its coming-to-be. — Since this is the nature of what exists, and to the extent that what exists has this nature for knowing, this knowing is not an activity which treats the content as alien. It is not a reflective turn into itself out of the content… [W]hile knowing sees the content return into its own inwardness, its activity is instead sunken into that content, for the activity is the immanent self of the content as having at the same time returned into itself, since this activity is pure self-equality in otherness” (p. 34).

Here we have a direct statement about what overcoming alienation ought to look like.

“Its determinateness at first seems to be only through its relating itself to an other, and its movement seems imposed on it by an alien power. However, … it has its otherness in itself…, for this is the self-moving and self-distinguishing thought, the thought which is its own inwardness, which is the pure concept. In that way, the intelligibility of the understanding is a coming-to-be, and as this coming-to-be, it is rationality” (p. 35).

Overcoming alienation is anything but the suppression of what is other. Neither is it a return to an original perfection. Rather, it consists in a non-ordinary sense of self that is not opposed to the other or to the field of otherness.

“Logical necessity in general consists in the nature of what it is [for something] to be its concept in its being. This alone is the rational, the rhythm of the organic whole, and it is just as much the knowing of the content as that content itself is the concept and the essence…. The concrete shape which sets itself into movement… is only this movement, and [its concrete existence] is immediately logical existence. It is therefore unnecessary to apply externally a formalism to the concrete content. That content is in its own self a transition into this formalism, but it ceases to be the latter external formalism because the form is the indigenous coming-to-be of the concrete content itself” (ibid).

In emphasizing the contentfulness of the concept rather than formal syntax as the true driver of logical necessity, he seems to be talking about something like what Sellars and Brandom call material inference.

“Although what is stated here expresses the concept, it cannot count as more than an anticipatory affirmation. Its truth does not lie in this narrative exposition” (p. 36, emphasis added).

Truth, once again, must lie in an extensive development that is never truly finished by us humans. This remark could reasonably apply to the whole Preface, but I am struck by the reference to the concept and by the place in which it occurs, just after an explicit reference to logic. Here he is looking forward not only to the main body of the Phenomenology, but even more so to what will become the Science of Logic.

He goes on to criticize “clever argumentative thinking” at length, and to contrast it with “comprehending thinking”.

“[C]lever argumentation amounts to freedom from content and to the vanity that stands above all content” (p. 36).

By Hegel’s high standards, any argument that assumes meanings are determined in advance at least tends toward the vanity and irresponsibility of what Plato and Aristotle denounced as sophistry.

Hegel wants to recommend instead that “This vanity is expected to give up this freedom, and, instead of being the arbitrary principle moving the content, it is supposed to let this freedom descend into the content and move itself by its own nature…. This refusal both to insert one’s own views into the immanent rhythm of the concept and to interfere arbitrarily with that rhythm by means of wisdom acquired elsewhere, or this abstinence, are all themselves an essential moment of attentiveness to the concept” (pp. 36-37).

Moreover, what plays the role of the subject of thought is not at all the same for comprehending thinking as it is for clever argumentation.

“[C]lever argumentative thinking is itself the self into which the content returns, and so too, the self in its positive cognition is a represented subject to which the content is related as accident or predicate. This subject constitutes the basis in which the content is bound and on the basis of which the movement runs back and forth” (p. 37).

He continues, “Comprehending thinking conducts itself in quite a different way. While the concept is the object’s own self, or the self which exhibits itself as the object’s coming-to-be, it is not a motionless subject tranquilly supporting the accidents; rather, it is the self-moving concept which takes its determinations back into itself. In this movement, the motionless subject itself breaks down; it enters into the differences and the content and constitutes the determinateness, which is to say, the distinguished content as well as the content’s movement, instead of continuing simply to confront that movement” (pp. 37-38).

Comprehending thinking “enters into the differences and the content”.

“[T]here is an obstacle based in the habit of grasping the speculative predicate according to the form of a proposition instead of grasping it as concept and essence” (p. 41).”

The form of a proposition is simply to be true or false. He may also have in mind the form of predication. Grasping something as concept and essence is treating it as articulable meaning to be interpreted, rather than as a mere thing to be pointed at.

“True thoughts and scientific insight can be won only by the labor of the concept. Concepts alone can produce the universality of knowing” (p. 44).

Otherness

I wanted to elaborate a bit on what I see “otherness” as doing in the part of Hegels’ text that formed the subject of the previous post. Cambridge University Press provided only a skimpy index, which scandalously includes no entry at all for this key term. I don’t specifically recall “otherness” being literally used in the main body of the Phenomenology, though it may well exist somewhere. What I find googling “Hegel otherness” seems entirely devoted to the relation of self-consciousness to other people. Quick review of top results failed to turn up a supporting quote from Hegel using the literal term “otherness” in that way, however. This leaves it unclear to me whether this more social usage of “otherness” is even literally Hegelian, or is rather a term interpolated by commentators.

Relations to other rational beings are essential to Hegelian self-consciousness, to the point where I have quipped that it might better be called other-consciousness. This social and ethical meaning of otherness is not irrelevant to the current context. However, I take Hegel’s use of “otherness” in the Phenomenology Preface to be primarily “logical” in his special sense, rather than social.

In the Preface, Hegel calls Anderssein (otherness; literally, “being-other”) the “element” and the “ether” in which knowing occurs. Hegel is using “knowing” in a very broad sense here, encompassing everything from the mere acquaintance of ordinary consciousness with objects, to the pinnacles of philosophy. He begins to develop otherness by way of implicit contrast with that other element of “familiarity” and “representation” that he mentions as an obstacle to the higher development of knowing.

He explicitly calls otherness the element of “science” (rational understanding) in knowing, while implying that familiarity and representation characterize a contrasting element of immediacy that he sees as an obstacle to “science”. For Hegel, “science” is first and foremost the “logic” that was to form the first part of the “system” the Phenomenology was to introduce, so it could equally be said that otherness here is the unfamiliar standpoint of Hegelian logic, for which the whole long detour of the Phenomenology is intended to gently and patiently prepare us.

Once again, I take a deflationary approach to his rhetoric about “science” and “system”. In general with Hegel, rather than starting with ordinary assumptions about what his terms mean, it is best to interpret them in light of what he does. Here otherness provides a first thematic anticipation of the general point of view Hegel wants to recommend, and in particular of what is at stake in Hegelian “logic”, “science”, and “system”.

As a first approximation then, we have otherness expressed as the “element of knowing” that the Science of Logic will later develop, initially expressed by way of a contrast with a point of view centered on immediacy, familiarity, and representation.

There seems to be a kind of analogy between this contrast and what I read as the Phenomenology‘s other big contrast between the standpoints of consciousness and self-consciousness. I think Hegel’s view is that neither of these latter is ever found entirely independent of the other in real life, but at the same time that the alienation inherent to the relation of ordinary consciousness to objects is eventually to be overcome by dwelling primarily in what he calls self-consciousness and spirit. The higher phases of self-consciousness and spirit will be characterized by an openness to otherness.

The contrast between the feeling of otherness and those of familiarity and immediacy gives us a first starting point that we can grasp even within the standpoint of the most naive ordinary consciousness. The second contrast between the standpoint of otherness and the standpoint of representation brings this into sharper focus.

In the Preface, Hegel only hints at his very strong reservations about the place of representation in early modern mainstream views of knowledge such as those of Descartes and Locke. But in the Consciousness chapter of the Phenomenology, the alienated relation of consciousness to objects broadly captures aspects of the views of Descartes and Locke, who were the two great representationalist promoters of “consciousness” in philosophy (literally in Locke, and its ancestor French conscience in Descartes; see Consciousness in Locke and Hegel).

We cannot communicate without representation, any more than we can exercise higher functions without consciousness. But Hegel’s implicit critique of representation in the Preface and his more developed critique of consciousness in the Consciousness chapter together constitute a vital thread of his argument. His repeated warnings against taking “fixed thoughts” at face value and against taking propositions in isolation are closely tied to this.

Otherness challenges both fixed representations of thoughts and an overly fixed notion of self. Self from the perspective of otherness is a contextual, relational and adverbial term, not an independently contentful noun with a reference fixed once and for all (see The Ambiguity of “Self”).

What I think he is suggesting is a strong conclusion that in explaining meaning, we ought as much as possible to subordinate the point of view associated with representation, consciousness, objects, immediacy, and familiarity, rather than treating all of these as foundational touchstones.

What we ought to subordinate them to is developed throughout the rest of the Phenomenology, but especially involves the actualization of self-consciousness, and of forms of spirit that are not merely what he calls substantial, but are self-conscious, and thus for Hegel depend essentially on relations of mutual recognition.

A fully developed self-consciousness will be “at home” in otherness.

Here in the Preface, I think he is suggesting an argument complementary to that of the Phenomenology‘s main thread. In the Preface, the accent seems to be on knowing as such, whereas I take the overall thrust of the main thread to be primarily ethical in intent. Here too, at least in a general sense the Preface is closer to the concerns of what Hegel calls “logical” inquiry. The critique of the classic early modern concept of representation falls in this area.

Foundational uses of representation are based on strong presuppositions about the identity of represented things (the “fixed thoughts” to which Hegel is objecting). Representationalist theories of meaning focus on the ways in which representations are supposed to unambiguously refer to objects, which basically reduces meaning to a kind of implicit pointing at things that are presumed to be unambiguously identifiable. But this is a huge presumption that Hegel wants to question.

Alternatively, the meaning of representations can be explained in terms of form, value, internal structure, and inter-relations, all of which I think for Hegel are potentially articulable complete in themselves “in the element of otherness”, without any pointing or presumption required. Otherness thus appears to stand for coherence over reference and difference over identity in the explanation of meaning. Again, that is not to suggest that reference is absent, just that it ought not to dominate or primarily drive our explanations.

Finally, Hegel would remind us that even pure difference or pure coherence also needs to be considered from the point of view of its becoming and not just one static view. Otherness as an orientation toward difference and coherence in their becoming gives us a first approximation of the concerns Hegel means to bring to the fore when he speaks of dialectic. (See also Pure Negativity?; Teleology After Kant.)

Cause

Aristotle flourished before the great flowering of Greek mathematics that gave us Euclid, Ptolemy, Apollonius, and Aristarchus. In his day, mathematics amounted to just arithmetic and simple geometry. In spite of the famous Pythagorean theorem that the square constructed from the hypotenuse of a right triangle is equal in area to the sum of the squares constructed from the other two sides, the historic reality of the Pythagorean movement had more to do with number mysticism, other superstitions, and curious injunctions like “don’t eat beans” than it did with real mathematics.

I think Aristotle was entirely right to conclude that arithmetic and simple geometry were of little use for explaining change in the natural world. I’ve characterized his physics as grounded in a kind of semantic inquiry that Aristotle pioneered. We are not used to thinking about science this way, as fundamentally involved with a very human inquiry about the meaning of experience in life, rather than predictive calculation. For Aristotle, the gap between natural science and thoughtful reflection about ordinary experience was much smaller than it is for us.

Aristotle invented the notion of cause as a semantic tool for expressing the reasons why changes occur. Aristotle’s notion is far more abstract than the metaphor of impulse or something pushing on something else that guided early modern mechanism. Even though the notion of cause was originally developed in a text included in Aristotle’s Physics, the “semantic” grounding of Aristotelian physics places it closer to logic than to modern physical inquiries.

I think the discussion of the kinds of causes could equally well have been grouped among his “logical” works. In fact, the form in which we have Aristotle’s works today is the result of the efforts of multiple ancient editors, who sometimes stitched together separate manuscripts, so there is room for a legitimate question whether the discussion of causes was originally a separate treatise. We tend to assume that there must be something inherently “physical” about the discussion of causes, but this is ultimately due to a circular argument from the fact that the more detailed version of it came down to us as part of the Physics (there is another, briefer one that came down to us as part of the Metaphysics).

Since Hume and especially since the later 19th century, many authors have debated about the role of causes in science. Bertrand Russell argued in the early 20th century that modern science does not in fact depend on what I have called the modern notion of cause.

More recently, Robert Brandom has argued that the purpose of logic is “to make explicit the inferential relations that articulate the semantic contents of the concepts expressed by the use of ordinary, nonlogical vocabulary”. I see Aristotelian causes in this light.

I want to recommend a return to a notion of causes in general as explanatory reasons rather than things that exert force. This can include all the mathematics used in modern science, as well as a broader range of reasons relevant to life. (See also Aristotelian Causes; Mechanical Metaphors; Causes: Real, Heuristic?; Effective vs “Driving”; Secondary Causes.)

Logic for Expression

In recent times, Robert Brandom has pioneered the idea that the role of logic is primarily expressive. In his 2018 essay “From Logical Expressivism to Expressivist Logic”, he says this means its purpose is “to make explicit the inferential relations that articulate the semantic contents of the concepts expressed by the use of ordinary, nonlogical vocabulary” (p. 70).

In my humble opinion, this is what logic was really supposed to be about in Aristotle, but the tradition did not follow Aristotle. Aristotle insisted that logic is a “tool” not a science, but most later authors have assumed the contrary — that logic was the “science” of correct reasoning, or perhaps the science of consequence relations. Several scholars have nonetheless rediscovered the idea that the purpose of logical demonstration in Aristotle is not to prove truths, but to express reasoned arguments as clearly as possible.

Brandom says that “the task of logic is to provide mathematical tools for articulating the structure of reasoning” (p. 71). People were reasoning in ordinary life long before logic was invented, and continue to do so. But the immensely fertile further development of logic in the late 19th and early 20th centuries was mostly geared toward the formalization of mathematics. Reasoning in most specialized disciplines — such as the empirical sciences, medicine, and law — actually resembles reasoning in ordinary life more than it does specifically mathematical reasoning.

According to Brandom, “The normative center of reasoning is the practice of assessing reasons for and against conclusions. Reasons for conclusions are normatively governed by relations of consequence or implication. Reasons against conclusions are normatively governed by relations of incompatibility. These relations of implication and incompatibility, which constrain normative assessment of giving reasons for and against claims, amount to the first significant level of structure of the practice of giving reasons for and against claims.”

“These are, in the first instance, what Sellars called ‘material’ relations of implication and incompatibility. That is, they do not depend on the presence of logical vocabulary or concepts, but only on the contents of non- or prelogical concepts. According to semantic inferentialism, these are the relations that articulate the conceptual contents expressed by the prelogical vocabulary that plays an essential role in formulating the premises and conclusions of inferences” (pp. 71-72).

“Material” relations of consequence and incompatibility have a different structure from formal ones. Formal consequence is monotonic, which means that adding new premises does not change the consequences of existing premises. Formal contradiction is “explosive”, in the sense that any contradiction whatsoever makes it possible to “prove” anything whatsoever (both true statements and their negations), thereby invalidating the very applicability of proof. But as Brandom reminds us, “outside of mathematics, almost all our actual reasoning is defeasible” (p. 72). Material consequence is nonmonotonic, which means that adding new premises could change the consequences of existing ones. Material incompatibilities can often be “fixed” by adding new, specialized premises. (As I somewhere heard Aquinas was supposed to have said, “When faced with a contradiction, introduce a distinction”.)

Brandom notes that “Ceteris paribus [“other things being equal”] clauses do not magically turn nonmonotonic implications into monotonic ones. (The proper term for a Latin phrase whose recitation can do that is ‘magic spell’.) The expressive function characteristic of ceteris paribus clauses is rather explicitly to mark and acknowledge the defeasibility, hence nonmonotonicity, of an implication codified in a conditional, not to cure it by fiat” (p. 73).

“There is no good reason to restrict the expressive ambitions with which we introduce logical vocabulary to making explicit the rare material relations of implication and incompatibility that are monotonic. Comfort with such impoverished ambition is a historical artifact of the contingent origins of modern logic in logicist and formalist programs aimed at codifying specifically mathematical reasoning. It is to be explained by appeal to historical causes, not good philosophical reasons” (ibid). On the other hand, making things explicit should be conservative in the sense of not changing existing implications.

“…[W]e should not emulate the drunk who looks for his lost keys under the lamp-post rather than where he actually dropped them, just because the light is better there. We should look to shine light where we need it most” (ibid).

For relations of material consequence, the classical principle of “explosion” should be replaced with the weaker one that “if [something] is not only materially incoherent (in the sense of explicitly containing incompatible premises) but persistently so, that is incurably, indefeasibly
incoherent, in that all of its supersets are also incoherent, then it implies everything” (p. 77).

“The logic of nonmonotonic consequence relations is itself monotonic. Yet it can express, in the logically extended object language, the nonmonotonic relations of implication and incompatibility that structure both the material, prelogical base language, and the logically compound sentences formed from them” (p. 82).

Material consequence relations themselves may or may not be monotonic. Instead of requiring monotonicity globally, it can be declared locally by means of a modal operator. “Logical expressivists want to introduce logical vocabulary that explicitly marks the difference between those implications and incompatibilities that are persistent under the addition of arbitrary auxiliary hypotheses or collateral commitments, and those that are not. Such vocabulary lets us draw explicit boundaries around the islands of monotonicity to be found surrounded by the sea of nonmonotonic material consequences and incompatibilities” (p. 83).

Ranges of subjunctive robustness can also be explicitly declared. “The underlying thought is that the most important information about a material implication is not whether or not it is monotonic — though that is something we indeed might want to know. It is rather under what circumstances it is robust and under what collateral circumstances it would be defeated” (p. 85).

“The space of material implications that articulates the contents of the nonlogical concepts those implications essentially depend upon has an intricate localized structure of subjunctive robustness and defeasibility. That is the structure we want our logical expressive tools to help us characterize. It is obscured by commitment to global structural monotonicity—however appropriate such a commitment might be for purely logical relations of implication and incompatibility” (pp. 85-86).

“Logic does not supply a canon of right reasoning, nor a standard of rationality. Rather, logic takes its place in the context of an already up-and-running rational enterprise of making claims and giving reasons for and against claims. Logic provides a distinctive organ of self-consciousness for such a rational practice. It provides expressive tools for talking and thinking, making claims, about the relations of implication and incompatibility that structure the giving of reasons for and against claims” (p. 87).

Real-World Reasoning

I think most people most of the time are more influenced by apprehended or assumed meanings than by formal logic. What makes us rational animals is first of all the simple fact that we have commitments articulated in language. The interplay of language and commitment opens us to dialogue and the possibility of mutual recognition, which simultaneously ground both values and objectivity. This opening, I’d like to suggest, is what Hegel called Spirit. (See also Interpretation.)

Predication

It is extremely common to see references to “predication” as if it were a central concept of Aristotelian logic. We are so used to a grammatical interpretation in terms of relations between subjects and predicates that it is hard to disengage from that. However, historically it was Aristotelian logic that influenced ancient Greek accounts of grammar, not vice versa.

Modern logicians distinguish between a neutral proposition — which might be merely mentioned, rather than asserted — and the assertion of the proposition. Grammatical predication in itself does not imply any logical assertion, only a normatively neutral syntactic relation between sentence components. But “said of” in Aristotle always refers to some kind of meaningful assertion that has a normative character, not to grammatical predication.

Aristotle talks about what we might call kinds of “sayables” (“categories”). He famously says that we can only have truth or falsity when one kind of sayable is “said of” another. Mere words or phrases by themselves don’t assert anything, and hence cannot be true or false; for that we need what modern writers have referred to as a “complete thought”.

The ordinary meaning of “to categorize” in ancient Greek was “to accuse in a court of law”. Aristotle used it to talk about assertions. It didn’t originally connote a classification. The modern connotation of classification seems to stem from the accident that independent of what “category” meant in his usage, Aristotle famously developed a classification of “categories”.

Aristotle also talks about logical “judgment” (apophansis, a different word from practical judgment or phronesis). Husserl for instance transliterated this to German, and followed the traditional association of logical judgment with “predication”. But the ordinary Greek verb apophainein just means to show or make known. Aristotle’s usage suggests a kind of definite assertion or expressive clarification related to demonstration, which makes sense, because demonstrations work by interrelating logical judgments.

All of Aristotle’s words and phrases that get translated with connotations of “predication” actually have to do with normative logical assertion, not any connecting of a grammatical subject with a grammatical predicate. Nietzsche and others have complained about the metaphysical status foisted on grammatical subjects, implicitly blaming Aristotle, but all these connotations are of later date.

The great 20th century scholar of ancient and medieval logic and semantics L. M. de Rijk in his Aristotle: Semantics and Ontology (2002) argued at length that Aristotle’s logical “is” and “is not” should be understood as not as binary operators connecting subjects and predicates, but as unary operators of assertion and negation on whole propositions formed from pairs of terms. (See also Aristotelian Propositions.)

As in similar cases, by no means do I wish to suggest that all the work done on the basis of the common translation of “predication” is valueless; far from it. But I think we can get additional clarity by carefully distinguishing the views and modes of expression of Aristotle himself from those of later commentators and logicians, and I think Aristotle’s own more unique perspectives are far fresher and more interesting than even good traditional readings would allow.

Formal and Material Interpretation

Human reasoning has two sides, that could be called formal and material. Any reasoning applicable to the real world necessarily involves the “material” side that is concerned with actual meaning “content”. It may also involve the “formal” side, which aims to express reasoning in terms of mechanically repeatable operations that are completely agnostic to the actual meanings they are used to operate on. Reasoning in some abstract contexts may rely entirely on the formal side.

Aristotle is usually credited with inventing formal logic, but he paid a lot of attention to the material side as well. In the Latin middle ages both sides were recognized, but the formal side was generally emphasized.

Formal mathematical logic underwent an immense development in the 20th century, somewhat like the earlier success story of mathematical physics. The syntactic devices of mathematical logic seemed so powerful that its rise led to a great neglect of the material, interpretive side of logic. Husserl was one of the few 20th century authors who questioned this from the start. More recently, Brandom has argued that Kant and Hegel were both fundamentally concerned with the material, interpretive side of logic, and that this is what Kant meant by “transcendental” logic (and what Hegel meant by “dialectic”).

Generally when I mention interpretation here, I have the material side in mind, but there is also such a thing as formal interpretation. Formal interpretation or “evaluation” of expressions in terms of other expressions is the most fundamental thing that interpreters and compilers for programming languages do. As with material interpretation, formal interpretation makes meanings explicit by expressing them in terms of more elementary distinctions and entailments, but it uses purely syntactic substitution and rewriting to do so.

Material interpretation can always potentially go on indefinitely, explaining real-world meanings by relating them to other meanings, and those in terms of others, and so on. In practice, we always cut it short at some point, once we achieve a relatively stable network of dependencies.

Formal interpretation on the other hand is usually engineered to be decidable, so that it actually does reach an end at some point. The fact that it reaches an end is closely related to the fact that precise formal models are always in some sense only approximations of a determination of reality that is actually open-ended. Formal models are a sort of syntactic reification of open-ended material interpretation. We may think we have taken them as far as they can go, but in real life it is always possible that some new case will come up that requires new detail in the model.

We also use a kind of formal interpretation alongside material interpretation in our spontaneous understanding of natural language. Natural language syntax helps us understand natural language meaning. It provides cues for how different clauses are intended to relate to one another. Is what is meant in this clause an exception? A consequence? A presupposition? A fact? A recommendation? Something being criticized? (See also Formal and Informal Language.)

Logic for People

Leading programming language theorist Robert Harper refers to so-called constructive or intuitionistic logic as “logic as if people mattered”. There is a fascinating convergence of ideas here. In the early 20th century, Dutch mathematician L. E. J. Brouwer developed a philosophy of mathematics called intuitionism. He emphasized that mathematics is a human activity, and held that every proof step should involve actual evidence discernible to a human. By contrast, mathematical Platonists hold that mathematical objects exist independent of any thought; formalists hold that mathematics is a meaningless game based on following rules; and logicists argue that mathematics is reducible to formal logic.

For Brouwer, a mathematical theorem is true if and only if we have a proof of it that we can exhibit, and each step of that proof can also be exhibited. In the later 19th century, many new results about infinity — and infinities of infinities — had been proved by what came to be called “classical” means, using proof by contradiction and the law of excluded middle. But from the time of Euclid, mathematicians have always regarded reproducible constructions as a better kind of proof. The law of excluded middle is a provable theorem in any finite context. When the law of excluded middle applies, you can conclude that if something is not false it must be true, and vice versa. But it is not possible to construct any infinite object.

The only infinity we actually experience is what Aristotle called “potential” infinity. We can, say, count a star and another and another, and continue as long as you like, but no actually infinite number or magnitude or thing is ever available for inspection. Aristotle famously defended the law of excluded middle, but in practice only applied it to finite cases.

In mathematics there are conjectures that are not known to be true or false. Brouwer would say, they are neither true nor false, until they are proved or disproved in a humanly verifiable way.

The fascinating convergence is that Brouwer’s humanly verifiable proofs turn out also to exactly characterize the part of mathematics that is computable, in the sense in which computer scientists use that term. Notwithstanding lingering 20th century prejudices, intuitionistic math actually turns out to be a perfect fit for computer science. I use this in my day job.

I am especially intrigued by what is called intuitionistic type theory, developed by Swedish mathematician-philosopher Per Martin-Löf. This is offered simultaneously as a foundation for mathematics, a higher-order intuitionistic logic, and a programming language. One might say it is concerned with explaining ultimate bases for abstraction and generalization, without any presuppositions. One of its distinctive features is that it uses no axioms, only inference rules. Truth is something emergent, rather than something presupposed. Type theory has deep connections with category theory, another truly marvelous area of abstract mathematics, concerned with how different kinds of things map to one another.

What especially fascinates me about this work are its implications for what logic actually is. On the one hand, it puts math before mathematical logic– rather than after it, as in the classic early 20th century program of Russell and Whitehead — and on the other, it provides opportunities to reconnect with logic in the different and broader, less formal senses of Aristotle and Kant, as still having something to say to us today.

Homotopy type theory (HoTT) is a leading-edge development that combines intuitionistic type theory with homotopy theory, which explores higher-order paths through topological spaces. Here my ignorance is vast, but it seems tantalizingly close to a grand unification of constructive principles with Cantor’s infinities of infinities. My interest is especially in what it says about the notion of identity, basically vindicating Leibniz’ thesis that what is identical is equivalent to what is practically indistinguishable. This is reflected in mathematician Vladimir Voevodsky’s emblematic axiom of univalence, “equivalence is equivalent to equality”, which legitimizes much actual mathematical practice.

So anyway, Robert Harper is working on a variant of this that actually works computationally, and uses some kind of more specific mapping through n-dimensional cubes to make univalence into a provable theorem. At the cost of some mathematical elegance, this avoids the need for the univalence axiom, saving Martin-Löf’s goal to avoid depending on any axioms. But again — finally getting to the point of this post — in a 2018 lecture, Harper says his current interest is in a type theory that is in the first instance computational rather than formal, and semantic rather than syntactic. Most people treat intuitionistic type theory as a theory that is both formal and syntactic. Harper recommends that we avoid strictly equating constructible types with formal propositions, arguing that types are more primitive than propositions, and semantics is more primitive than syntax.

Harper disavows any deep philosophy, but I find this idea of starting from a type theory and then treating it as first of all informal and semantic rather than formal and syntactic to be highly provocative. In real life, we experience types as accessibly evidenced semantic distinctions before they become posited syntactic ones. Types are first of all implicit specifications of real behavior, in terms of distinctions and entailments between things that are more primitive than identities of things.

Things Themselves

Husserl continues his Logical Investigations with a long critical discussion of the then-current tendency to reduce logic to psychological “laws” of mental operations, which are in turn supposed to be reducible to empirically discoverable facts. He then begins to discuss what a pure logic ought to be. “We are rather interested in what makes science science, which is certainly not its psychology, nor any real context into which acts of thought are fitted, but a certain objective or ideal interconnection which gives these acts a unitary relevance, and, in such unitary relevance, an ideal validity” (p. 225).

To do this, we need to look at both things and truths from the point of view of their interconnections. In his famous phrase, we need to go “to the things themselves”. As Aristotle emphasized before, we need to look carefully at distinctions of meaning.

Expressive meanings are not the same thing as indicative signs. Meaning for Husserl is not reducible to what it refers to; it originates in a kind of act, though it is not to be identified with the act, either. Verbal expressions have an “intimating” function. “To understand an intimation is not to have conceptual knowledge of it… it consists simply in the fact that the hearer intuitively takes the speaker to be a person who is expressing this or that” (p. 277). “Mutual understanding demands a certain correlation among the acts mutually unfolded in intimation…, but not at all in their exact resemblance” (p. 278). “In virtue of such acts, the expression is more than a sounded word. It means something, and insofar as it means something, it relates to what is objective” (p. 280). “The function of a word… is to awaken a sense-conferring act in ourselves” (p. 282).

“Our interest, our intention, our thought — mere synonyms if taken in sufficiently wide senses — point exclusively to the thing meant in the sense-giving act” (p. 283). “[A]ll objects and relations among objects only are what they are for us, through acts of thought essentially different from them, in which they become present to us, in which they stand before us as unitary items that we mean” (ibid).

“Each expression not merely says something, but says it of something: it not only has a meaning, but refers to certain objects” (p. 287). “Two names can differ in meaning but can name the same object” (ibid). “It can happen, conversely, that two expressions have the same meaning but a different objective reference” (p. 288). “[A]n expression only refers to an objective correlate because it means something, it can rightly be said to signify or name the object through its meaning” (p. 289). “[T]he essence of an expression lies solely in its meaning” (ibid).

“Expressions and their meaning-intentions do not take their measure, in contexts of thought and knowledge, from mere intuition — I mean phenomena of external or internal sensibility — but from the varying intellectual forms through which intuited objects first become intelligibly determined, mutually related objects” (ibid). Meanings do not have to do with mental images.

“It should be quite clear that over most of the range both of ordinary, relaxed thought and the strict thought of science, illustrative imagery plays a small part or no part at all…. Signs are in fact not objects of our thought at all, even surrogatively; we rather live entirely in the consciousness of meaning, of understanding, which does not lapse when accompanying imagery does so” (p. 304). “[A]ny grasp is in a sense an understanding and an interpretation” (p. 309).

“Pure logic, wherever it deals with concepts, judgments, and syllogisms, is exclusively concerned with the ideal unities that we here call ‘meanings'” (p. 322). “[L]ogic is the science of meanings as such, of their essential sorts and differences, as also of the ideal laws which rest purely on the latter” (p. 323). “Propositions are not constructed out of mental acts of presentation or belief: when not constructed out of other propositions, they ultimately point back to concepts…. The relation of necessary consequence in which the form of an inference consists, is not an empirical-psychological connection among judgements as experiences, but an ideal relation among possible statement-meanings” (p. 324).

“Though the scientific investigator may have no reason to draw express distinctions between words and symbols, on the one hand, and meaningful thought-objects, on the other, he well knows that expressions are contingent, and that the thought, the ideally selfsame meaning, is what is essential. He knows, too, that he does not make the objective validity of thoughts and thought-connections, … but that he sees them, discovers them” (p. 325).

“All theoretical science consists, in its objective content, of one homogeneous stuff: it is an ideal fabric of meanings” (ibid). “[M]eaning, rather than the act of meaning, concept and proposition, rather than idea and judgement, are what is essential and germane in science” (ibid). “The essence of meaning is seen by us, not in the meaning-conferring experience, but in its ‘content'” (p. 327).

Husserl on Normativity

Translator J. N. Findlay ranks Husserl (1859-1938) with Plato, Aristotle, Kant, and Hegel, and calls Husserl’s Logical Investigations (1899-1901) his greatest work. My previous acquaintance with Husserl has been limited to his later, explicitly “phenomenological” period.

In the first two chapters, Husserl surveys and criticizes the then-dominant views of Utilitarian John Stuart Mill and his followers on the nature of logic, objecting that they reduced it to a “technology dependent on psychology” (p. 56). Frege had already introduced mathematical logic, but the great flowering of the latter had not occurred yet. Husserl in these chapters is particularly concerned with the objectivity of knowledge, and with principles of validation.

I was initially confused by his polemic against the claim that logic is a “normative discipline”. To me, “normative” means “axiological”, i.e., concerned with value judgments. I take the Aristotelian view that judgment refers first of all to a process of evaluation, rather than a conclusion. In this sense, judgment and normativity inherently involve a Socratic dimension of genuinely open inquiry about what is good.

All versions of normativity involve a “should”. But it turns out that the view Husserl is polemicizing against treated a “normative discipline” as one that takes some particular and predetermined end for granted, and is only concerned with what we “should” do to realize that predetermined end. On this view, “normativity” is only concerned with necessary and/or sufficient conditions for achieving predetermined ends. Thus Husserl associates it with a sort of technology, rather than with something ultimately ethical. So, what he is doing here is rejecting a merely technological view of normativity.

There is also a theoretical-versus-practical axis to Husserl’s argument. Aristotle had contrasted the ability to successfully perform an operation with the ability to explain the principles governing it. One does not necessarily imply the other. Husserl notes how many activities in life are merely oriented toward operational success, and says that most of the practice of modern sciences — including mathematics — has a mainly operational character.

Elsewhere I have contrasted “tool-like” reason with what I like to call ethical reason, but I don’t think they are mutually exclusive, and my notion of “tool-like” reason has potentially rather more positive connotations than that toward which Husserl seems to be leading. I don’t take the fact that engineering tends to drive science to be inherently bad. I think engineering can drive science in a good way, involving an integral consideration of ends; a concern with good design guided by those ends and the best practices we can come up with; and a recognition that the real world doesn’t always cooperate with our intentions.

On the other hand, I also find that the best engineering relies more on fundamental theoretical insight and well-rounded judgment than on sheer technology. This is a perspective that is simultaneously “practical” and concerned with first principles. When Husserl argues for the priority of theoretical disciplines over practical ones, he is mainly arguing for the importance of a concern for first principles. While I generally prefer the Kantian/Brandomian primacy of practical reason, I find common ground with Husserl in the concern for principles.