Reason Relations

“The construction gestured at so far foreshadows an argument for understanding reason relations of consequence and incompatibility as constituting a structure common to representational meaning and to practical use, to truth-making and to justificatory practices, to the objective world talked about and to the activities of talking about it, to what is represented and to the representing of it. That these same reason relations show up from the two otherwise disparate perspectives afforded by (the right kind of) semantic and pragmatic metavocabularies offers some reason to think of those relations as central to language or discourse as such” (Brandom in Hlobil and Brandom, p. 11).

Hlobil and Brandom’s Reasons for Logic presents major new results. In the technical part, Hlobil presents not just one but two very detailed new isomorphisms that unexpectedly seem to unify previously disparate areas of research in a convincing way. I will barely skim the surface of all that is afoot here. My goal is just to work through a few more pages of the motivational part, which also briefly summarizes the whole.

This notion of reason relations is already quite fascinating.

“Such an approach is unusual, and so perhaps surprising in how it discerns rational forms amphibious between these different dimensions” (p. 12).

It is the “amphibious” or hylomorphic character of what is going on here that is so amazing. This is not just something on the horizon offered to aim at as a goal, but an actual concrete accomplishment. This could make it possible to specify in detail what the substantiality of reason will amount to in particular cases. Brandom’s work has clearly taken on a life of its own, and is now being carried forward by others in new ways.

One of the many ideas afoot here is a suggestion that relations come before “things” in the order of explanation. This has been one of my favorite themes throughout the years. It even appears that this amphibious character of reason relations could enable us to say what constitutes objectivity in particular cases, and not merely gesture at it. If so this is huge, from the point of view of perennial human deficits and conflicts. It could be as big a leap for talking animals as the introduction of Platonic dialogue. Of course, we should anticipate that people will still find things to argue about.

Earlier, it was Brandom who convinced me to take Kant and Hegel seriously, and to take analytic philosophy seriously as actual philosophy and not just a technical endeavor. This greatly elevated appraisal, especially of Kant and Hegel, naturally led me to direct attention to Kant and Hegel themselves. In this context, I almost came to think of Brandom primarily as a very innovative expositor of their work. The products of this collaboration in the Research Group on Logical Expressivism that are reported here leave no doubt that there is much more to Brandom’s work than that.

“One important criterion of adequacy for both semantic and pragmatic metavocabularies as we understand them is that they offer expressive resources sufficient to provide explanations of the reason relations of arbitrary base vocabularies. They are able to say, each in their own distinctive idiom, both what it means for some sentences to stand to others in relations of implication or incompatibility and why they do…. Our preferred version of semantics offers, in effect, truth-makers for the claims that Γ#A (Γ is incompatible with A) and Γ|~A (Γ implies A) in alethic modal terms of the impossibility of fusions of truth-making states of A, and truth-making states Γ with falsifying states of A, respectively — that is, in terms of how the sentences of Γ and A represent the world to be. Our preferred version of pragmatics specifies how one must use sentences in order thereby to count as practically taking or treating them as standing in relations of implication or incompatibility. It does that in deontic normative terms of constellations of commitments to accept and reject the claimables they express being improper, inappropriate, or ‘out of bounds’ ” (ibid, emphasis in original).

“Because both of these kinds of metavocabulary appeal to conceptual resources beyond those intrinsic to the base vocabularies of which they are the metavocabularies, and do so in service not just of characterizing the reason relations of those base vocabularies but of explaining them, the sorts of semantic and pragmatic metavocabulary we consider can be denominated ‘extrinsic-explanatory’ rational metavocabularies” (pp. 12-13).

“In addition to extrinsic-explanatory rational metavocabularies, there are also intrinsic-explicative ones. This latter kind of metavocabulary for reason relations restricts itself to the conceptual resources supplied by the base vocabularies whose reason relations it characterizes, and is used to make explicit those reason relations and the conceptual contents they articulate, rather than to explain why they are as they are, or what it is for them to be what they are. The principal phenomenon we initially seek to understand in these terms is logic. The first way logical vocabulary differs from the semantic and pragmatic metavocabularies considered so far is that it is an intrinsic, rather than an extrinsic metavocabulary for codifying reason relations. The rules by which logical vocabulary is introduced to extend any arbitrary nonlogical base vocabulary appeal to nothing more than the reason relations sentences of the base vocabulary stand in to one another” (p. 13, emphasis in original).

“Gentzen’s basic innovation was to treat reason relations, paradigmatically implications, as objects, called ‘sequents’, that can be referred to and manipulated, and their metainferential relations made explicit in a mathematical metavocabulary. The sequent-calculus metavocabulary can be thought of as applying to an arbitrary nonlogical base vocabulary…. This sequent-calculus metavocabulary allows for efficient expression of the reason relations that hold in any base vocabulary, including metainferential relations. But it is essentially just a notation, requiring no substantial additional conceptual resources beyond what is provided by the base vocabulary whose nonlogical implications and incompatibilities it specifies explicitly.”

“Perhaps surprisingly, the spare sequent-calculus notation… turns out to be sufficient to formulate rules for adding logical vocabulary to any arbitrary base vocabulary, and (most importantly), computing the reason relations of the extended vocabulary from those of the base…. The idea is first to extend the lexicon of the base vocabulary, by syntactic rules that specify that the base lexicon is included in the logically extended lexicon, and that if A and B are sentences in the extended lexicon, then so are [A implies B, A and B, and A or B]…. The complete logically extended vocabulary… can then be computed from the base vocabulary. We say that a corresponding logically extended vocabulary can be elaborated from any arbitrary base vocabulary. Implications and incompatibilities (and metainferences involving them) that hold in every logical extension of a base vocabulary, no matter what base vocabulary it is elaborated from, can then be said to hold in virtue of logic alone” (pp. 13-14).

“The sequent-calculus vocabulary is accordingly a rational metavocabulary — a vocabulary for specifying the reason relations of some other vocabulary — that has the special feature that it permits the elaboration of arbitrary base vocabularies over lexicons that extend the lexicons of the base vocabularies by adding logically complex sentences formed by combining the sentences of the base vocabulary with logical operators. Rules for those operators formulated in the sequent-calculus vocabulary conservatively extend the reason relations of the base vocabulary, in the sense that the implications and incompatibilties that hold among logically atomic sentences in the logically extended vocabulary are just those that already held among them in the base vocabulary. And the connective rules formulated in the sequent-calculus vocabulary do this while appealing to no resources outside of those provided already by the reason relations of the base vocabularies” (p. 15, emphasis added).

“”That is, sequent-calculus metavocabularies are intrinsic rational metavocabularies…. And they elaborate all the reason relations of the extended vocabulary solely from the reason relations of the base vocabulary…. When the reason relations of the logically extended vocabulary are suitably elaborated from those of a base vocabulary, it becomes possible for the first time to say explicitly, in the extended vocabulary, what implications and incompatibilities hold in that base, and also in its logical extension” (ibid, emphasis in original).

“The constellation of the sequent calculus metavocabulary and the logical vocabulary it introduces stands in an intrinsic-explicative relation to the reason relations of any base vocabulary whatsoever…. The rules of the logics we propose can be shown to be expressively complete in a strong sense…. [A]lmost all extant logics either presuppose that the base vocabularies they extend satisfy strong global structural constraints — paradigmatically the monotonicity and transitivity at the core of traditional understandings of specifically logical consequence as a kind of closure operator — or retroactively impose some such global structure, thereby failing to be conservative over some substructural base vocabularies. While we believe that specifically logical consequence does have a global closure structure (and that logical consistency is monotonic), we argue that this is not in general true of nonlogical reason relations” (p. 16, emphasis in original).

Next in this series: Implication Spaces

An Isomorphism

“The present point is that if the claim that it is possible to identify a rational structure common to what is expressed in pragmatic and semantic metavocabularies could be made out in detail, it would cast light on issues of much wider philosophical significance. For we can look at the relations between what is expressed in normative pragmatic and representational semantic metavocabularies in another way: as articulating the relations between the activities of talking and thinking, and what is being talked or thought about. This is the intentional nexus between subjects and objects, between mind and the world, knowers and the known.” (Brandom in Hlobil and Brandom, Reasons for Logic, p. 8).

Brandom uses the term intentionality in a non-psychological sense that he elsewhere attributes to Kant. We are implicitly in what I think of as Aristotelian-Hegelian territory, where a Cartesian-style division into Subject and Object is not assumed. Brandom’s low-key summary of what to me are the rather dramatic stakes in this issue focuses on the American pragmatists, whom he discussed in the recent Pragmatism and Idealism lectures.

“The American Pragmatists inherited from the German Idealists — who in turn inherited it from Romantic critics of the Enlightenment — the idea that the Cartesian tradition failed structurally, making itself a patsy for skepticism, by attempting to define subjects and objects independently of one another, and then later on facing the problem of how to bolt together things understood as having wholly disparate natures…. The better strategy, they thought, was to start with a conception of intentionality as successful cognition (and action)…. One way to work out such a strategy begins with the thought that there is a kind of structure common to what normative pragmatic metavocabularies make it possible to say about the practices of discursive subjects using declarative sentences to manifest practical attitudes and undertake commitments, on the one hand, and what representational semantic metavocabularies make it possible to say about the modal relations among matter-of-factual states of the world those sentences come to represent by being so used, on the other” (ibid).

Here he references the classic pragmatist emphasis on “successful” thought and action. But especially since he is about to explicitly invoke an Aristotelian (and Scholastic) connection on the next page, this suggests to me that even a very elementary mainstream notion of pragmatism could be recast as evincing a kind of Aristotelian teleological concern with ends and that-for-the-sake-of-which, but in language that hides this angle and is suited to survive in the climate of uncomprehending modern antipathy to Aristotle. The main difference is that Aristotle says much more clearly that the ends that matter are those that are sought for their own sake, and not as means to other ends.

I used to think that logical and linguistic pragmatics as a field of study had nothing in particular to do with pragmatism as a view of the world. Brandom’s recent writings provocatively suggest that there is indeed a connection.

The emphasis on structure is also significant. Although Brandom does not identify with it as I did especially in my youth, French so-called structuralism and poststructuralism represent another major strand of non-Cartesian, non-subject-centered thought in the 20th century. Brandom’s usage seems closer to mathematical structuralism, and perhaps to the structural functionalism of the sociologist Talcott Parsons and the cognitive psychologist Jean Piaget that attracted Jürgen Habermas, whom Brandom has called a personal hero.

“For the worldly version of the relations that articulate the structure we are calling ‘conceptual’ are relations of necessity and impossibility whose existence owes nothing to the activities of discursive practitioners. They are objective relations, specified in the alethic modal vocabulary used to state laws of nature, and more generally to specify subjunctively robust relations” (pp. 8-9).

Brandom has consistently highlighted the significance of modality and modal logic for formulating what he likes to call subjunctive robustness. Next he invokes non-Cartesian strands within analytic philosophy.

“We take the view we develop to be a way of understanding what Frege means when he says ‘A fact is a thought that is true’. It is also one way of understanding the Tractarian [early Wittgenstein] claim that the world is the totality of facts…. John McDowell (1996) explores the same sort of conceptual realist view in Mind and World under the slogan ‘The conceptual has no outer boundary’.”

While I am highly sympathetic to the non-Cartesian ambitions here, I think that facts alone are too shallow a basis to constitute a world. I am not a Wittgenstein scholar, but I think he later moved away from this attempt to ground everything on atomic facts. But what else is needed is something like the subjunctive robustness or modal aspect of things that Brandom dwells upon. This emerges naturally as we move from world-as-totality-of-fact to the idea of a world constituted from implications and distinctions (the latter being my preferred way of thinking about what Brandom calls incompatibilities).

“These are deep waters. These pronouncements by great philosophers are mentioned to indicate that the stakes are high for the enterprise of explicating any form of conceptual realism. Here is a sketch of how we go about it. One of the key arguments we appeal to in filling in this neo-Aristotelian metalinguistic bimodal conceptual realism is a technical result…. Greg Restall and David Ripley have worked out what they call a ‘bilateral’ normative pragmatic understanding of the turnstile that marks implication relations in multisuccedent sequent calculi [which looks approximately like |~ and means that if all formulae on the left (often represented as a context capital gamma Γ) are true, then at least one formula on the right is true.]…. The Restall-Ripley bilateral normative pragmatic metavocabulary turns out to be related in surprising ways to what we take to be the most sophisticated contemporary heir of Tarskian model theory and later intensional semantics in terms of possible worlds (Lewis, out of Kripke, out of Carnap), namely Kit Fine’s truth-maker semantic framework…. The representational content of declarative sentences is then understood in terms of assignments to them of sets of states as truth-makers and falsifiers. Global structural conditions on modally partitioned state spaces (for instance requiring that all the mereological parts of possible states be possible) interact with conditions on assignments of truth-makers and falsifiers (for instance forbidding the truth-makers and falsifiers of logically atomic sentences to be overlapping sets).”

Sequent calculi are proof-theoretic notations due to Gerhard Gentzen in the 1930s. They generalize Gentzen’s system of natural deduction. In sequent calculi, every line is a conditional or sequent, rather than an unconditional assertion. In effect, the primitive terms are implications. This is a formal analogue of Brandom’s idea that the common structure of the world and of thought is at root constituted out of implications (and distinctions) rather than simple facts. Hlobil and Brandom’s book shows that it is general enough to support radically nonmonotonic and nontransitive cases.

“We show below that if one defines semantic consequence in just the right way, a powerful, fruitful, and detailed isomorphism can be constructed relating truth-maker modal semantic metavocabularies and bilateral normative pragmatic vocabularies” (pp. 9-10).

Serious logicians mainly study the properties of different logical systems, or logics, and develop new ones. Alternate logics have hugely proliferated since the first half of the 20th century. He is alluding to the fact that many differently detailed notions of logical consequence have been proposed. What is the “right” one depends in part on its conditions of use.

An isomorphism is a structure-preserving mapping that works bidirectionally. The existence of an isomorphism — like the one mentioned further below between algebra and geometry, or the one Brandom is talking about immediately below, between semantics and pragmatics — is an extremely nonrandom, rare occurrence, and therefore is often taken to be deeply significant.

“Assertion and denial line up with truth and falsity, combinations of commitments (to accept and reject) in a position line up with fusion of truth-making and falsifying states, and normative out-of-boundness (preclusion of entitlement to the commitments incurred by those assertions and denials) of a compound practical position lines up with the modal impossibility of such a fusion state.”

“When Spinoza looked back on the relations between algebraic equations and geometric shapes on which Descartes modeled mind-world relations, he saw that the key feature distinguishing that new, more abstract notion of representation from earlier atomistic resemblance-based conceptions is the existence of a global isomorphism between the algebraic and geometrical vocabularies. Spinoza’s slogan for the holistic insight that animated the representational revolution was ‘The order and connection of ideas is the same as the order and connection of things’ (Spinoza, Ethics II, Prop. vii). The isomorphism between normative pragmatic and alethic representational metavocabularies turns out to make possible in our setting a precise, tractable, and productive specification of that shared rational ‘order and connection’. We think this is a good way to rationally reconstruct some central aspects of Aristotelian (and Scholastic) intelligible forms. This isomorphism is the core of our version of bimodal (deontic/alethic) metalinguistic conceptual realism” (p. 11).

Brandom has been a consistent critic of standard versions of representationalism, but he has always been careful not to reject too much. The more affirmative reference to representation and Tarskian model theory here specifically involves not just any representation but an inferentialist semantics that undoes many conventional assumptions. Apparently there is a formal result to the effect that inferentialist semantics can be expressed not only in terms derived from Gentzen’s proof theory, but also in terms of an evolved variant of Tarski’s model theory in which the things represented are implications.

Next in this series: Quick Note on Proof Theory

Narrative Identity, Substance

Narrative identity for Ricoeur is intended as a kind of mean between ordinary logical identity or sameness, which he calls idem identity, and a kind of mediated reflexivity, which he calls ipse identity. Ordinary logical identity is rigid and static, but worse than that, it is often taken for granted. On the cutting edge of its home ground of mathematics, however, it has become recognized that criteria for logical identity of each type of thing need to be explicitly defined. Logical identity then effectively reduces to isomorphism. Sameness effectively reduces to sameness of form, and Leibniz’s thesis of the indiscernability of indiscernability and identity is vindicated.

I have argued, however, that Aristotle’s notion of identity as applied to so-called “substance” not only implicitly anticipates this thesis of Leibniz, but also ultimately circumscribes it with a further processual dimension accommodating continuity through change over time. Independent of the considerations of narrative developed by Ricoeur but potentially interpretable in similar terms, the “identity” of a “substance” for Aristotle is already extended to continuity through change. This kind of situationally appropriate, delimited relaxation of identity criteria allows Aristotle to accommodate “realistic” nuances in the application of common-sense reasoning or material inference that cannot be justified by purely formal logic. Judgments of real-world “identity” are practical judgments, with all the usual caveats.

While Aristotle was very process-oriented, the processes with which he was concerned were short- and medium-term processes, generally not extending beyond the scope of a life. History for Aristotle is mainly an accumulation of accidents, and thus in Aristotle’s sense intelligible mainly in the register of materiality. To the extent that he thinks about history, he treats it in terms of delimited “histories” rather than an enveloping “History”.

Within that accumulation of accidents, however, we can potentially explicate other levels Aristotle left unexplored, like Ricoeur’s historical explanation or Foucault’s “archaeology”. Foucault developed a meta-level account aimed at articulating underlying forms implicit in something like Aristotle’s delimited accumulations of accidents, while I think that after the detour of historical explanation, Ricoeur ultimately wanted to cultivate signposts for an enveloping “History” as metaphors expressing a broader “meaning of life”. In a very general way, Ricoeur’s aim thus resembles Brandom’s “Hegelian genealogy”.

Split Subject, Contradiction

The Žižekians, referencing Lacan, like to talk about a “split subject” that is noncoincident with itself. In broad terms, I think this is useful. What we call subjectivity is divided, and lacking in strong unity. (See also Pure Negativity?; Acts in Brandom and Žižek.) But it seems to me that if we try to speak carefully about this, we should not then go on using singular articles like “the” or “a”.

I tend to think subjectivity is not just fractured or un-whole, but also actually consists of a complex overlay of different things that we tend to blur together. In particular, it seems clear to me that a common-sense, biographical “self” whose relative unity over time is trackable by relation to the “same” physical body — or by Lockean continuity of memory — is not the same as what we might in a given moment view from a distance as an individualized ethos, or up close as a unity of apperception. This is, I believe, the same distinction that Brandom discusses in terms of sentience and sapience.

Ethos and unity of apperception, and their constituent values and conceptions — the very things that most properly say “I”, and play the functional role of an ethical “subject”, or of a subject of knowledge — are profoundly involved with language, social relations, and what Lacan in his earlier work called the Symbolic and the “Other”. These instances of sapience are pure forms whose identity can only be expressed in terms of sameness of form — nonempirical, but inseparable from a larger ethical world — and simultaneously intimate to us, but by no means strictly “ours”. (See also Self, Subject.)

Where I am still a bit torn is that I also feel that emotions — which I’ve been locating on the former, “self” side — are fundamental to subjectivity as a whole, but I have theoretically separated them from the main locus of transcendental ethical and epistemic subjectivity, even though they play an essential role in making it possible. One logical solution would be to say this just means subjectivity as a whole is more than just ethical and epistemic. Another would be to say that there is a separate kind of emotional subjectivity. I’m not entirely satisfied yet, because I think feeling combines these, but the noncoincidence of our factual selves with our ethical and epistemic being seems very important in understanding how we overcome empirical limitations.

The Žižekians will perhaps remind us that they were not talking about a split between self and subject, but about a split within the subject. I think we habitually overstate the degree of unity and identity we attribute to selves, subjects, and things in general, so I’m fine with that, too. They also want to expand this into a general “ontological” point, which I see as a semantic point.

Perhaps the Žižekians are more comfortable talking about “a” or “the” subject in part due to their doctrine of the ubiquity of contradiction. Todd McGowan in Emancipation After Hegel (2019) nicely distinguishes the Žižekian notion from the old confusion between contradiction and conflict or polarity — and from immediate self-contradiction — but still wants to maintain that the standard logical law of noncontradiction ultimately “refutes itself”, and that Hegel thought this as well. This argument combines a laudable awareness of some of the practical issues with identity, with a logically invalid use of the distinction between explicit and implicit self-contradiction.

Hegel meditated profoundly on the difficulties of applying logic to meaningful content and to real life. He strained language to the breaking point trying to express his conclusions.

On the frontiers of mathematical logic today, the so-called law of identity has been replaced by a requirement to specify identity criteria for each formally defined type, and identity in general has been weakened to isomorphism. (See also Form as a Unique Thing.)

Real-world applications of strong identity typically involve loose “extensional” reference to things assumed to be the same, and a lot of forgetting. The linchpin of old “identity thinking” was inattention to difficulties of formalization from ordinary language — basically an illegitimate moving back and forth between formal and informal domains, resulting in lots of homogenizing confusion of things that ought to be distinct. Weaker, “intensional” assertions about identity as specifiable sameness of form make it the exception rather than the rule. What come first conceptually are distinctions within the manifold, not pre-synthesized things already possessed of identity. Where things are not the same to begin with, contradiction — far from being omnipresent — is not even potentially at issue. (See also Self-Evidence?)

Meanwhile, Sellars and Brandom have revived material inference about meant realities in contrast to formal logic, which deals with purely syntactic relations between presumed extensional “things” with presumed identity. Things Kant and Hegel said about Understanding and Reason can be nicely understood in terms of the relation between syntactic inference about symbolic terms standing for formless extensional “things” and substantive, material inference about the actual form of meant realities. Especially in the reading of Hegel, not having the resource of this distinction available now seems positively crippling.

Finally, Aristotle, who originated the law of noncontradiction as a kind of ethical imperative, and stands in the background to all of Hegel’s discussions of logic, was himself rather cautious and tentative about applying identity to real things, and in his logic was also mainly concerned with (composition of) material inferences, which have more to do with the actual form of things .

Hegel never violated Aristotle’s imperative not to say opposite things about the same thing said in the same way. What he did was to constantly point out the gap between reality and traditional semi-formal logic applied to ordinary language — not to encourage us to reject logic, but rather to refine and sublimate it. (See also Aristotelian and Hegelian Dialectic.)

Aristotle and Brandom?

For the second time, I think I discovered a significant new insight into a major Aristotelian concept by thinking it through in Brandomian terms. When I began this effort, Aristotle and Brandom were just the two philosophers with whom I was most engaged, who seemed to me to share my overarching concern with the ethical import of reasons and things said, but it is growing to be something more.

(To some, this might seem a strange pairing. However, in spite of his own lack of direct engagement with Aristotle, Brandom has commented that a number of his best interlocutors (unnamed) were what he called neo-Aristotelians. Certainly, Hegel — the historic philosopher with whom Brandom has been most engaged — makes major use of Aristotle, and Brandom’s co-thinkers on Hegel, Robert Pippin and Terry Pinkard, have highlighted this.)

Earlier, I noted a kind of isomorphism between Aristotelian potentiality and Brandomian modally robust counterfactual inference, which then turned into a three-way correspondence with the structuralist concept of structure, and helped illuminate the old synchronic/diachronic issue associated with structuralism.

The other day, I noted a second isomorphism, between canonical Aristotelian proposition-forming combination and separation and Brandomian material consequence and material incompatibility. The result is that Aristotle’s canonical conception of logical truth seems very consistent with what Brandom recommends, in terms of using goodness of material inference to explain truth rather than using truth to explain inference.

Brandom has referred to this sort of interpretation as a recollective genealogy, grounded in Hegel’s way of retrospectively interpreting past philosophers in light of the present. Obviously there is a creative element to such an endeavor. The important and delicate point is that it not be an arbitrary imposition, but something that yields genuine insight that is both relevant to the present and honestly compatible with the best historiographic objectivity we can fallibly attain. In the two cases mentioned above, I think that has been achieved.

Going in the other direction, developing an Aristotelian interpretation of Brandom’s distinction between sentience and sapience has helped me to achieve full sympathy with this notion, and with several of Kant’s apparently dualistic moments as well.

Somewhat ambidextrously, it seems to me that Brandomian commitments, together with the sort of pattern of performance with respect to responsibility measured by Brandomian deontic scorekeeping, make up the ethical character or culture that Aristotle called ethos. (See also Ethos; Aristotelian Subjectivity; Brandomian Choice.)