Conditionals and Conceptual Roles

Saying something is more than the material fact of emitting sounds in conventionalized patterns. We ought to be able to say more about that “more”.

This is part two of a look at an early programmatic document in which Brandom first develops his highly original approach to meaning and logic. Brandom’s “logical expressivism” treats logic as a tool for explaining meaning, rather than a discipline with its own distinctive subject matter. That logic is such a tool and not a science is an Aristotelian view (or, I would say, insight) that has been mostly ignored by subsequent traditions.

The dominant modern tradition treats meaning as representation by pointing or reference. But pointing is rather trivial and uninformative. By contrast, I normally think of meaning in terms of something to be interpreted. But this hermeneutic approach tends to focus attention on concrete details. Brandom ambitiously wants to say meaningful things about meaning in general, and I think he succeeds.

As in the first installment, I will continue to focus on the discursive parts of the text, while skirting around the formal development. (There is more formal logical development in this text than anywhere else in Brandom’s corpus, at least until this year’s publication of the collaborative work Reasons for Logic, Logic for Reasons, which returns to the current text’s aim of implementing his program of logical expressivism.)

Brandom begins with the early work of Frege, who pioneered modern mathematical logic.

“To make out the claim that the systems of social practices we have described implicitly define assertion, we need to supplement that account of assertings with a story about the contents which are thereby asserted. Our starting point is Frege’s discussion in the Begriffschrift, where the distinction between force and content was first established…. First, Frege identifies conceptual content with inferential role or potential. It is his project to find a notation which will allow us to express these precisely. Second, sentences have conceptual contents in virtue of facts about the appropriateness of material inferences involving them. The task of the logical apparatus of the conceptual notation which Frege goes on to develop is to make it possible to specify explicitly the conceptual contents which are implicit in a set of possible inferences which are presupposed when Frege’s logician comes on the scene. The task of logic is thus set as an expressive one, to codify just those aspects of sentences which affect their inferential potential in some pre-existing system” (“Assertion and Conceptual Roles”, p. 21).

Meaningful “content” is to be identified with the inferential roles of things said, which are each in turn defined by the pair consisting of the conditions of their application and the consequences of their application. The novelty of what is expressed here is tactfully understated by the reference to “facts” about the appropriateness of material inferences. This tends to downplay the “fact” that the inquiry into conditions of application is really a normative inquiry into judgments about appropriateness more than an inquiry into facts.

What is being said here also needs to be sharply distinguished from the nihilistic claim that there are no facts. There are facts, and they need to be respected. The point is that this respect for facts ought to be opposed to taking them for granted.

“We will derive conceptual contents from the systems of practices of inference, justification, and assertion described above. Following the Fregean philosophy of logic, we do so by introducing formal logical concepts as codifications of material inferential practices. First we show how conditionals can be introduced into a set of practices of using basic sentences, so as to state explicitly the inference license which the assertion of one sentence which becomes the antecedent of the conditional can issue for the assertion of another (the consequent of the conditional). With conditionals constructed so as to capture formally the material inferential potential of basic sentences, we then show how conceptual contents expressed in terms of such conditionals can be associated with basic sentences on the model of the introduction and elimination rules for compound sentence forms like the conditional” (ibid).

Introduction and elimination rules are characteristic of the natural deduction and sequent calculi due to Gentzen. This style of formalization — common in proof theory, type theory, and the theory of programming languages — is distinctive in that it is formulated entirely in terms of specified inference rules, without any axioms or assumed truths.

Until Sellars and Brandom, modern logic was considered to be entirely about formal inference. Brandom argues that the early Frege correctly treated it instead as about the formalization of material inference. Brandom also endorses Quine’s logical holism against atomistic bottom-up views like that defended by Russell.

“We cannot in general talk about ‘the consequences’ of a claim (for instance, that the moon is made of green cheese) without somehow specifying a context of other claims against the background of which such consequences can be drawn. (Can we use what we know about the mammalian origins of cheese and take as a consequence that at one time the moon was made of milk, for instance?) Quine, in “Two Dogmas [of Empiricism]”, may be seen as arguing against the possibility of an atomistic theory of meaning (e.g. one which assigns to every sentence its ‘conceptual content’) that such meanings must at least determine the inferential roles of sentences, and that the roles of each sentence in a ‘web of belief’ depends on what other sentences inhabit that same web. In particular, whether anything counts as evidence for or against a certain claim … depends on what other sentences are held concurrently. Given any sentence, … and given any second sentence there will be some webs in which the second counts as evidence for the first, and some where it counts as evidence against the first, where what ‘web of belief’ is considered determines what other sentences are available as auxiliary hypotheses for inferences. Accepting the general Fregean line that meanings as theoretical constructs are postulated to express inferential potentials, Quine reminds us of basic facts about our inferential practices … to impugn the comprehensibility of assignments of conceptual role to individual sentences, unrelativized to some doxastic context. Conceptual roles can only be specified relative to a set of other sentences which are all and only those which can be used as auxiliary hypotheses, that is, as Quine puts it, at the level of whole theories-cum-languages, not at the level of individual sentences” (pp. 22-23).

Much of the ensuing discussion will revolve around conditionals, and what logicians call the implicational fragment of a logic, in which only implication is considered. This is a kind of minimal form for what constitutes a logic — if you specify a notion of implication, you have a logic. But the common modern truth-table definition of implication has been criticized from many quarters. Much work has been done on the precise definition of alternate or “better” notions of implication. This is one of the things Brandom will be doing here.

One of the most important questions about implication is whether it is “primitive” — i.e., something in terms of which other things are defined, which is itself considered to be defined only operationally (indirectly, by its use) — or whether it is to be defined in terms of something else, such as a truth table. For instance, category theory (by which all of mathematics can be interpreted) can be elaborated entirely in terms of primitive “arrows” or morphisms, which generalize both the notion of a mathematical function and that of logical implication. Arrow logics, which generalize modal logic, also start from a primitive notion of arrows. Later in this text, Brandom will develop his own notion of arrows as a primitive, alternate form of implication.

In the context of the debate about holism and atomism, it is interesting to consider the scholastic practice of debating for and against individual propositions. At top level, it seems atomistic, in that the propositions are taken up one at a time. But at a detailed level, the arguments turn out to be mostly about the consequences of accepting or rejecting the proposition under discussion. Brandom will argue that propositions are to be understood by the combination of their consequences and their conditions of appropriate use.

He turns to the question of what assertion is. The novelty here is that assertion will be explained in terms of primitive conditionals, rather than treated as primitive.

“The first step in our account of the semantic contents or conceptual roles sentences acquire in virtue of being used according to the practices expressed in some idiom is the introduction of some logical vocabulary. We understand the inference-licensing function of assertion by our model of justificatory systems of social practices. We will introduce the conditional as a compound sentence-form constructed out of the basic sentences on which some idiom is defined. The conceptual content of the conditionals will be stipulated; a sentence of the form p→q is to have as content the inference-license of a statement of the appropriateness of an inference from the assertion of p to the assertion of q. Various formal inferential connections between such conditional sentences will then be elicited. For these formal principles to comprise a logic is for them to make possible the explicit formal codification of the material inferential and justificatory practices of some conceptual idiom. This is the task Frege sets for logic in the Begriffschrift — although in that work he succeeded only in completely codifying the formal inferences involving his logical constructions, his discussion makes clear that the ultimate criterion of adequacy for his conceptual notation is its capacity to express explicitly and precisely the contextual material inferences which define the conceptual roles of non-logical sentences” (p. 23).

We see here too some of the motivation for focusing on compound sentences — all sentences that include explicit conditionals are compound. But according to his analysis, it will turn out that simple sentences of the form “A is B” implicitly express a sort of minimal form of material inference.

I would suggest that the allegedly unconditional or categorical judgment “A is B” is best understood as a kind of shorthand for a judgment like A(x)→B(x). Aristotle’s concern with sayings leads him to treat the sentences that express propositions in a non-atomic way. He glosses “A is B” as expressing “combination” and “A is not B” as expressing “separation”. I have suggested that “combination” could be read as a relation of material consequence, and “separation” as a relation of material incompatibility. This means that for Aristotle too, a proposition can be considered a kind of minimal material inference. (See Aristotelian Propositions.)

“Once the conditional has been introduced as codifying the consequence relation implicit in material inferential practice, and its formal logical properties have been presented, we will use such conditionals both as models for the conceptual roles of non-logical sentences (which will have analogues of introduction and elimination rules, and will be given content as licensing inferences from their circumstances of appropriate application to the consequences of such application) and as tools for making those roles explicit” (ibid).

Treating conditionals as models for the conceptual roles of simple “non-logical” sentences like “A is B” begins from the intuition that these simple assertions are the potential antecedents or consequents of inferences, and that this role in possible inferences is what gives them specifiable meaning.

“We may think of the relation between basic and extended repertoires in a conceptual idiom as defining a consequence function on admissible sets of sentences. For the extended repertoire … comprises just those sentences which an individual would socially be held responsible for (in the sense that the relevant community members would recognize anaphoric deference of justificatory responsibility for claims of those types to that individual) in virtue of the dispositions that individual displays explicitly to undertake such responsibility for the sentences in his basic repertoire. The extended repertoire consists of those claims the community takes him to be committed to by being prepared to assert the claims in his basic repertoire. These community practices thus induce a consequence function which takes any admissible basic repertoire and assigns to it its consequence extension. The function only represents the consequences of individual sentences relative to some context, since we know what the consequences are of p together with all the other sentences in a basic repertoire containing p, but so far have no handle on which of these various consequences might ‘belong’ to p. Thus we have just the sort of material inferential relations Frege presupposes when he talks of the inferences which can be drawn from a given judgment ‘when combined with certain other ones’…. The idiom also expresses a material consistency relation…. The sets which are not idiomatically admissible repertoires are sets of sentences which one cannot have the right simultaneously to be disposed to assert, according to the practices … of the community from which the idiom is abstracted. The final component of a conceptual idiom as we have defined it is the conversational accessibility relation between repertoires” (pp. 23-24).

The accessibility relation will turn out to correspond to whether a sentence makes sense or is categorial nonsense like “Colorless green ideas sleep furiously”.

“Given such an idiom defined on a set of non-logical sentences, we will add conditional sentences p → q to each of the consequence-extended repertoires in which, intuitively, p is inferentially sufficient for q, in such a way that the newly minted sentences have the standard inferential consequences of conditionals such that this formal swelling of the original repertoires is inferentially conservative, that is does not permit any material inferences which were not already permitted in the original idiom” (p. 24).

He defines an idiom as a triple consisting of a set of sets of sentences or basic repertoires, a function from basic repertoires to their consequence extensions, and a function from repertoires to the other repertoires “accessible” from each.

“Recalling the constitutive role of recognitions by accessible community members in determining consequence relations, we may further define p as juridically (inferentially) stronger than q at some repertoire R just in case p is actually stronger than q at every repertoire S accessible from R. This natural modal version of inferential sufficiency will be our semantic introduction rule for conditional sentences…. The conditional thus has a particular content in the context of a given repertoire, a content determined by the inferential roles played by its antecedent and consequent” (p. 25).

“We must show that the important formal properties of idioms are preserved by the introduction of conditionals, and that the conditionals so introduced have appropriate properties. In order to permit sentences with more than one arrow in them, we must swell the basic idiom with conditionals first, and then iterate the process adding conditionals which can have first-order conditionals as antecedents or consequents, and so on, showing that the relevant properties of conceptual idioms are preserved at each stage. Our procedure is this. Starting with a basic idiom …, we define a new idiom … with repertoires defined not just over the original set of non-logical sentences, but also containing first-order conditionals, as well as consequence and accessibility relations between them. The same procedure is repeated, and eventually we collect all the results” (ibid).

“The properties of conceptual idioms which must be preserved at each stage in this construction are these. First is the extension condition, that for any admissible repertoire R, R [is a subset of its consequence extension]. The motive for this condition is that the consequence extension c(R) of R is to represent those claims one is taken to be committed to in virtue of being prepared explicitly to take responsibility for the members of R, and certainly one has committed oneself to the claim one asserts, and licenses the trivial inference which is re-assertion justified by anaphoric deferral to one’s original performance. Second of the properties of conceptual idioms which we make use of is the interpolation condition, which specifies that any basic repertoire R which can be exhibited as the result of adding to some other repertoire S sentences each of which is contained in the consequence extension of S, has as its consequence extension c(R) just the set c(S).” (pp. 25-26).

“The idempotence of the consequence function, that for all [repertoires in the domain], c(c(R)) = c(R), is a consequence of the interpolation property. Of course this is a desirable circumstance, since we want idempotence in the relation which is interpreted as the closure under material inference (as constituted by social attributions of justificatory responsibility) of admissible basic repertoires” (p. 26).

“The consequence relation is contextual, in that a change in the total evidence which merely adds to that evidence may entail the denial of some claims which were consequences of the evidential subset. Allowing such a possibility is crucial for codifying material inferential practices, which are almost always defeasible by the introduction of some auxiliary hypothesis or other…. [B]oth ‘If I strike this match, it will light’, and ‘If I strike this match and I am under water, it will not light’, can be true and justified. Denying monotonicity (that if [one repertoire is a subset of another], then [its consequence extension is a subset of the consequence extension of the other]) forces our logic to take account of the relativity of material inference to total evidence at the outset, with relativity to context made an explicit part of the formalism instead of leaving that phenomenon to the embarrassed care of ceteris paribus [other things being equal] clauses because standard conditionals capture only formal inference, which is not context-sensitive” (p. 27).

Real things are in general sensitive to context, whereas formal logical tautologies are not.

Monotonicity is a property of logics such that if a conclusion follows from a set of premises, no addition of another premise will invalidate it. This is good for pure mathematics, but does not hold for material inference or any kind of causal reasoning, where context matters. The match will light if you strike it, but not if you strike it and it is wet, and so on.

“We are now in a position to investigate the logic of the arrow which this formal, non-substantive expansion of the basic idiom induces. To do so, we look at the sentences which are idiomatically valid, in that every repertoire in the formally expanded idiom contains these sentences in its consequence extension. First, and as an example, we show that if p is in some consequence-extended repertoire, and p→q is also in that repertoire, then so is q, that is, that modus ponens is supported by the arrow” (p. 29).

What he calls a basic repertoire is defined by some set of simple beliefs, assumptions, or presumed facts, with no specifically logical operations defined on it. Non-substantive expansion leaves these unchanged, but adds logical operations or rules.

At this point he proves that modus ponens (the rule that p and (p implies q) implies q, which he elsewhere refers to as “detachment” of q) applies to the conditional as he has specified it. Additional theorems are proved in an appendix.

“[T]he most unusual feature of the resulting logic is its two-class structure, treating conditionals whose antecedents are other conditionals rather differently from the way in which it treats conditionals involving only basic sentences. This feature is a direct consequence of the introduction of first-order conditionals based on material inferential circumstances of the repertoire in question, and higher-order conditionals according to purely formal, materially conservative criteria. Thus it is obvious from inspection of the … steps of our construction of the hierarchy of conditionals that the complement of basic sentences in a consequence extended repertoire is never altered during that construction, and that the novel repertoires introduced always have first-order restrictions which are elements of the original set…. Higher-order conditionals, of course, are what are added to the original idiom, and … those conditionals obey a standard modal logic. The principles governing conditionals with basic sentences as antecedents or consequents, however, are those of the pure implicational fragment of Belnap and Anderson’s system EI of entailment” (ibid).

Belnap and Anderson worked on relevance logic, which restricts valid inference to the case where premises are relevant to the conclusion. The premises of a material inference are always “relevant” in this sense. Formal inference on the other hand doesn’t care what the underlying terms or propositions are. It is entirely governed by the abstractly specified behavior of the formal operators, whereas material inference is entirely governed by the “content” of constituent terms or propositions.

That there would be two distinct kinds of conditionals — first-order ones that formally codify material inferences, and higher-order ones that operate on other conditionals in a purely formal way — seems consonant with other cases in which there is a qualitative difference between first-order things and second-order things, but no qualitative difference between second-order and nth-order for any finite n.

“We may view the conditionals which end up included in the consequence extensions of formally extended repertoires as partially ordering all of the sentences of the (syntactically specified) language. Since according to our introduction rule, a repertoire will contain conditionals whose antecedents and consequents are not contained in that (extended) repertoire, the ordering so induced is not limited to the sentences of the repertoire from which the ordering conditionals are drawn. Although the conditional induces an appropriately transitive and reflexive relation on the sentences of the language, the ordering will not be total (since for some p, q and R [in the domain], it may be that neither p→q nor q→p is in c(R)), and it will not be complete, in that sentences appearing only in inaccessible repertoires will have only trivial implication relations (e.g. p→p)” (ibid).

“The conditionals which do not have antecedents in c(R) are counterfactual with respect to R. These are of three kinds: i) those taken true by the theory codified in the repertoire, that is, counterfactuals in c(R), ii) those taken not to be true, i.e. conditionals not in c(R) but on which R induces non-trivial entailments, and iii) inaccessible counterfactuals, assigned no significance by the extended repertoire (e.g. ‘If the number seventeen were a dry, well-made match’, an antecedent generating counterfactuals which, with respect to a certain set of beliefs or repertoire simply makes no sense). Entailment relations between counterfactuals of the first two kinds and between each of them and base sentences will be underwritten by the induced partial ordering, all depending on the original material inferential practices involving only base sentences” (pp. 29-30).

There are many counterfactuals that we take to be true. For example, if I had left earlier, I would have arrived earlier. In fact counterfactuals are essential to any truth that has any robustness. Without counterfactuals, what Brandom is calling an idiom could apply only to some exactly specified set of facts or true statements. This would makes it very brittle and narrowly applicable. For example, any kind of causal reasoning requires counterfactuals, because causes are expected to operate under a range of circumstances, which by definition cannot all hold at the same time. Counterfactuals play an important role in Brandom’s later work.

“The repertoire which induces such a partial ordering by its conditionals will then be a distinguished subset of the sentences it orders, one which Theorem 1 assures us is deductively closed under modus ponens. Each repertoire is in short a theory or set of beliefs, embedded in a larger linguistic structure defining the implications of the sentences in that theory. Not only do different repertoires codify different theories, but they assign different significances to syntactically type-identical sentences of those theories, in that p as an element of c(R) may have one set of inferential consequences, and as an element of c(R’) have a different set of consequences. The repertoires ordered by their indigenous implication relations thus deserve to be called ‘webs of belief’ in Quine’s sense, as the smallest units of analysis within which sentences have significance. The idiom, comprising all of these repertorial structures of implicational significance and embedded belief, is not a set of meanings common and antecedent to the repertoires, but is the structure within which each such web of belief is a linguistic perspective made possible by a justificatory system of social practices” (p. 30).

Each repertoire counts as a “theory” or set of beliefs.

“The systematic variation of the significance of those sentences from one individual to another expressed in a formally expanded idiom then exactly answers to whatever communication is going on in the original set of practices. The possibility of communication consists in [a] kind of coordination of significances across repertoires codified in a formally expanded idiom” (p. 31).

The success or failure of communication depends on something like a kind of translation from your repertoire to mine.

“We have described the practical origins and effects of elements of extended repertoires which are first-order sentences of the language, in terms of attributions and undertakings of justificatory responsibility and the issuing and recognition of inferential authority. What, in these terms, should we take to be the significance of a conditional p→q? The presence of such a conditional in the formally expanded consequence extension of the repertoire exhibited by an individual should signify, first, that that individual recognizes others who are prepared to assert p as licensing the inference to q, and, second, that he recognizes the assertion of p as justifying the assertion of q” (p. 32).

“So if all those recognized by the individual exhibiting R are responsible for the conditional p→q and p [is in] c(R), then q [is in] c(R), which means that p→q plays the proper role as codifying the recognition of inferential licensing and appropriate justification of q by p” (ibid).

“Finally, we state a more general condition under which the arrow we have defined will be a practically complete expression of a justificatory system” (ibid).

Next in this series: Anaphora and Prosentences

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

Quick Note on Proof Theory

I read Aristotelian demonstration as more a making explicit than a proof of truths. The logical expressivism of the author of Making It Explicit (Robert Brandom) does something similar with modern logic. Nonetheless it would be very wrong to conclude that proof theory has no philosophical relevance.

To begin with, proof theory is itself not concerned with proving this or that truth. It is the study of proofs, the beginning of which is to recognize that proofs and proof calculi are themselves mathematical objects. Proofs are functions from premises to conclusions. This has profound consequences.

At an utterly simple level, one small piece of a far larger result is that the notion of an implication A => B is at a certain level formally interchangeable with the notion of a mathematical function A => B. Frege very explicitly treats logical predication as a function as well.

Category theory builds all of mathematics on such morphisms, starting from a single basic operation of composition of arrows. Homotopy type theory suggests that we think of the arrows as paths through spaces. All this is an elaboration and abstraction of the utterly simple but crucial notion of “follows from”, or what Brandom calls subjunctive robustness.

Then an Aristotelian syllogism can be seen on the model of the composition of two predications or functions or morphisms or arrows or paths A => B and B => C around a common type or middle term B that is the output of one and the input of the other. This is not intended to capture a sophisticated result like a mathematical theorem, but rather to express sound reasoning in the simplest, most perspicuous, and most universal way possible.

Next in this series: Reason Relations

The Role of Reasons

In a brand-new book co-authored with logician Ulf Hlobil — Reasons for Logic, Logic for Reasons: Pragmatics, Semantics, and Conceptual Roles (2025) — Robert Brandom introduces results from the Research Group on Logical Expressivism, which is inspired by a major strand of his work. Logical expressivism is a highly innovative approach that takes the primary purpose of formal logic to be not the proving of truths, but a kind of making explicit of all kinds of real-world reasoning that are carried out in natural language.

The book introduces quite a number of big ideas — among them logical expressivism, reason relations, implication spaces, conceptual roles, and important new technical results that highlight the importance of nonmonotonic logic and substructural logic. Established Brandomian themes such as normativity and its relation to modality, inferentialism, material inference, and the close connection between semantics and pragmatics also show up here in new light. Brandom has written the more philosophical chapters, and Hlobil the more technical ones.

One interesting surprise is that Brandom explicitly calls the new approach “neo-Aristotelian”. This “neo-Aristotlian metalinguistic bimodal conceptual realism” will be “hylomorphic in a recognizably Aristotelian sense. For it identifies a kind of rational form that is understood as common to thoughts and things…. [T]he relations of consequence and incompatibility that show up in different guises in a whole constellation of intimately interrelated metavocabularies… are those that in the end underwrite practices of reasoning, by determining what is a reason for and against what” (p. 9, emphasis in original).

This is well short of the more full-blooded re-visioning of an open Aristotelianism that I have been suggesting here, but within its scope it does seem genuinely Aristotelian to me — particularly the idea that there are forms common to thought, things, language, and practices of reasoning. This is a nice vindication of the “Aristotle and Brandom” theme with which I began this blog almost six years ago.

“[T]he strategy of addressing philosophy’s perennial concern with the nature of understanding or reason in general by investigating language…. has been developed in two quite different directions…. The first, dominant, better worked out tradition focused on logic, and later, also formal semantics, as perspicuous mathematical metalanguages…. The other tradition focused rather on language as a kind of social practice” (p. 1, emphasis in original).

Brandom has always been interested in both of these. At the beginning of his career he worked on logic, but for most of his maturity he has tended to favor the pragmatic side. Here at one point he ends up suggesting that they may be equally important. The book presents new results in mathematical logic that help bridge the gap.

“Where the formalist tradition is oriented by a conception of understanding and reasons as codified in artificial logical calculi and semantic metalanguages, the pragmatist tradition looks instead directly to natural languages, thought of as social practices and forms of life. In place of the exclusively monological character of reasoning as deriving, modeled on proof, characteristic of the other tradition, understanding shows up in this tradition as a social achievement, and reasoning as essentially dialogical: a matter of discursive practices of giving and asking for reasons, defending and challenging claims that amount to taking up positions in a contestable, public, normative space” (p. 2).

“The two traditions ought by rights to be understood as focusing on different aspects of language: roughly, on the meanings of linguistic expressions, and on their use. In suitably broad senses, we might understand semantics as the study of meaning, and pragmatics as the study of use or discursive practices and abilities. So understood, semantics (even a semantics inspired by and paradigmatically applicable to logic) and pragmatics show up as complementary theoretical endeavors. The goal should be to synthesize semantic and pragmatic theories…. Perhaps the combination of those thoughts recommends rather a more balanced view that eschews claims of explanatory priority in favor of understanding each aspect as in principle intelligible only in terms of its relation to the other” (pp. 2-3, emphasis in original).

“The lesson that emerges, we will argue, is a kind of discursive or linguistic rationalism. Language becomes visible as at base the medium of reasons, and reasoning as the beating heart of language. On the side of pragmatics, the fundamental speech act is that of making claims. The basic speech act of making claims, asserting, is to be understood in terms of practices of defending and challenging those claims, by making other claims that have the practical significance of giving reasons for and against them. Understanding claiming this way provides a path to understanding the claimable contents expressed by declarative sentences in terms of the role they play in relations of being a reason for or against — what we will call ‘reason relations” (p. 3, emphasis in original).

He continues, “On the side of semantics, worldly represented states show up as what determines the reason relations of consequence and incompatibility that the sentences whose truth-makers and falsifiers they are stand in to one another: their roles in reason relations. By understanding the common topic that semantic and pragmatic metalanguages articulate aspects of, not just under the vague rubric of ‘language’, but more specifically as the implicit reason relations that distinguish discursive practices as such, we can better understand not only the relations between the meaning and the use of linguistic expressions, but also the relations between truth (the central concept of traditional semantics) and justification (the central concept of pragmatics, according to linguistic rationalism), in the form of practices of defending claims by giving reasons for them and challenging claims by giving reasons against them” (pp. 3-4, emphasis in original).

“At the core of this book, then, is the rationalist explanatory strategy of understanding the nature of language in terms of what we will call ‘reason relations’. As addressed here, that is a genus with two principal species: implication and incompatibility. They correspond to being a reason for and being a reason against” (p.4).

“A closely related term of art is ‘vocabulary’. We use it in a technical sense, to mean a lexicon or set of declarative sentences, together with an implication relation and an incompatibility relation defined on those sentences. To begin with, we can think of an implication relation as holding between a set of sentences that are its premises and a single sentence that is a conclusion that follows from, is a consequence of, or is implied by those premises. An incompatibility relation holds between a set of premises and a further sentence that those premises exclude, or rule out, or are incompatible with” (p. 5).

He continues, “By calling them (declarative) ‘sentences’ we just mean that they are what in the first instance stand to one another in reason relations of implication and incompatibility…. In virtue of standing to one another in reason relations of implication and incompatibility, what thereby count as declarative sentences express conceptual contents. Those contents can be thought of as the functional roles the sentences play in constellations of implications and incompatibilities” (ibid).

“According to this order of explanation, the key question is: what do we mean by talk of reason relations of implication and incompatibility? In virtue of what does something deserve to count as a consequence or incompatibility relation?” (ibid).

“The idea is to identify reason relations in terms of the various vocabularies that can be used to specify them. Because these are vocabularies for talking about (the reason relations of) other vocabularies, they are metavocabularies. Because it is in particular the reason relations of base vocabularies that they address, we can call them rational metavocabularies” (pp. 5-6, emphasis in original).

“Semantic metavocabularies explain reason relations of implication and incompatibility by specifying what the sentences that stand in those relations mean, in the sense of how the world must be for what they say to be true. The sentences stand to one another in relations of implication and incompatibility because the objective states of affairs that are their semantic truth conditions stand to one another in modally robust relations of necessitation and noncompossibility” (p. 6).

“Pragmatic vocabularies explain what is expressed by reason relations of base vocabularies by saying what features of the discursive practice of using those sentences it is, in virtue of which practitioners count as practically taking or treating the sentences as standing to one another in relations of implication and incompatibility. Pragmatic metavocabularies make it possible to say what it is that language users do in virtue of which they are properly to be understood as practically taking or treating some sentences as implying others in the sense of taking assertion or acceptance of the premises as providing reasons for asserting or accepting the conclusions, and practically taking or treating some sentences as incompatible with others in the sense of taking assertion or acceptance of the premises as providing reasons against asserting or accepting the conclusions. Reason relations show up from the expressive perspective provided by pragmatic metavocabularies as normative standards for assessment of the correctness of rational defenses of and challenges to claims, made by offering other claims as reasons for or reasons against those claims” (p. 6).

“As we will see later in much more detail, to do their job properly, semantic metavocabularies must use alethic modal vocabulary to make claims about what states and combinations of states of the world the base vocabulary talks about are and are not possible. To do their job properly, pragmatic metavocabularies must use deontic normative vocabulary to make claims about what acts, practical attitudes, and combinations of them are and are not appropriate, and what other acts and attitudes would and would not entitle an interlocutor to them. What can be said in alethic modal terms is substantially and importantly different from what can be said in deontic normative terms. The one concerns features of the objective world, the other features of the practice of discursive subjects. These are the two poles of the intentional nexus that links knowers and the known, minds and the world they understand and act in, representings and what is represented. We want to understand both kinds of thing, and the important relations between them” (p. 7).

“Alethic” is from the Greek aletheia, for truth. The parallelism or isomorphism between the “alethic modal” notion of measuring the subjunctive robustness of assertions, and a “deontic normative” Kantian articulation of the compelling or necessary character of ethical conclusions, which Brandom has long stressed, is very substantially elaborated in the new book.

“In the terms used above to introduce the idea of reason relations we propose to understand the alethic modal semantic metavocabulary and the deontic normative pragmatic metavocabulary as offering different (meta)conceptual perspectives on a common object: the incompatibility of what is expressed by the declarative sentence p and what is expressed by the declarative sentence q. Corresponding claims apply to reason relations of consequence or implication” (pp. 7-8, emphasis in original).

Next in this series: An Isomorphism

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).