This will conclude an examination of Brandom’s early programmatic work “Assertion and Conceptual Roles”. At one point he pithily comments that he is developing an account of saying that does not depend on a prior account of naming. Once again, at a broad level I think that is also something that Aristotle does. Saying viewed this way is more oriented toward valuation than toward representation.

I would suggest that naming is a kind of shorthand for a description or classification that is sufficient to pick something out from other things in the applicable context. What a name cannot be counted on to do is to unambiguously specify an essence or an adequate definition. The very first topic raised in Aristotle’s Categories — which was traditionally placed first in the order of instruction — is “things said in many ways”.

The young Brandom says, “Our strategy now is to use the conditionals we have constructed to develop precise representations of the conceptual contents sentences acquire in virtue of playing a material inferential role in some justificatory system. The most sophisticated use of the notion of a conceptual role has been made by Sellars, who in Science and Metaphysics and elsewhere develops a theory of meaning couched in terms of dot-quoted expressions, where such dot-quotation of an expression results in a term referring to the conceptual (inferential-justificatory) role of that expression” (p. 34).

Every concept worth its salt carries its justification with it. We don’t properly understand an expression if we are unable to justify its use. As Aristotle says, the mark of knowing something is the ability to explain why it is the case. I would maintain that there isn’t any knowing “never you mind how”. The latter is rather the mark of what Plato calls mere opinion.

“According to the present view, it is the defining task of a logic or logical construction that it make possible the explicit codification in a conceptual role of what is implicit in the inferential and justificatory employment of an expression…. [C]onceptual roles in Frege’s and Sellars’ sense can be expressed, using the conditionals of our formal logic not only as the means of expression of roles, but also as providing the model according to which we understand such roles.”

On this view, ordinary if-then reasoning turns out to be a kind of key to understanding meaning. But considerable care is required in working out the details. The conditional that codifies material inferences has different detailed behavior than the common one based on a truth table, and that is a good thing, because the truth table one has significant defects.

“The key to this line of thought is the observation that the only sentences whose roles we understand explicitly are the conditionals. We understand them because we constructed them, stipulating their introduction conditions, and deriving the consequences of such introduction (the validity of detachment)” (ibid).

If-then conditionals allow us to explicitly express the reasons and dependencies that implicitly guide judgment and thought.

“We propose to generalize this clear case, and conceive the mastery of the use of an expression which one must exhibit in order to properly be said to understand it (‘grasp’ its conceptual role) as consisting of two parts, knowing when one is entitled to apply the expression, and knowing what the appropriate consequences of such application are (what justifies using the expression, and what inferences one licenses by so doing). Applying the expression is thus assimilated to performing an inference from the circumstances of appropriate application of the expression to the consequences of its application” (ibid).

But “applying the expression” is just what assertion is. By these lights, every asserting is an inferring.

“On this model, suggested by the later Carnap’s use of partial reduction forms, the conceptual role of any expression is the pair of its circumstances of appropriate application and the consequences of such application, that is, of its (individually) sufficient conditions and of its (jointly) necessary conditions. The application of that expression is to be thought of as an inference from the former to the latter. Assertion thus becomes a limiting case of inference” (p. 35).

It is inference that grounds assertion, not the reverse. Only through inference can anyone understand the significance of an assertion.

“More must be said, however, about the ramifications of taking conditionals to be the models for the conceptual roles of basic sentences, insasmuch as our strategy has been to construct a conditional as stating explicitly (as a license) what is implicit in an inference from its antecedent to its consequent, and then to assimilate the content of basic statements to the model of these constructed conditional statements” (ibid).

“In general, one might think that it was incoherent or circular to define the contents of the categorical sentences of an idiom in terms of the contents of hypothetical sentences of that idiom…. Our construction avoids this worry, since we define conditionals in terms of the contents of basic sentences only in the sense in which those contents are implicit in the informal inferential practices which are the use of the basic sentences.” (pp. 35-36).

Kant already questioned the primitiveness of categorical judgments. My take is that they constitute a form of shorthand for what are really reasonings or interpretations.

“Nor is there anything peculiar about taking a sub-class of sentences as the paradigms to which all others are assimilated in a theory of meaning. Frege, for instance, treats all sentences as implicit identity statements (involving names of the True or the False)…. Thus Frege constructs a theory of meaning based on terms explicated with the logical device of identity, where we base our account on sentences explicated by means of the logical device of conditionals” (p. 36).

Brandom has a complex relation to Frege, championing some of his early work and questioning some of his later work.

“We attempt to give a direct account of saying and what is said which does not appeal to naming and what is named” (ibid).

“This is the essential difference between conceptual role semantics inspired by the sort of concerns articulated by the later Wittgenstein, and referential semantics inspired by Frege” (ibid).

“As Dummett points out, the later Frege broke from previous logicians in treating logic not as the study of inference, but of a special kind of truth…. This view seems to have been motivated by his presentation of logic as an axiomatic system, where some truths are stipulated and other truths are derived from them by a minimum of purely formal inferential principles. The philosophical critique in terms of linguistic practice of the distinction between meaning-constitutive stipulated truths and empirically discovered truths, together with Gentzen’s achievement of parity of formal power between proof-theoretic methods of studying consequence relations and the truth-oriented methods epitomized by matrix interpretations … require us to reassess the relations of explanatory priority between the notions of inference and truth” (p. 36).

Brandom makes a good case for seeing the early Frege as a proto-inferentialist concerned with the formalization of material inference. The later Frege propounded an original and rather strange notion of truth and truth-values as foundational. He held that truth is a (unique) object referred to by all true statements, rather than a property.

“One of Frege’s achievements is his formulation of the principle of semantic explanation, according to which the appropriateness of a form of inference is to be accounted for by showing that it never leads from true premises to conclusions which are not true. The usual way in which to exploit this principle is to begin with an account of truth (typically in representational or referential terms) and partition a space of abstractly possible inferences and forms of inference into those which are appropriate and those which are not appropriate according to the semantic principle, as Frege does in the Begriffschrift. Our approach in effect reverses this order of explanation, beginning analysis with a set of appropriate inferences and explaining semantic interpretants, including truth-values, in terms of them” (pp. 36-37).

The idea of this “principle of explanation” is that sound reasoning from true premises cannot yield a false conclusion. This is not a fact, but a definition that also has characteristics of a Kantian imperative. It is up to us to make it true.

He considers possible objections to the idea of treating hypothetical judgments as more originary than categorical judgments. This should not be taken to apply at the level of truths. In a similar vein, he also says that what our words mean does not determine what we believe.

“Just as it is implausible to take what is possible as determining what is actual, so it is implausible to take the totality of conditional truths as determining the totality of unconditional truths. Indeed, the possession by a formal system of this semantic property would be a strong reason to take its conditional as not a reasonable rendering of the English hypothetical construction ‘if … then’. Embarrassingly enough, the standard truth-functional (mis-named ‘material’) conditional which Frege employs has just this property, namely that if the truth-values of all of the conditionals of the language are settled, then the truth-values of all the sentences of the language are settled. This is proven in Appendix II” (p. 37).

This surprising proof really turns things around. I suppose this result is related to the concerns about “logical omniscience” in classical logic. It is not reasonable to suppose that if a human knows A, then she necessarily knows all the consequences of A. But this is independent of the question of whether we really know anything unconditionally (I tend to think not). There is a also question whether we are properly said to “know” abstract tautologies like A = A, without necessarily knowing what A is (I am inclined to use some other word than knowledge for these cases).

“Our genuine conditional, introduced as codifying a set of non-formal inferences, will not have this undesirable property…. We avoid that result by taking the principle that appropriate inference should never lead from true premises to conclusions which are not true as a necessary, but not sufficient condition for appropriateness of inference. The truth-functional conditional results from taking the principle to provide sufficient conditions as well” (ibid).

Again, this falls within the tradition of alternative, “better” definitions of implication.

“Taking Frege’s semantic explanatory principle as a necessary condition on an account of inferential relations settles that the primary semantic notion will be whatever it is that is preserved by appropriate inferences. Frege calls this ‘truth’, but abstractly there are other properties which could also play this role (e.g., justificatory responsibility) and there are good reasons to expect an adequate semantic theory to account as well for the preservation of ‘relevance’ of some kind by appropriate inferences. This primary semantic notion, however, pertains only to the use of a sentence as a free-standing assertive utterance. A full notion of sentential content must specify as well the role a sentence has as a component in other, compound, sentences, paradigmatically in conditionals. It cannot be determined a priori that these two roles coincide. If with Frege we take the first semantic property to be a truth-value either possessed or not by any sentence, then the assumption that the second or componential notion coincides with the first results in classic two-valued truth-functional logic” (p. 38).

It is noteworthy that even the later Frege’s concern in this context was with “whatever it is that is preserved by appropriate inferences”.

He has previously used the term “designatedness”, which names that “whatever it is that inference preserves” that plays a role in multi-valued logics broadly analogous to that played by truth in two-valued logics.

“[M]any-valued semantics requires the assignment to each sentence of two different sorts of semantic interpretant: a designatedness value indicating possession or lack by a sentence used as a free-standing utterance of the property which appropriate inference must preserve, and a multivalue codifying the contribution the sentence makes to the designatedness value of compound sentences containing it, according to the principle … Two sentences have the same multivalue if and only if they are intersubstitutable salva designatedness value in every sort of compound sentence” (p. 39).

He relates the current development to technical work on the algebraic interpretation of logics.

“A matrix is characteristic for a logic if it verifies just the theorems of that logic. Lindenbaum showed that every logic has a characteristic matrix, namely the one gotten by taking the set of multivalues to be classes of inferentially equivalent sentences, and the designated multivalues to be the theorems of the logic in question” (ibid).

“We are now in a position to notice that a repertoire, together with the partial ordering induced on the sentences of a repertoire by the conditionals contained in its formally expanded consequence extension constitute such a Lindenbaum matrix” (ibid).

The conditional as Brandom has defined it provably meets Frege’s criterion of inference preservation. Brandom has extended algebraic logic to include patterns of material inference.

“Theorem 1 above shows that modus ponens preserves designatedness, that is membership in the extended repertoire. Or, to put the same point another way, that result shows that our constructed conditional satisfies Frege’s semantic explanatory principle when membership in a repertoire is taken as the prime semantic notion, and social practice determines an antecedent class of appropriate material inferences. The formally extended repertoire thus is, in a precise sense, the characteristic semantic matrix not for a logic or a set of formal inferences, but for a set of material inferences” (p. 40).

“There are three specific points which should be made concerning this interpretation. First, what is captured by semantic matrices is taken to be a matter of formal inferences first, and logical truths verified by the matrix only second, although this is not how such matrices are usually thought of. Second, we generalize the notion of a characteristic matrix for a set of formal inferences to apply to material inferences as well. Finally, notice that in addition to the structure of material inference codified in each repertoire-matrix we can in fact identify a logic with regard to the whole idiom, insofar as some complicated conditionals will appear in all repertoires. We have not constructed a characteristic matrix for this logic by ordering the sentences of the language according to repertoire-designated conditionals. In some ways it is accordingly more appropriate to say that each repertoire expresses a single matrix valuation characteristic of a set of material inferences, and that the whole idiom comprising all admissible repertoires is characteristic of the formal or logical inferences involving the conditional we used to make explicit the materially appropriate inferences” (ibid).

“In this way, then, we can exploit Frege’s semantic explanatory principle and the truth-oriented matrix semantics it inspired as theoretical auxiliaries useful in the formal analysis of a socially specified set of appropriate inferences” (ibid).

“Seeing logic in the way I have been recommending, however, as a formal tool for the explicit expression of inferential roles, obviates the need for appealing to prior notions of truth or truth-value. We have interpreted Frege’s truth-values as they figure in his semantic principle first as the designatedness values of multivalued logic, and then moving from concern with the codification of formal inference to concern with the codification of material inference, interpreted as expressing membership in a repertoire. Recalling the social practical origins of these repertoires, it would be appropriate to call the two circumstances of membership and non-membership in a particular repertoire assertibility values with respect to that repertoire. We have given a much more precise sense to this term than semantic theorists who advocate the primacy of assertibility over truth typically manage to do, however” (pp. 40-41).

“We represent the matrix valuation on the language induced by a formally expanded repertoire by associating with each sentence its repertoire-relative conceptual role, consisting of inferential circumstances and consequences of assertion. It is clear that this is an adequate representation in that this set of roles, together with the repertoire generating them, determines the partial order of the language by the conditional which is the Lindenbaum matrix. These conceptual roles are thus taken as multivalues, with repertoire membership identified as designatedness with respect to the semantic principle. The multivalues must, of course, determine compounding behavior according to our motivation…. It is … a criterion of adequacy of this representation that sentences with the same conceptual role, that is, multivalue, should be intersubstitutable in conditionals preserving both designatedness values and multivalues” (p. 41).

So far he has focused on a notion of the conditional that is a primitive “arrow” rather than something defined by a truth table. He briefly considers how to define other connectives that work off of the designatedness that plays a truth-like role in multi-valued logics, but again affirms the special importance of conditionals.

” ‘Truth-functional’ connectives can now be introduced using designatedness values as the extensions of sentences…. We would like to be able to semantically interpret all forms of sentence compounding by means of functions taking conceptual roles, or sets of them, into conceptual roles, as we can do for conditionals…. Our use of the conditional as both the model of and a tool for the expression of conceptual roles embodies the belief that the contribution a sentence makes to the roles of conditional it is a component in suffices to determine its role in other compounds” (p. 42).

He quotes Frege saying that the kernel of the problem of judgment splits into that of truth and that of what he calls “a thought”, which refers to some declarative content. Given Frege’s unitary view of “truth”, this thought-content identified with saying and conceptual roles has to be responsible for all differentiation.

“By a thought, Frege makes clear, is intended what is referred to in English by that-p clauses. We have identified these judged contents as conceptual roles. In what follows, we try to exhibit a representative variety of uses of such that-p clauses in terms of conceptual roles” (p. 43).

Finally we come to prosentences.

“Our starting point is the prosentential theory of truth of Grover, Camp, and Belnap. That account can best be sketched as the product of three different lines of thought: i) the redundancy theory of Ramsey and others, which says that the conceptual content of ‘it is true that-p‘ is always just the same as that of p…. ii) an account of truth in terms of infinite conjunctions and disjunctions…. [T]he best succinct statement of this view is in Putnam’s Meaning and the Moral Sciences…. ‘If we had a meta-language with infinite conjunctions and infinite disjunctions (countable infinite) we wouldn’t need “true”!…. [F]or example, we could say … “He said ‘P₁‘ & P₁” (ibid).

“iii) Finally, and this is what is distinctive to the view under discussion, it is observed that pronouns serve two sorts of purposes. In their lazy use, … they may simply be replaced by their antecedents (salva conceptual role). In their quantificational use, as in ‘Each positive number is such that if it is even, adding it to 1 yields an odd number’, the semantic role of the pronoun is determined by a set of admissible substituends (in turn determined by the pronomial antecedent)” (p. 44).

“Thus ‘Everything he said is true’ is construed as a quantificational prosentence, which picks up from its anaphoric antecedent a set of admissible substituends (things that he said), and is semantically equivalent to their conjunction” (ibid).

“The authors of the prosentential theory are concerned that ‘is true’ be taken to be a fragment of a prosentence, not a predicate which characterizes sentence-nominalization…. The authors are worried that if the first part of a sentence of the form ‘X is true’ is taken to be a referring sentential nominalization that, first, ‘is true’ will inevitably be taken to be a predicate, and second, the anaphoric prosentential reference of the whole sentence will be passed over in favor of the view that the nominalization does all the referring that gets done, and would vitiate the view” (p. 45).

“In fact this is a situation in which we can have our cake and eat it too. We consider ‘X is true’ as composed of a sentence nominalization X which refers to sentences, and a prosentence-forming operator ‘is true’.” (ibid).

“Our construction of conceptual roles in terms of conditionals of course presents natural criteria of adequacy for translation functions between repertoires contained in a single idiom, or which are members of different idioms” (p. 51).

“We show now how those semantic facts about the idiom can be expressed explicitly as the content of claims made within that idiom. We use the logical vocabulary of conditionals and repertoire attributions we have already constructed to define a further bit of expressive machinery, that-clauses, which will thus have a logical function in making explicit semantic features implicit in the idiom” (p. 53).

“[T]he account of conceptual roles is novel in being entirely non-representational. In the formal idiom we develop, it is not a necessary feature of a saying that-p that the sentence involved represent some state of affairs. Of course sentences used to say things may also be representations, and this fact might be crucial for the understanding of the use of language in empirical inquiry. But our model is broader, and we may hope that it can find application in the explication of other forms of discourse (e.g., literary and political discourse) where the representational paradigm is less apt than it perhaps is for scientific idioms” (p. 55).

“Perhaps the most important feature of our account is the crucial place given to logic, as providing the formal means by which an idiom can come to express explicitly crucial semantic facts which are implicit in the system of justificatory practices which are the use of a language. We argued that the function thus assigned to logic as a formal auxiliary in a theory of meaning is that which Frege originally envisioned and pursued. Our own development looked at he codification of inferential practices in conditionals in some detail, and somewhat less closely at the codification of repertoires in prosentences containing ‘is true’ and in propositional attitudes, and at the codification of roles in ‘that’-clauses. The basic claim here is that logic must not be restricted to the analysis of the meanings sentences acquire in virtue of the formal inferences they are subject to, as is the usual procedure). Logic should not be viewed as an autonomous discipline in this way, but as a tool for the analysis of material inference, and for making explicit the roles played by sentences in systems of material inferential practice. Using logical devices so interpreted, we were able to specify not only what role a performance needs to play in a system of social practices in order to be a saying (asserting, professing, claiming, etc.) that-p, but also to show what it is about that system of practices in virtue of which the content of such a saying can be that someone else has said (asserted, etc.) something. Indeed the only sort of ‘aboutness’ we ever employ is the reference of one bit of discourse to another (anaphoric reference if performance or sentence tokens are at issue, and mediated by conceptual roles otherwise)” (pp. 55-56).

When Aristotle discusses saying something about something, implicitly that second something is also something said. This phrase refers to that phrase. The kind of reference that is most relevant in all this is what I think of as constitutive cross-reference, or as Brandom calls it, back-reference or anaphora. Less adequately, it has been called “self” reference, but if we examine this closely, it does not involve a unitary self or a pure undifferentiated reflexivity, but rather parts referring to other parts.

Conceptual content emerges out of a sea of cross-reference. A constitutive molecular cross-reference of Fregean declarative “thoughts” or “content” or Aristotelian “sayings” precedes sedimentation into molar subjects and objects.

Epilogue to this series: Anaphora and Reason Relations