Anaphora and Prosentences

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

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