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

Identity, Difference, Reflection

Reflection is also the key to Hegel’s often misunderstood views on identity and difference.

“Reflection is the shining of essence within itself. Essence, as infinite immanent turning back is not immediate simplicity, but negative simplicity; it is a movement across moments that are distinct, is absolute mediation with itself. But in these moments it shines; the moments are, therefore, themselves determinations reflected into themselves” (Hegel, Logic, di Giovanni trans., p. 354, emphasis in original).

He goes on to discuss identity, difference, and the notorious “contradiction” as principal moments or determinations of reflection. Sometimes he uses these terms in the conventional way — of which he is highly critical — and sometimes he gives them his own meaning.

On Aristotelian grounds, I have long had doubts about appeals to an implicitly immediate simplicity or “identity” of substance in traditional metaphysics. I take these to be a form of Platonizing that originated in the neoplatonic commentaries on Aristotle. Hegel’s alternative suggestion of a “negative simplicity” seems much more plausible generally, as well as more consistent with the Aristotelian texts. We just have to get past the difficulty of Hegel’s idiosyncratic metaphorical straining of language about “negation”, and recognize that he is inventing ways to talk about the limits of representation, rather than grossly abusing the “classical” negation of formal logic.

Hegel’s remarks about identity are actually pretty clear, and worth quoting at length. As with negation, in Hegel identity, difference, and “contradiction” only have the meanings that they have in classical logic when he is pointing out their limitations. The alternative meanings that he actually endorses deeply reflect his critique of representationalism.

“In its positive formulation, A = A, [the principle of identity in classical logic] is at first no more than the expression of empty tautology. It is rightly said, therefore, that this law of thought is without content and that it leads nowhere. It is thus to an empty identity that they cling, those who take it to be something true, insisting that identity is not difference but that the two are different. They do not see that in saying, ‘Identity is different from difference’, they have thereby already said that identity is something different. And since this must also be conceded as the nature of identity, the implication is that to be different belongs to identity not externally, but within it, in its nature. — But, further, inasmuch as these same individuals hold firm to their unmoved identity, of which the opposite is difference, they do not see that they have thereby reduced it to a one-sided determinateness which, as such, has no truth. They are conceding that the principle of identity only expresses a one-sided determinateness, that it only contains formal truth, truth abstract and incomplete. — Immediately implied in this correct judgment, however, is that the truth is complete only in the unity of identity and difference, and, consequently, that it only consists in this unity. When asserting that formal identity is incomplete, there is vaguely present to one’s mind the totality, measured against which that identity is incomplete; but the moment one insists that identity is absolutely separate from difference and in this separation takes it to be something essential, valid, true, then what transpires from these two contradictory claims is only the failure to reconcile these two thoughts: that identity as abstract identity is essential, but that, as such, it is equally incomplete. What is lacking is awareness of the negative moment as [that by] which, in these claims, identity itself is displayed. — Or when this is said, that identity is identity essentially as separation from difference or in the separation from difference, then right there we have the expressed truth about it, namely that [formal] identity consists in being separation as such, or in being essentially in the separation, that is, it is nothing for itself but is rather moment of separation.”

“As to the other confirmation of the absolute truth of the principle of identity, this is made to rest on experience in so far as appeal is made to the experience of every consciousness; for anyone presented with this proposition, ‘A is A’, ‘a tree is a tree’, immediately grants it and is satisfied that the proposition is self-evident and in need of no further justification or demonstration.”

“On the one hand, this appeal to experience, that every consciousness acknowledges the principle universally, is a mere manner of speaking. For nobody will want to say that the abstract proposition, ‘A is A’, has actually been tried out in every consciousness. The appeal to actual experience is therefore not in earnest but is rather only an assurance that, if the experiment were made, universal acknowledgement of the proposition would be the result. — And if it is not the abstract proposition as such that is meant, but the proposition in concrete application, from which application the abstract proposition would then have to be developed, then the claim to the universality and immediacy of the latter would consist in the fact that every consciousness assumes it or implies it as a foundation, and indeed does so in every utterance. But the concrete and the application are precisely in the reference that connects simple identity with a manifold which is different from it. Expressed as a proposition, the concrete would be first of all a synthetic proposition. From this concrete itself, or from the synthetic proposition expressing it, abstraction could indeed extract the principle of identity through analysis; but, in actual fact, it would not then leave experience as it is but would have altered it, since in experience the identity was rather in unity with difference. And this is the immediate refutation if the claim that abstract identity is as such something true, for what transpires in experience is the verry opposite, namely identity only united with difference” (pp. 358-359, emphasis in original).

“Identity, instead of being in itself the truth and the absolute truth, is thus rather the opposite; instead of being the unmoved simple, it surpasses itself into the dissolution of itself.”

“More is entailed, therefore, in the form of the proposition expressing identity than simple, abstract identity; entailed by it is this pure movement of reflection in the course of which there emerges the other, but only as reflective shine, as immediately disappearing…. The propositional form can be regarded as the hidden necessity of adding to abstract identity the extra factor of that movement…. Consequently, if appeal is made to what appearance indicates, then the result is this: that in the expression of identity, difference also immediately emerges” (p. 360, emphasis in original).

“From this it is clear that the principle of identity itself, and still more the principle of contradiction, are not of merely analytical but of synthetic nature” (ibid, emphasis in original).

Here he uses Kant’s distinction of analytic from synthetic judgments. Analytic judgments are purely formal and tautological; canonically, the predicate is considered to be literally implied by the subject. Synthetic judgments on the other hand go beyond what is already implied by the subject or premises. This includes most judgments in ordinary experience. Synthetic judgments involve the material inference that Robert Brandom has particularly expounded in recent times.

“Thus the result of this consideration is this: (1) the principle of identity or contradiction, when meant to express merely abstract identity in opposition to difference as a truth, is not a law of thought but expresses rather the opposite of it; (2) these two principles contain more than is meant by them, namely this opposite, absolute difference itself” (p. 361, emphasis in original).

On Reason

Plato, Aristotle, Kant, and Hegel all give reason an ethical mission that goes far beyond what formal logic can do. Formal logic has immense value in specialized contexts like the engineering of systems, but does not necessarily or directly yield any philosophical insight. As Aristotle said, it is a tool and not an independent source of knowledge.

As Paul Ricoeur has pointed out, a wide variety of such technical tools can also indirectly serve to sharpen a hermeneutic understanding. But for that to be possible, a hermeneutic project must already be underway.

Nevertheless I would argue that for Plato, Aristotle, Kant, and Hegel among others, such a project is precisely what “reason” primarily amounts to. More than anything else, reason is constituted by reflective interpretive judgment. Only secondarily is it concerned to deduce consequences from premises. Every such deduction either implicitly presupposes reflective interpretive judgments about meaning, or simply assumes that meanings are already somehow pre-given.

But reason at its core is the ramified and expansive understanding of meaning, as well as the process of aiming at and achieving such understanding, and the putting of that understanding into practice. And “reasonableness” is a matter of an emotional constitution that allows judgments to be made without bias or prejudice or excessive self-centeredness, so that one’s actions reflect and manifest this kind of understanding.

Pure Reason?

Hegel’s “logic” takes what Kant calls pure reason as its subject matter. Hegel regards Kantian pure reason as a world-changing revolution, because in contrast to early modern views, it seeks not to imitate the formal character of mathematical reasoning, but rather to achieve the discipline of a kind of self-sufficiency that does not appeal to anything external to it. Kant and Hegel differ on the scope of this self-sufficiency, but that is a different matter.

Early modern views of the world generally rely on many substantive assumptions. There is strong motivation for them to do so, because in order to yield any substantive conclusions, reasoning of a broadly formal kind requires substantive assumptions. The assumptions are typically of a sort analogous to those that Aquinas regards as grounded in the natural light of reason, which is not itself reason, but a kind of originating intuition of truth given to us by God. Descartes, for example, explicitly appeals to a variant of the Thomistic doctrine of natural light.

(The strong Thomistic notion of the natural light of reason and of reason’s relative autonomy from the simple dictates of authority is itself a development of almost inestimable importance, compared to completely authority-bound views of religion such as present-day fundamentalism. Indeed, something like the natural light of reason was never completely absent from the earlier medieval tradition either.)

But for Kant, reason is purely discursive, and cannot appeal to any intuitive source of truth like a natural light. Pure reason is nonetheless supposed to be able to stand on its own. In Kant’s language, it is “autonomous” (see also Kant’s Groundwork; Self-Legislation?). Kant’s critique of dogmatism especially targets assumptions that are naively realistic in the sense of claiming direct knowledge of external or inner objects, but it is broader than that.

Kant in the Critique of Pure Reason is most directly responding to empiricist views broadly associated with Locke, which were dominant in England and France, and popular in Germany in his day; but even more so to the rationalist system of Christian Wolff (1679-1754), which then dominated German academic teaching. (Wolff was an accomplished mathematician who had corresponded with Leibniz, and greatly contributed to popularizing the part of Leibniz’s philosophy that Leibniz had published in his own lifetime. Like Leibniz, he is associated with moderate Enlightenment, while at the same time showing a degree of sympathy for scholastic philosophy.)

Kantian pure reason effectively aims to be free of unnecessary assumptions, especially those of the Wolffian system, but also those of the empiricists. Kant also criticizes Wolff’s and Spinoza’s idea that philosophical reasoning should as much as possible resemble mathematical reasoning. What makes it possible for Kant to avoid assumptions beyond the famous “God, freedom, and immortality” (and for Hegel to avoid any assumptions at all) is a move away from the early modern ideal of reason as formal.

Without ever explicitly saying so, Kant in fact takes up and works with a notion of reason that is close to aspects of Plato and Aristotle that were generally neglected in the intervening tradition. Reason in Plato, Aristotle, Kant, and Hegel is not limited to formal reasoning. It includes what in more recent times Sellars and Brandom have elaborated under the name of material inference.

Formal reasoning is called formal because it is supposed to apply to all things, independent of any analysis of meaning. But this makes it dependent on assumptions in order to yield conclusions. Material inference — which was also present as a minor theme in scholastic logic — is on the contrary grounded in the interpretation of meaning. It is this reflective grounding that can enable reason to be autonomous and “pure”, with no reliance on anything external to it.

Sellars illustrates material inference with examples like “there are dark rain clouds in the sky, so I should take my umbrella when I go out”. Brandom elaborates with an account of how such judgments may be successively refined based on additional information. In general, if I strike a match correctly, it will light. But under certain conditions, it will not light. But under yet more specific additional conditions, it will in fact still light.

Both Sellars and Brandom — working within the tradition of contemporary analytic philosophy — tend to reach for examples that involve empirical facts, and relations of cause and effect in the broad modern sense. But material inference is more general than that. It is grounded in meaning as we encounter it in real life. Its scope is not limited to any particular kind of meaning, nor does it assume any particular theory of meaning.

Pure reason, then — far from excluding meaning, as formal logic does — is concerned with the progressive self-clarification of meaning — or Kantian “taking as”, or judgment — in a reflective context.

For Hegel, “logic is to be understood as the system of pure reason, as the realm of pure thought” (Science of Logic, di Giovanni trans., introduction, p. 29). This is what he calls the “concept of science”, and also “absolute knowledge” (p. 28). As I’ve pointed out before, in Hegel these terms have specialized meanings that are far from their ordinary connotations in English. Science need not be empirical, and “absolute” in this context just means the same thing as “pure” or “autonomous” — that reflective judgment need presuppose nothing outside itself.

For Hegel, the standpoint of pure reason (or “science”, or “absolute” knowing) is that of reflective judgment. The whole effort of the Phenomenology of Spirit is required to reach this point, which he then uses as a starting point in the Logic.

“Pure science thus presupposes the liberation from the opposition of consciousness [between itself and its object]…. As science, truth is pure self-consciousness as it develops itself and has the shape of a self, so that that which exists in and for itself is the conscious concept and the concept as such is that which exists in and for itself” (p. 29, emphasis in original).

The reflective concept has the shape of a “self” — a reflexivity — that is not to be identified with our empirical self, but rather is related to the reflective character of self-consciousness, which overcomes the simple opposition between consciousness and its object.

“This objective thinking is thus the content of pure science. Consequently, far from being formal, far from lacking the matter for an actual and true cognition, it is the content which alone has absolute truth” (ibid).

He calls reflective judgment objective thinking, precisely because it begins only after the separation of consciousness from its object ends. Reflective judgment and self-consciousness will not be separated from “the concept” in which they are embodied. Rather, we have here a case of the Aristotelian identity of pure thinking with what it thinks.

“Logic has nothing to do with a thought about something which stands outside by itself as the base of thought; nor does it have to do with forms meant to provide mere markings of the truth; rather, the necessary forms of thinking, and its specific determinations, are the content and the ultimate forms of truth itself.”

“To get at least some inkling of this, one must put aside the notion that truth must be something tangible. Such tangibility, for example, is carried over even into the ideas of Plato which are in God’s thought, as if they were, so to speak, things that exist but in another world or region, and a world of actuality were to be found outside them which has a substantiality distinct from those ideas and is real only because of this distinctness” (pp. 29-30).

Truths are not objects, and they are not given to us in the way that ordinary consciousness takes objects to be. For Hegel, moreover, spiritual values do not have to do with turning away from this world in favor of another one. They are intended to guide us in life.

“There will always be the possibility that someone else will adduce a case, an instance, in which something more and different must be understood by some term or other” (p. 28).

Reflection and interpretation are inherently open-ended.

“How could I possibly pretend that the method that I follow in this system of logic, or rather the method that the system itself follows within, would not be capable of greater perfection, of greater elaboration of detail? Yet I know that it is the one true method. This is made obvious by the fact that this method is not something distinct from its subject matter and content — for it is the content in itself, the dialectic which it possesses within itself, which moves the subject matter forward. It is clear that no expositions can be accepted as scientifically valid that do not follow the progression of this method and are not in tune with its simple rhythm, for it is the course of the fact [Sache] itself” (p. 33).

Translator di Giovanni comments in his glossary, “In non-technical contexts, [Sache] can and should be translated in a variety of ways, such as ‘substance’, or even ‘thing’. As category, however, ‘fact’ seems to be the best rendering. Sache, like ‘fact’, denotes a thing or a situation which we understand to implicitly contain all the factors required for an explanation of its existence. Its presence therefore cannot be doubted even when those factors have yet to be made explicit. The related word, Tatsache, was first coined… in order to translate the English term ‘matter of fact'” (pp. lxxi-lxxii).

To me, these sound like reasons for calling Hegel’s Sache something other than “fact”. Especially in a work of “logic” that invokes “science”, the English word “fact” would most commonly be taken taken to mean an unambiguous empirical truth. Both what I think Hegel means and the explanation di Giovanni gives of it seem better suited by the more open connotations of an English phrase like “the concrete case” or “the matter at hand”. The Sache is something objective, but it is objective in the indefinite sense of a Gegenstand [“object” in the sense of something standing over and against us, but whose nature has yet to be determined].

I used to think that reason that would be applicable to life (or to anything like Hegel’s Sache) could not possibly be pure. I now think that with the inclusive character of reflective judgment and material inference, it can be pure.

Heidegger, Sartre, Aquinas?

The heyday of existential Thomism is well past, but Etienne Gilson and others were certainly not wrong to take note of a close connection, despite other large differences.

Heidegger in Being and Time (1926) famously claimed that philosophers since Plato had been preoccupied with questions about beings and had lost sight of the central importance of Being writ large. Many 20th century Thomists partially accepted this argument, but contended that Aquinas was an obvious exception, citing Aquinas’ identification of God with pure Being. Heidegger rejected that identification, and would have insisted that Being was not a being at all, not even the unique one in which essence and existence were identified. Nonetheless there is a broad parallel, to the extent that Heidegger and Aquinas each in their own way stress the dependency of beings on Being.

In some circles, Aquinas has been criticized for promoting a “philosophers’ God”. But according to Burrell, Aquinas argued in effect that on the assumption that there is only one God, the God of Summa Theologica and the God of common doctrine must be acknowledged to have the same referent even if they have different senses, like Frege’s example of the morning star and the evening star.

Sartre in his 1945 lecture “Existentialism is a Humanism” put forth the formula that “existence precedes essence”. Aquinas in Being and Essence had argued that God has no essence other than existence. Sartre argued in effect that the human has no essence other than existence. In his context, this is to say either that the human essence consists only in matters of fact, or that there is simply no such thing as a human essence.

Sartre’s use of the word “essence” reflects a straw-man caricature of bad essentialism. Whatever we may say that essence really is, contrary to Sartre’s usage it is supposed to be distinguished from simple matters of fact. On the other hand, in formal logic, existence does reduce to matters of fact.

What Aquinas, Heidegger, and Sartre have in common is that they all want to treat existence as something that transcends the merely factual and formal-logical. Speaking schematically, it is rather the analogues of essence that transcend the merely factual in the Platonic and Aristotelian traditions. Thus Aquinas made a major innovation in inventing a new, unprecedented concept of existence that transcends the factual. I’m inclined, however, to sympathize with Dietrich of Freiberg’s argument that the concept of essence could already do all the work that Aquinas’ new supercharged concept of existence was supposed to do.

What is important for practical purposes is that there is something that transcends the merely factual. I think the close connection of “essence” with form and ends makes it an ideal candidate. The big difference between form and ends on the one hand and facts on the other is that logically speaking, facts can be arbitrary, whereas any form or end or essence necessarily implies some nonarbitrary order.

For Aquinas, God is simultaneously a fact and more than fact, and is unique in this regard. Nothing else has this dual status. Sartre transferred this unique dual status to the human. By contrast, the neoplatonic One is strictly more than fact — in traditional language, the One as source of being was said to be “beyond being” altogether. The 20th century theologian Paul Tillich quipped that it could be considered blasphemy to say that God exists (because “existence” is mundane and factual).

The “To-Be itself” of Aquinas, while profoundly innovative with respect to previous tradition and certainly not strictly Aristotelian, is nonetheless arguably more Aristotelian in spirit than the neoplatonic One, insofar as it is less ambiguous about the goodness of the actual world. Plotinus struggled mightily to reconcile a commitment to the goodness and beauty of this actual world with an ascetic tendency to devalue all finite things in face of the infinite One. In Aquinas there is still some tension between the reality of secondary causes and the absolute dependence of everything on God, but I think it is fair to say that the way Aquinas sets up the problem makes the reconciliation easier to achieve. This was a huge accomplishment. Nonetheless, taking into account other factors like assertions about the place of omnipotence and sheer power in the scheme of things, my overall sympathies lie more with the neoplatonic “strictly more than fact” perspective, and even more so with Aristotle’s more modest view that the “First” cause is strictly a final cause.