All the Way Down

Once of the things I’ve most appreciated about Brandom has been his unwillingness to reduce normativity and value judgments to non-normative factors. Repeatedly in Making It Explicit, he speaks of norms “all the way down”. There is even a subheading for “all the way down” in the index entry for “norms” (p. 732). But in conjunction with this, he repeatedly suggests that the relation between pragmatics and semantics, while symmetrical in many respects, also includes an asymmetry, according to which it is more appropriate to say that normative pragmatics grounds representational semantics than vice versa. This is in distinction both to common views that privilege representation over inference and semantics over pragmatics, and to the purely symmetrical view of semantics and pragmatics that he seems to propound in Reasons for Logic, Logic for Reasons.

The symmetrical view can be seen in the favorable light of other symmetries that Hegel argues for in his campaign against “one-sidedness”. But it also implies that there is no sense in which normative pragmatics ought to be seen as coming before representational semantics.

Brandom’s 1976 dissertation, which is partly framed as the elaboration of a new form of pragmatism, makes links between the pragmatism it advocates, and a priority of pragmatics over semantics in philosophy of language. But as mentioned above, this year’s Reasons for Logic, Logic for Reasons, while applying inferentialist explanation to semantics in new ways, and while remaining as much as ever committed to an inferentialist order of explanation in general, nonetheless seems to back off from claiming any priority for pragmatics over semantics.

My worry is that this new symmetry and parity between pragmatics and semantics could end up weakening the commitment to “normativity all the way down”. The new thesis of full symmetry builds on his previous analogy between normativity and modality or subjunctive robustness, which I take to be sound. It may be that normativity all the way down does not really require the relative priority of pragmatics over semantics that Brandom claims in the dissertation and Making It Explicit, but I think more on this needs to be said.

Implication Spaces

“Logical vocabularies make reason relations explicit in terms that appeal only to the conceptual resources supplied by the base vocabularies from which they are conservatively elaborated. They are in that sense intrinsic vocabularies for specifying reason relations. Logical vocabularies, however, are not purely metavocabularies, in the sense in which semantic and pragmatic rational vocabularies are. The sequent-calculus vocabularies in which we say how to elaborate arbitrary base vocabularies into logically extended vocabularies with the capacity to codify reason relations are genuine metavocabularies in that sense. Like semantic and pragmatic metavocabularies, they do not extend the base vocabularies for which they are metavocabularies. They are purely metalinguistic, talking about expressions in the base vocabulary, rather than using them” (Brandom in Hlobil and Brandom, Reasons for Logic, pp. 17-18, emphasis in original).

“Logical vocabulary is a hybrid or mongrel kind of metavocabulary. It plays the expressive role of explicitating reason relations: making them explicit, constructing sentences intelligible as saying that relations of implication and incompatibility hold. That is a broadly metalinguistic function. But logical vocabulary performs that explicative expressive function by using the sentences whose reason relation it articulates, rather than by talking about them (mentioning them).”

“These observations raise the question whether there is a purely intrinsic-explicative vocabulary for specifying reason relations that is a rational metavocabulary in the sense of being genuinely and wholly metalinguistic. The answer is ‘yes’…. Our candidate, informed by work due to Dan Kaplan (2022), is an implication-space metavocabulary for specifying both reason relations and the conceptual role sentences play in virtue of standing to one another in such reason relations. Very roughly, where Gentzen’s sequent-calculus metavocabulary treats implications as basic objects in a proof-theoretic formalism, Kaplan’s implication-space metavocabulary treats them as basic objects in a model-theoretic formalism. It represents the current state of the art in inferentialist semantics.”

“Inferentialists have long thought that the universe from which semantic interpretants are drawn or from which those interpretants are built — the analogue of the universe of mereologically structured worldly states out of which semantic interpretants (propositions) as pairs of sets of truth-making states and falsifying states are built — should consist of implications (including incompatibilities coded as implications) and sets of them” (p. 18, emphasis in original).

This is vital stuff. At the risk of sounding dogmatic in the Kantian sense myself, I have long thought that the world is made of implications. What this really means is that the determinacy in it is made of implications.

“Kaplan’s (2022) first conceptual innovation was the idea that thoroughgoing inferentialists ought to treat the most basic units being interpreted, no less than the semantic interpretants assigned to them, as being implications, rather than the sentences that make up their premises and conclusions. Only at a second, subsequent stage would semantic interpretation be extended from implications to the sentences they contain. He accordingly begins with a universe of candidate implications, together with a partition of that universe into a distinguished set of good implications — ones whose conclusions really follow from their premises — and the rest. This universe of candidate implications with a distinguished subset is an implication space.” (p. 19).

Note that he speaks of implications containing sentences, rather than of sentences “having” implications. This reflects the implication-first point of view: implications are “the most basic units being interpreted”.

“Any base vocabulary determines such an implication space, since the lexicon of the vocabulary suffices to define the points (candidate implications as ordered pairs of sets of sentences of the lexicon), and the reason relations of the vocabulary suffice to determine the distinguished set of good implications” (ibid).

“We are exploring the idea of understanding meaning to begin with in terms of reasons instead of understanding it in terms of truth. That is to understand meaning in terms of a dyadic relation (between sets of sentences) instead of in terms of a monadic property (of sentences). On the approach that takes truth as basic, one starts with assignments to sentences of a truth value: as true or false, correct or incorrect, good or bad (as a representation). However, although assignments of truth values are the beginning of semantic interpretation on this approach, they are not the end. To get a notion of meaning that corresponds to what one grasps (however imperfectly) when one understands a sentence, one must advance from consideration of truth values to consideration of truth conditions. (One must add to a semantic conception of Fregean Bedeutung of a sentence a semantic concept of its Fregean Sinn.)” (pp. 19-20; see also Brandom on Truth).

When we contrast appeal to reasons with direct appeals to truth, the problem with direct appeals to truth is that there is no good way to separate them from what Kant would call dogmatic assertions.

It seems to me that the truth-first approaches to meaning inevitably end up assuming particular truths. Such assumptions may be entirely innocent and tentative, or not, and there is no way to easily distinguish the innocent ones. On many traditional views, the necessity of such assumptions is simply taken for granted. Here is an alternative to all of that that respects natural language, but can also be made mathematically rigorous. I did not expect such a thing to even be possible.

I think Aristotle and Plato already took a reasons-first approach, but it was purely hermeneutic, without mathematical underpinning, in spite of Plato’s great interest in mathematics.

Ultimately I do more hermeneutics than mathematics myself, but for quite some years I was keenly interested in mathematics. In my day job, I implicitly lean on both constructive mathematics and a kind of hermeneutics on an everyday basis, in doing a kind of logically oriented engineering modeling of “real world” use cases. So whereas records in a database may be taken as expressing sentences that are supposed to be true, I do all my design in terms of the functional dependencies of one thing on another (where the value of one is a simple mathematical function, fully determined by others that can be finitely enumerated and are usually very few). These can be thought of as if-then rules that apply to all practically relevant cases, without claiming to represent universal truth. This applies a kind of lightly formalized inferentialism in the engineering world, which can also be very pragmatic and adaptable to new hypotheses. I do indeed find that these practical judgments (even well outside of the broadly ethical domain that I am mainly concerned with here) have all the characteristics that Brandom talks about. So naturally I found Brandom’s explicit inferentialism very appealing.

“At the extensional semantic ground level, one can say that a sentence is true, and in the reason-based setting one correspondingly can say at the extensional semantic ground level that an implication is good or an incompatibility holds. Given that analogy, the question becomes: what stands to implication (reason relation) values (good/not-good) as truth conditions stand to truth values?”

This is a distinction that Aristotle also makes in his own way. The more elementary stages of inquiry are concerned with a preliminary mapping out that some characterization of something in the domain is at least pragmatically true. The more advanced stages are concerned with why it is true, or what makes it true.

“The idea behind truth conditions (and Fine’s generalization to truth-makers and falsifiers) is that apart from the question of whether a truth-candidate actually is true or false, there is the question of what it would take to make it true — what things would have to be like for it to count as correct in this distinctive semantic sense. The idea behind the first stage of implication-space semantics is that apart from the question of whether a candidate implication actually is good (according to the partition of the space of candidate implications into good and bad determined by the underlying base vocabulary), there is the question of what it would take to make it good. In the special case of reason relations that already do hold, candidate implications that are good, this takes the form of asking about the circumstances under which it would remain good. That is the range of subjunctive robustness of the implication” (p. 20).

This notion of a scale of subjunctive robustness is where the hermeneutics meets the math.

“The range of subjunctive robustness of a candidate implication is its semantic counterpart in the form of its good-makers, as in Fine’s truth-based semantic setting the semantic interpretants are their truth-makers (and falsifiers).

“Grasping ranges of subjunctive robustness in this sense is an essential part of understanding reason relations in ordinary vocabularies” (pp. 20-21).

“The ranges of subjunctive robustness of candidate implications are their ‘goodness’ conditions, as truth conditions are the ‘goodness’ conditions of sentences. For an implication to be good in the reasons-first semantic setting is for its premises to provide reasons for its conclusion (or reasons against, in the case of incompatibilities), while for a sentence to be good in the truth-first semantic setting is for it to be true. The advance from a conception of semantic goodness to a conception of meaning is the advance to consideration of circumstances under which a reason relation or sentence would be good….. In the implication-space setting, the circumstances are additional premises (and, in the fully general multisuccedent case also additional conclusions) that would make or keep the reason relation good. By contrast to the truth-maker setting, in the implication-space setting, those further premises and conclusions are just more sentences of the lexicon of the base vocabulary. That is why implication-space semantics counts as intrinsic” (pp. 21-22, emphasis in original).

“In this way, a model-theoretic inferentialist semantics becomes available that is sound and complete for the aforementioned expressive logic NMMS [NonMonotonic MultiSuccedent logic]. The implication-space semantics shows how to compute the conceptual roles of arbitrary logically complex sentences from the conceptual roles of logically atomic sentences of any base vocabulary — even when the base vocabulary, and so its (conservative) logical extension, are radically substructural, including those that do not satisfy the metainferential structural closure conditions of monotonicity and transitivity. To do this, the implication-space rational metavocabulary must make explicit the conceptual roles played by sentences of all those base vocabularies, as well as their logical extensions. It is universally explicative of sentential conceptual roles. And since implication spaces can be constructed using no resources other than those supplied by the spare specifications of arbitrary, even substructural base vocabularies — just sentences and set-theoretic constructions from them representing their reason relations — the implication-space model-theoretic semantics qualifies as a universal intrinsic-explicative rational metavocabulary” (pp. 22-23, emphasis in original).

“Metainferences of various kinds can be defined precisely, systematic combinations of them recursively constructed, and the effects of those combinations computed. The result is a principled botanization of constellations of metainference that offers revealing characterizations of a number of logics that have been the subject of intense interest among logicians and philosophers of logic over the past few decades…. In treating metainferential relations among conceptual roles as objects that can be combined and manipulated, this calculus stands to conceptual roles as the sequent calculus stands to the sentences that are the relata of the implication relations it codifies as sequents. This intrinsic rational metavocabulary, built on top of the implication-space inferentialist model-theoretic semantics for conceptual roles, provides the expressive power to make explicit a hitherto unexplored level of metainferential reason relations among those roles, and thereby offers an illuminating new semantic perspective on the relations among a variety of well-studied logics.”

“The implication-space metavocabulary provides a model-theoretic semantics for the conceptual roles sentences play in virtue of standing to one another in reason relations of implication and incompatibility. It is a reason-based inferentialist semantics, rather than a truth-based representational semantics like truth-maker semantics. By contrast to the proof-theoretic treatment of reason relations by the sequent calculus, the implication-space metavocabulary assigns sets of implications as the semantic interpretants of sentences, and set-theoretic constructions out of those sets as the semantic interpretants of sentences, and then operates on and manipulates those semantic interpretants to codify reason relations and conceptual roles. In fact, it does so in a way that can be shown to be isomorphic to truth-maker model-theoretic semantics…. In both cases, the universe is taken to be structured by a commutative monoid (fusion of states and a corresponding operation combining candidate implications according to their ranges of subjunctive robustness). Nonetheless, the implication-space metavocabulary provides an intrinsic semantics, since it appeals to nothing that is not made available by the base vocabulary to which it is applied: sets of sentences and their reason relations. Implication-space semantics is something like the intrinsification of truth-maker semantics — in a way formally analogous to, but expressively more powerful than, Fine’s use of intrinsic ‘canonical models'” (pp. 23-24).

The abstract algebraic notion of a monoid is also ubiquitous in contemporary functional programming. Per Wikipedia, a monoid is a set equipped with an associative binary operation and an identity element. One easy example is the set of positive integers with addition as the associative operation and 0 as the identity element, but there are a great many others as well.

“When this structural isomorphism of implication-space and truth-maker semantics — which holds between the universes from which semantic interpretants are drawn, the interpretants themselves, and the way reason relations of consequence and incompatibility are determined for sentences in terms of their semantic interpretants — is appreciated in detail, and considered in context with the orthogonal isomorphism at the level of reason relations between the truth-maker alethic modal semantic metavocabulary and the deontic normative bilateral pragmatic metavocabulary, it becomes clear that the implication-space semantics makes explicit the abstract rational forms common to those two extrinsic-explanatory metavocabularies of meaning and use. Those rational forms are just the conceptual roles the implication-space semantics characterizes” (p. 24).

Epilogues to this series: Anaphora and Reason Relations; All the Way Down

An Isomorphism

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Next in this series: Quick Note on Proof Theory

Logic for Expression

In recent times, Robert Brandom has pioneered the idea that the role of logic is primarily expressive. In his 2018 essay “From Logical Expressivism to Expressivist Logic”, he says this means its purpose is “to make explicit the inferential relations that articulate the semantic contents of the concepts expressed by the use of ordinary, nonlogical vocabulary” (p. 70).

In my humble opinion, this is what logic was really supposed to be about in Aristotle, but the tradition did not follow Aristotle. Aristotle insisted that logic is a “tool” not a science, but most later authors have assumed the contrary — that logic was the “science” of correct reasoning, or perhaps the science of consequence relations. Several scholars have nonetheless rediscovered the idea that the purpose of logical demonstration in Aristotle is not to prove truths, but to express reasoned arguments as clearly as possible.

Brandom says that “the task of logic is to provide mathematical tools for articulating the structure of reasoning” (p. 71). People were reasoning in ordinary life long before logic was invented, and continue to do so. But the immensely fertile further development of logic in the late 19th and early 20th centuries was mostly geared toward the formalization of mathematics. Reasoning in most specialized disciplines — such as the empirical sciences, medicine, and law — actually resembles reasoning in ordinary life more than it does specifically mathematical reasoning.

According to Brandom, “The normative center of reasoning is the practice of assessing reasons for and against conclusions. Reasons for conclusions are normatively governed by relations of consequence or implication. Reasons against conclusions are normatively governed by relations of incompatibility. These relations of implication and incompatibility, which constrain normative assessment of giving reasons for and against claims, amount to the first significant level of structure of the practice of giving reasons for and against claims.”

“These are, in the first instance, what Sellars called ‘material’ relations of implication and incompatibility. That is, they do not depend on the presence of logical vocabulary or concepts, but only on the contents of non- or prelogical concepts. According to semantic inferentialism, these are the relations that articulate the conceptual contents expressed by the prelogical vocabulary that plays an essential role in formulating the premises and conclusions of inferences” (pp. 71-72).

“Material” relations of consequence and incompatibility have a different structure from formal ones. Formal consequence is monotonic, which means that adding new premises does not change the consequences of existing premises. Formal contradiction is “explosive”, in the sense that any contradiction whatsoever makes it possible to “prove” anything whatsoever (both true statements and their negations), thereby invalidating the very applicability of proof. But as Brandom reminds us, “outside of mathematics, almost all our actual reasoning is defeasible” (p. 72). Material consequence is nonmonotonic, which means that adding new premises could change the consequences of existing ones. Material incompatibilities can often be “fixed” by adding new, specialized premises. (As I somewhere heard Aquinas was supposed to have said, “When faced with a contradiction, introduce a distinction”.)

Brandom notes that “Ceteris paribus [“other things being equal”] clauses do not magically turn nonmonotonic implications into monotonic ones. (The proper term for a Latin phrase whose recitation can do that is ‘magic spell’.) The expressive function characteristic of ceteris paribus clauses is rather explicitly to mark and acknowledge the defeasibility, hence nonmonotonicity, of an implication codified in a conditional, not to cure it by fiat” (p. 73).

“There is no good reason to restrict the expressive ambitions with which we introduce logical vocabulary to making explicit the rare material relations of implication and incompatibility that are monotonic. Comfort with such impoverished ambition is a historical artifact of the contingent origins of modern logic in logicist and formalist programs aimed at codifying specifically mathematical reasoning. It is to be explained by appeal to historical causes, not good philosophical reasons” (ibid). On the other hand, making things explicit should be conservative in the sense of not changing existing implications.

“…[W]e should not emulate the drunk who looks for his lost keys under the lamp-post rather than where he actually dropped them, just because the light is better there. We should look to shine light where we need it most” (ibid).

For relations of material consequence, the classical principle of “explosion” should be replaced with the weaker one that “if [something] is not only materially incoherent (in the sense of explicitly containing incompatible premises) but persistently so, that is incurably, indefeasibly
incoherent, in that all of its supersets are also incoherent, then it implies everything” (p. 77).

“The logic of nonmonotonic consequence relations is itself monotonic. Yet it can express, in the logically extended object language, the nonmonotonic relations of implication and incompatibility that structure both the material, prelogical base language, and the logically compound sentences formed from them” (p. 82).

Material consequence relations themselves may or may not be monotonic. Instead of requiring monotonicity globally, it can be declared locally by means of a modal operator. “Logical expressivists want to introduce logical vocabulary that explicitly marks the difference between those implications and incompatibilities that are persistent under the addition of arbitrary auxiliary hypotheses or collateral commitments, and those that are not. Such vocabulary lets us draw explicit boundaries around the islands of monotonicity to be found surrounded by the sea of nonmonotonic material consequences and incompatibilities” (p. 83).

Ranges of subjunctive robustness can also be explicitly declared. “The underlying thought is that the most important information about a material implication is not whether or not it is monotonic — though that is something we indeed might want to know. It is rather under what circumstances it is robust and under what collateral circumstances it would be defeated” (p. 85).

“The space of material implications that articulates the contents of the nonlogical concepts those implications essentially depend upon has an intricate localized structure of subjunctive robustness and defeasibility. That is the structure we want our logical expressive tools to help us characterize. It is obscured by commitment to global structural monotonicity—however appropriate such a commitment might be for purely logical relations of implication and incompatibility” (pp. 85-86).

“Logic does not supply a canon of right reasoning, nor a standard of rationality. Rather, logic takes its place in the context of an already up-and-running rational enterprise of making claims and giving reasons for and against claims. Logic provides a distinctive organ of self-consciousness for such a rational practice. It provides expressive tools for talking and thinking, making claims, about the relations of implication and incompatibility that structure the giving of reasons for and against claims” (p. 87).