“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