Inferentialism and
Logical Expressivism

Core, Penumbra, and Convexity: Toward a Two-Level Semantics of Open-Textured Legal Concepts

Standard accounts of open-textured legal concepts often rely on Hart’s classic spatial metaphor of core and penumbra. While intuitively appealing, this metaphor fails to explain why concepts display dynamic boundaries or how argumentative practice shapes legal meaning over time. This article proposes a novel two-level semantics specifically designed for modern legal contexts, integrating John Horty’s reason-based model of precedent with Peter Gärdenfors’s theory of conceptual spaces. The proposed framework models a dynamic feedback loop between a conceptualization level (represented by Euclidean domain spaces, prototypes, and convex hulls) and a reasoning level (governed by reason-based rule priorities). The conceptualization level extracts discrete facts and factors from continuous features of a case, transmitting them to the reasoning level. In turn, legal decisions at the reasoning level constrain and modify the geometry of conceptual spaces. By incorporating the geometric notions of distance from prototypes, core regions, and convex hulls, this two-level approach provides a unified explanation of six characteristic phenomena of legal concepts, including the increasing justificatory burden as cases diverge from paradigmatic instances and the subtle behaviors observed at the frontiers of legal application. Rather than offering a general semantics for natural language, this model highlights the distinctively pragmatic, reason-driven nature of legal adjudication. Ultimately, it demonstrates that legal concepts maintain both stability and flexibility through the continuous interface between reason-giving arguments and geometric conceptual structures, offering a rigorous theoretical foundation for understanding the phenomenological meaning of legal concepts.

Reason Relations Rehabilitated, Relevantly

Brandom and Hlobil’s Reasons for Logic, Logic for Reasons aims to understand the meanings of linguistic expressions in terms of the reason relations in which they figure. Yet, in exchanges following the book’s publication, Hlobil (writing on behalf of both authors) appears to retreat from this ambition: the relations their framework characterizes are necessary but not sufficient for something’s being a reason in the sense that underwrites good reasoning. The core difficulty leading Hlobil to this retreat concerns the transition from incoherence to consequence. A set of assertions and denials can be incoherent even when some of its members make no contribution to that incoherence. Consequently, the incoherence of asserting certain sentences while denying another does not, by itself, establish that the former provide a reason to assert the latter. Drawing inspiration from Brandom’s earlier suggestion that the appropriate logic for his account of discursive practice should be a relevant one, I show how a small modification of the framework can address this difficulty. I distinguish merely incoherent sets of speech acts from minimally incoherent ones, introduce a proof-theoretic parameter to track contribution to incoherence, and restrict the transition from incoherence to consequence accordingly. The resulting logic is the minimal modification of Hlobil and Brandom's supra-classical logic that restores relevance, and the purely logical fragment is the monotonic closure of the non-transitive relevant logic NTR (Verdée et al., 2019). I argue that this relevant logic is better suited to be called a "Logic for Reasons."

Does Brandom’s Hegel Dispel Habermas’s Concern? Kantian vs. Hegelian Pragmatism

Habermas regards R. Brandom’s Making It Explicit (1994; hereafter MIE) as advancing the philosophy of language from Kant to Hegel. However, Habermas’s engagement is fundamentally critical. In Truth and Justification (Wahrheit und Rechtfertigung, 1999) Habermas regards Brandom’s Hegelian move as excessive and instead advocates a return toward Kant. In short, Habermas endorses a Kantian pragmatism (Habermas 2004, 16). In A Spirit of Trust: A Reading of Hegel’s Phenomenology (2019), Brandom characterizes Hegel’s Absolute Idealism as comprising three elements: conceptual realism, objective idealism, and conceptual idealism. Notably, Habermas had already identified conceptual realism and objective idealism in MIE, criticizing what he took to be its excessive Hegelianism. This suggests a plausible interpretation: in A Spirit of Trust, Brandom implicitly responds to Habermas’s critique by embracing these very features and defending them as central to both Hegel’s philosophy and his own. In this way, Habermas’s critique of MIE helps to illuminate the systematic intent of A Spirit of Trust. In this talk I will argue that Habermas’s Kantian pragmatism and weak realism—both of which he claims to be superior to Brandom’s Hegelian pragmatism (including conceptual realism)—are in fact undermined in A Spirit of Trust, but Brandom’s supposed response remains insufficient in its application to moral theory.

Keywords; R.B.Brandom; pragmatism, J.Habermas, I.Kant, G.W.F.Hegel, realism, idealism, morality

Normativity of Meaning for Truth-Inapt Sentences

Our previous paper (Shimamura and Tiisala 2023) introduced and deployed a new strategy to defend the normativity of meaning, specifically the thesis of ME normativism that there cannot be meanings without norms. This new paper is a sequel, in which we redeem the promissory note that our approach, unlike its alternatives, can be extended to sentences that are not truth-apt. There is a wide range of different kinds of meaningful truth-inapt sentences, including expressives (‘Oops!’), requests (‘Close the window, please’) and greetings (‘Good afternoon’). They have not been considered at all in the debate, but we argue that they are just as essential to the normativity of meaning as declarative sentences. In our original argument, too, the normativity of meaning was formulated as a prohibition against the incursion of some particular pair of incompatible commitments, which a subject understands in virtue of understanding the meaning of any declarative sentence. In this paper, we extend that argument to truth-inapt sentences by locating the relevant incompatibility relation by reference to the use-conditions of such sentences. Thus, a competent speaker who sincerely says ‘Oops!’, thereby understands some sentence to be incompatible with it, which constitutes a prohibition against combining the two sentences.

Bilateralism and Normative Pragmatics

In Reasons for Logic, Logic for Reasons (2025), Ulf Hlobil and Robert Brandom develop a bilateralist normative pragmatics in which logical consequence is understood in terms of normative incompatibility between assertion and denial. Their framework departs from more standard forms of bilateralism by allowing consequence relations that are nontransitive and even non-reflexive. In this talk, I critically examine this departure by comparing their framework with the bilateralist account developed by Restall. I argue that a central advantage of bilateralism lies in its ability to distinguish pragmatic consequence from semantic consequence while also explaining how the two are connected. This explanatory role depends crucially on structural principles such as Cut and Identity. By abandoning these principles, Hlobil and Brandom lose an important motivation for adopting bilateralism in the first place. I conclude by contrasting two conceptions of the normativity of logic: the prooftheoretic tradition, going back to Dummett and developed in Restall’s bilateralism, which treats the construction of a semantics itself as a normative enterprise, and Brandom’s approach, on which the role of logic is to make explicit the inferential structure already implicit in material reasoning.

An implementation of the implication space semantics with predicates

In this talk, I will demonstrate how to use an open-source tool I've developed which accepts implication frames as inputs and evaluates logical queries on their respective implication spaces (as described in Reasons for Logic, Logic for Reasons). I will address some topics of theoretical interest (allowing consequence relations with multisets rather than sets, colorless and colorful objects, quantifiers, linear logic exponentials) via concrete examples which run on a computer.

What is Recollective Responsibility? A Case Study from Business Ethics

Robert B. Brandom’s inferentialism is thoroughly grounded in the concept of responsibility. Normative pragmatics is based on task responsibility, and furthermore, recollective responsibility is proposed as a solution to problems such as the intelligibility of conceptual content, the attitude-dependence of normative statuses, and the rule-following paradox. I focus on recollective responsibility: what is the function of this responsibility? While it is also referred to as forgiveness, does it have ethical implications? Drawing on cases from business ethics, this presentation shows that this responsibility well explains what victims of corporate accidents refer to as "victims' responsibility," yet argues that it merely indicates an epistemic constraint on the evaluator of an action, distinct from ethical forgiveness.

Saying Something Without Speaking: An Inferentialist Account of Charged Silence

Silence is not one thing. We may fall silent due to a lack of concepts, because nothing was asked of us, or merely as a strategic choice. But some silences are moves in the game of giving and asking for reasons, made not by choice but by necessity. The lover who answers an accusation with silence, or the mother who meets her child's apology without words, says something by not saying it, because neither answer is open to her. I offer an inferentialist account of such charged silence. It arises when the claim at issue is one the context expects the agent to address, and her own commitments render both asserting and denying it incoherent. Silence is then mandated by her position. It also has consequences mere non-assertion lacks: the addressee may draw conclusions from it that the silent person cannot assert without breaking their silence. The same commitments can thus be coherent for the addressee and incoherent for the silent person, suggesting that coherence is relative to one's place in a discursive exchange. What fixes what the silence says? I suggest an inferentialist answer: it is the commitments that make it mandatory in the first place.

Two Kinds of Defeasibility: The Expressive Advantage of Multiple-Conclusion Inferentialism

Inferentialists have sought to explain the defeasibility of reasons and material inferences in terms of nonmonotonicity. Yet defeasibility takes at least two forms: rebutting and undercutting. Because both are nonmonotonic, the existing inferentialist toolkit cannot distinguish between them and is therefore inadequate to capture the nature of reasons and of reasoning practices, which essentially involve comparing and evaluating reasons. I characterize these two forms of defeasibility using the multiple-conclusion framework of bilateral inferentialism recently developed in the seminal work of Hlobil and Brandom. This analysis aims to show that, although the independently developed single-conclusion framework has its own merits, the multiple-conclusion framework can make distinct aspects of our normative-pragmatic practices explicit.

Modality, invalidity, and anti-validity (joint work with Andreas Fjellstad)

In this talk, we aim to provide an inferentialist non-circular account of central modal notions. In particular, we explore the relationship between the modal logics S4, S5, and C on the one hand, and complementary calculi for invalidities and anti-validities, on the other hand. We work in the setting of Golan’s S4 calculus (Golan, 2025), which integrates into a regular sequent calculus the possibility to assume and discharge sequents, in a natural deduction-like manner. Golan’s rules for the S4 necessity operator are:

If we add to those rules Golan’s anti-sequent rules (Golan, 2024), with the addition of the classical reduction rule (the one in the center):

As well as anti-sequent axioms of the form Γ ⊬ Δ whenever Γ ∩ Δ = ∅, we obtain a calculus for the validities and anti-validities of the modal logic C. To get an S5 calculus, we drop all the anti-sequent axioms, in which case the negated turnstile represents anti-validity statements, rather than invalidity statements.

In conclusion, the difference between the above calculi make explicit the relationship between S4 and S5 in terms of the relation between intuitionistic and classical meta-theoretic negation. Furthermore, the account provides a proof-theoretic distinction between C and S5 in terms of whether anti-sequents represent invalidity statements or anti-validity statements.

• Golan, R. 2024. A Hybrid Calculus for the Validities and Invalidities of Classical Propositional Logic. Journal of Philosophical Logic 53(6), 1701–1716.
• Golan, R. 2025. Simple sequent systems for the modal logics K, D, T, and S4. Logic Journal of the IGPL 33(3), 1–24.

Topological Operators in Phase Semantics: An Inferentialist Interpretation

In this paper I develop implication space semantics, presented by R. Brandom and U. Hlobil, to provide a sound and complete semantics to full linear logic so that I can offer an interpretation of the topological operators used in phase semantics: the closure operator defined in terms of the set of antiphases and the interior operator used to interpret the exponentials. The idea is that the former gives us what may be called an inferential closure of a given set of inferences, whilst the latter corresponds to the restriction of a given set of inferences to a set of inferences with some desired structural properties. This interpretation in turn provides us with a natural way to understand all the logical connectives in linear logic.

On the Priorities of Reason and Duty in Determining What to Do

A debate intersecting with core topics in the philosophies of mind, language, and agency concerns the relative priority of the dictates of practical rationality in determining what we ought to do. To a first pass, this debate turns on whether the reasons we have to do something make it such that we fall under the corresponding duties, expressed with claims about what we ought to do, or whether instead it is our having the duties we do that gives us the reasons we have to infer to the corresponding claims about what to do. On the basis of an inferentialist treatment of deontic modal operators, I argue for the following thesis: the reasons we have to infer to deontic claims are meaning-constituting for those claims, while it is by our obeying their dictates qua duties that we manifest practical rationality in doing what we ought. The result is an intuitive view about the way we reason to and from our instrumental and moral duties, and which is plausible given the rest of what we understand of human cognition, agency, and natural language semantics.

Between Thinking and Imitating: An Inferentialist Demarcation

According to inferentialism, meaning is a matter of inferential role --- how a given linguistic expression is related to other expressions in inferences. Several authors point out that this view works in favor of the view that large language models, trained on linguistic data, are capable of understanding the meanings of linguistic expressions --- and, more generally, of saying and thinking meaningful things. Among those, Simonelli’s inferentialist argument is one of the most sophisticated and developed. In this paper, I argue against the claim above by criticizing Simonelli’s argument. More specifically, I argue that there are reasons even for inferentialists to distinguish thinking from merely imitating to think, which is, I claim, what LLMs amount to doing.

Inference and Object

I present an account of names and predicates in terms of assertoric sentences and their inferential roles. The account can be translated into an account of objects and properties in terms of worldly propositions (which are things that are made true or false by states) and the alethic modal relations among them. The account exploits symmetries of modal structures. In a slogan: properties are what is common among symmetric points in modal space; objects are what differs among symmetric points in modal space.