A Category-theoretic Reconstruction of Logical Expressivism

Kris Brown

c Press s for speaker notes

6/30/26

Outline

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What is “logical expressivism”?

• Motivation: representationalism vs inferentialism
• As presented in (Hlobil and Brandom 2025)
• Examples: with and without contraction

Free Girard quantales

• from incompatibility frames
• from implication frames
• from reflexive implication frames
• from ‘containment’ implication frames

Relation to traditional / categorical logic

In this talk we’ll view some philosophy of language through the lens of category theory. In particular, I’ll focus on Chapter 5 of this book. I’ll first attempt to briefly summarize the problem this book is addressing. Their solution is at all ad hoc from a philosophical point of view, but it can be bolstered by a mathematical perspective of how its many definitions arise naturally. We can view their core data structure, the “implication frame”, as an object of category. Then, their framework for pragmatist logic shakes out as the free addition of the structure of thin *-autonomous categories.

Motivation: representationalism vs inferentialism

Core philosophical question: relation between mind language and world. That some word has such-and-such semantic properties demands some kind of explanation:¹

• Representationalism: meaning grounded by reference to the world (atomistic)
  • Historical tradition: Augustinian picture,² British empiricism, behaviorist psychology
  • What is represented? How is the representational relation determined? Many answers.
  • Pro: compositional, thus explaining the productivity and systematicity of language
  • Con: Violates Frege’s context principle,³ risk of skepticism
• Inferentialism: meaning is determined by inferential role within a language (holistic)
  • Pro: Solves Frege’s puzzle, naturally handles non-referring vocabulary, no Cartesian gap.
  • Con: Holism means noncompositional, risks of solipsism/idealism/relativism.

Let me provide context of the philosophical debate which this work fits into.

• compositional: the meaning of its syntactically complex expressions is a function of their syntactic structures together with the meanings of their syntactic constituents.
• productivity: Chomskyian point about radical novelty of almost every spoken sentence
• systematicity: any natural language that can express the proposition P will also be able to express many propositions that are semantically close to P.
• Frege’s puzzle: “the morning star is the evening star” is of the form “A=A” if the meanings are identified with their reference.

1. “Why Meaning (Probably) Isn’t Conceptual Role” (Fodor and Lepore 1993).

2. According to Wittgenstein: “The individual words in a language name objects—sentences are combinations of such names. In this picture of language, we find the roots of the following idea: Every word has a meaning. This meaning is correlated with the word. It is the object for which the word stands.”

3. Only in the context of a sentence does a word have meaning. The proposition is the smallest unit of pragmatic force. Relatedly, see Wittgenstein: ‘to understand a sentence is to understand a language’.

Formal model of a norm / social practice

Core data: there is a set of things one can claim, and some notion of what follows from what.

Implication frame

An implication frame (L,\bot) is a set of propositional atoms and a subset {\bot\subseteq \mathbb{N}^{L+L}}.

An idempotent implication frame has a distinguished subset \bot\subseteq \mathbb{B}^{L+L}.

Each element of \mathbb{N}^{L+L} is a candidate implication. \bot is the set of good implications, according to the frame.

a \vdash a,b\quad a,b \vdash c\quad a \nvdash c\quad a\vdash a

Bilateralism: a way to interpret the turnstile (Restall 2005)

\underline{\color{blue}{\text{Assertions}}}\qquad \underline{\color{black}{\text{Incompatibility}}}\qquad \underline{\color{red}{\text{Denials}}} \color{blue}{A_1,}\color{blue}{A_2,A_3} \qquad \color{black}\vdash\qquad \color{red}{D_1,}\color{red}{D_2}

“\color{red}{\Delta} follows from \color{blue}\Gamma” = “It is out-of-bounds / incompatible to assert everything in \color{blue}\Gamma while denying everything in \color{red}\Delta”

Logical consequence relations are a source of \vdash relations. Logicians often assume:

• monotonicity (weakening, portability of reasoning)

• transitivity¹ (cut, composability of reasoning).

When we think about language holistically, there is a shfit in focus from truth to inference: rather than what makes an individual sentence true, we just care about what sentences justify other ones.

One way we can look at the turnstile which makes it perfectly natural to allow it to be an arbitrary subset is bilateralism. If you read it as truth preservation, you’d be hard pressed to explain how it could be not transitive. Bilateralism takes two kinds of speech acts as primitive and derives a notion of implication. It’s a way to read the turnstile as a norm governing assertions and denials. This is in contrast with the turnstile as truth preservation (which could be seen as a special case).

One source of these norms (just to further stress that we don’t have to think of the turnstile as social) comes from logical consequence relations.

There is no particular reason why a \vdash in the wild should obey these constraints. However, it is desirable to have such relations because of the portability and composability of reasoning.

1. “A sequent calculus without cut-elimination is like a car without engine” (Girard 1995)

Logical Elaboration + NMMS calculus

Process of taking some ‘base’ implication frame and deriving one whose propositional atoms are logically complex: (X,\bot) \rightarrow ({\rm BooleanFormulas}(X), \bot').

The logical elaboration of a frame freely extends its atoms with connectives. The below bidirectional rules determine whether the new sequent is in the elaborated \bot' based on \bot.

\boxed{\begin{align*} \Gamma &\vdash A, \Delta \\ \hline \Gamma, \neg A &\vdash \Delta \\ \end{align*}}\\

\boxed{\begin{align*} \Gamma, A &\vdash \Delta \\ \hline \Gamma &\vdash \neg A, \Delta \\ \end{align*}}\\

\boxed{\begin{align*} \Gamma, A,B &\vdash \Delta \\ \hline \Gamma, A\wedge B &\vdash \Delta \\ \end{align*}}\\

\boxed{\begin{align*} \Gamma&\vdash A,B,\Delta \\ \hline \Gamma &\vdash A\vee B, \Delta \\ \end{align*}}\\

\boxed{\begin{align*} \Gamma,A\vdash \Delta\quad \Gamma,B\vdash &\Delta\quad \Gamma,A,B\vdash \Delta \\ \hline \Gamma, A\vee B, &\vdash \Delta \\ \end{align*}}\\

\boxed{\begin{align*} \Gamma\vdash A,\Delta\quad &\Gamma\vdash B,\Delta\quad \Gamma\vdash A,B,\Delta \\ \hline \Gamma &\vdash A\wedge B, \Delta \\ \end{align*}}\\

Logical expressivism

The characteristic function of logic is to express features of some antecedent, prelogical system of implications. Here express means to make explicit, i.e. internalize, features of \bot that, were originally only describable in a metavocabulary.

NMMS - Nonmonotonic, multisuccedent logic

Any formal model is ultimately responsible to the thing it is trying to model as its standard of correctness. When we try to represent a domain with classical, or intuitionistic, or paraconsistent, or linear logic, whether or not we made the right choice ultimately depends on the structural features of the domain we’re trying to model. Viewing logic as constitutive of what we mean (rather than descriptive of it) is confusing the map for the territory.

Implication space semantics

The following definitions are all parameterized by a background frame \mathcal{X}:=(X,\bot).

Range of subjunctive robustness operation

The range of subjunctive robustness function, \operatorname{RSR}\colon {\mathcal{P}[\mathbb{N}[X+X]]\to\mathcal{P}[\mathbb{N}[X+X]]},¹ sends a set of candidate implications, e.g. \{(\Gamma_1,\Delta_1),...,(\Gamma_i,\Delta_i)\}, to the set {\{(\Theta,\Omega)\ |\ \forall i\colon (\Gamma_i\cup \Theta,\Delta_i\cup \Omega) \in \bot\}}. We will abbreviate \operatorname{RSR}(A) as A^\bot.

Implicational roles and conceptual contents

The set of implicational roles is \small \R:={\rm im}(\operatorname{RSR}).

The set of conceptual contents is \mathbb{C}:=\R^2.

Symjunction and adjunction of roles A,B \in \R

Symjunction: A \sqcap B:=(A\cup B)^{\bot\bot}

Adjunction: \small A\sqcup B:=\{{(\Gamma\cup \Gamma',\ \Delta\cup\Delta')}\ |\ (\Gamma,\Delta) \in A,\ (\Gamma',\Delta') \in B \}^{\bot\bot}

Semantic consequence + base interpretation

\vec{\texttt{A}}\vDash\vec{\texttt{B}} := (\bigsqcup_{i} \texttt{a}_{i+})\sqcup (\bigsqcup_{j}\texttt{b}_{j-}) \subseteq \bot

For all atoms a \in X:

[\![ a ]\!]:=\langle \{a^+\}^{\bot\bot},\ \{a^-\}^{\bot\bot}\rangle

Semantic clauses for classical and linear logic connectives

\begin{align*} [\![ \neg A ]\!]&:=\langle \texttt{a}_-,\texttt{a}_+\rangle & [\![ A \wedge B ]\!]&:=\langle \texttt{a}_+\sqcup \texttt{b}_+,\ \texttt{a}_- \sqcap \texttt{b}_- \sqcap (\texttt{a}_-\sqcup \texttt{b}_-)\rangle \\ [\![ A\vee B ]\!]&:=[\![ \neg(\neg A\wedge \neg B) ]\!] & [\![ A\to B ]\!]&:=[\![ \neg A\vee B ]\!] \\ [\![ A \otimes B ]\!] &:= \langle \texttt{a}_+\sqcup \texttt{b}_+,\ (\texttt{a}_-^\bot \sqcup \texttt{b}_-^\bot)^\bot\rangle & [\![ A \oplus B ]\!] &:= {\langle \texttt{a}_+ \sqcap \texttt{b}_+,\ (\texttt{a}_-^\bot\sqcap \texttt{b}_-^\bot)^\bot\rangle} \\ [\![ A\text{⅋} B ]\!]&:=[\![ \neg(\neg A \otimes \neg B) ]\!]& [\![ A\& B ]\!]&:=[\![ \neg(\neg A \oplus \neg B) ]\!] \end{align*}

The (-)^{\bot\bot} operation is a closure operator on the lattice of subsets.

The pairs that make up a content are called the premisory role and the conclusory rule.

Once we have fixed a frame, there is a canonical semantic space to interpret (logically complex combinations of) the propositional atoms of the frame and a canonical interpretation function of these complexes.

For the basic sentences in X, we are representing some element a via a pair of sets, the set of al

Despite compelling philosophical justifications for all of these definitions, I was interested in what sort of mathematical perspective these decisions would seem natural from. Before I get into that, let me introduce a small example.

1. Or {\mathcal{P}[\mathcal{P}[X+X]]\to\mathcal{P}[\mathcal{P}[X+X]]} in the idempotent setting.

Idempotent Example (1/3)

Let \mathcal{X}:=(X,\bot_\mathfrak{B}) where X=\{a,b\} and \bot_\mathfrak{B} is given by the following table.

E.g. a\vdash a,b and a\nvdash b for this frame.

\begin{array}{||c||c|c|c|c||} \hline\hline \bot_{\mathfrak{B}} & 0 & a^- & b^- & a^-b^- \\ \hline\hline 0 & \checkmark & \checkmark & \times & \checkmark \\ \hline a^+ & \times & \checkmark & \times & \checkmark \\ \hline b^+ & \times & \times & \checkmark & \checkmark \\ \hline a^+b^+ & \checkmark & \checkmark & \checkmark & \checkmark \\ \hline\hline \end{array}

Here is an individual RSR computation \begin{array}{||c||c|c|c|c||} \hline\hline \{a^+\}^\bot & 0 & a^- & b^- & a^-b^- \\ \hline\hline 0 & \times & \checkmark & \times & \checkmark \\ \hline a^+ & \times & \checkmark & \times & \checkmark \\ \hline b^+ & \checkmark & \checkmark & \checkmark & \checkmark \\ \hline a^+b^+ & \checkmark & \checkmark & \checkmark & \checkmark \\ \hline\hline \end{array}

Here are all of the singleton RSRs:

\begin{array}{||c||c|c|c|c||} \hline\hline (-)^\bot & 0 & a^- & b^- & a^-b^- \\ \hline\hline 0 & \bot_\mathfrak{B}& X_b & X_\pm & \top \\ \hline a^+ & X_\pm & \top & X_\pm & \top \\ \hline b^+ & X_\mp & X_\mp & \top & \top \\ \hline a^+b^+ & \top & \top & \top & \top \\ \hline\hline \end{array}

X_\pm:=\top\setminus\mathcal{P}[\{a^+,b^-\}] \qquad X_b:=\top \setminus \{b^+,b^+a^-\} X_\mp :=\top\setminus\mathcal{P}[\{a^-,b^+\}] \qquad \top:=\mathcal{P}[X+X]

Let’s do a small example with a language where there are two possible things that can be asserted, a and b. Furthermore, we’ll assume this frame is idempotent, so our candidate implications (which are either simply good or simply bad) have sets rather than multisets on either side of the turnstile. The advantage of idempotent frames is that the set of possible candidate implications is finite: there are just 16 for two propositions. I’ve drawn this to the right, where a + means on the left hand side (the assertions) and a - means the right hand side (the denials). We’ll derive everything in this example mechanically, starting from just this matrix of raw data.

Idempotent Example (2/3)

Now we can derive more inferential roles in \R by taking intersections of the singleton roles from the previous table, but this just yields one new role X_\bot=\{a^+,b^+\}^\bot = X_\pm\cap X_\mp.

\begin{array}{||c||c|c|c|c||} \hline\hline (-)^{\bot\bot} & 0 & a^- & b^- & a^-b^- \\ \hline\hline 0 & X_b & \bot_\mathfrak{B}& X_\mp & X_\bot \\ \hline a^+ & X_\mp & \bot_\mathfrak{B}& X_\mp & X_\bot \\ \hline b^+ & X_\pm & X_\pm & X_\bot & X_\bot \\ \hline a^+b^+ & X_\bot & X_\bot & X_\bot & X_\bot \\ \hline\hline \end{array}

\begin{array}{||c||c|c|c|c|c|c||} \hline\hline \vee & X_b & X_\bot & \bot_\mathfrak{B}& X_\pm & X_\mp & \top \\ \hline\hline X_b & X_b & X_b & X_b & \top & X_b & \top \\ \hline % 1 X_\bot & X_b & X_\bot & \bot_\mathfrak{B}& X_\pm & X_\mp & \top \\ \hline % 2 \bot_\mathfrak{B}& X_b & \bot_\mathfrak{B}& \bot_\mathfrak{B}& X_\pm & X_b & \top \\ \hline % 3 X_\pm & \top & X_\pm & X_\pm & X_\pm & \top & \top \\ \hline % 4 X_\mp & X_b & X_\mp & X_b & \top & X_\mp & \top \\ \hline % 5 \top & \top & \top & \top & \top & \top & \top \\ \hline\hline % 6 \end{array}

\begin{array}{||c||c|c|c|c|c|c||} \hline\hline \otimes & X_b & X_\bot & \bot_\mathfrak{B}& X_\pm & X_\mp & \top \\ \hline\hline X_b & X_b & X_\bot & \bot_\mathfrak{B}& X_\pm & X_\mp & \top \\ \hline % 1 X_\bot & X_\bot & X_\bot & X_\bot & X_\bot & X_\bot & X_\bot \\ \hline % 2 \bot_\mathfrak{B}& \bot_\mathfrak{B}& X_\bot & \bot_\mathfrak{B}& X_\pm & X_\bot & X_\pm \\ \hline % 3 X_\pm & X_\pm & X_\bot & X_\pm & X_\pm & X_\bot & X_\pm \\ \hline % 4 X_\mp & X_\mp & X_\bot & X_\bot & X_\bot & X_\mp & X_\mp \\ \hline % 5 \top & \top & X_\bot & X_\pm & X_\pm & X_\mp & \top \\ \hline\hline % 6 \end{array}

The Hasse diagram for the six values of \R are shown.

We can compute the two ways to combine inferential roles.

Idempotent Example (3/3)

We have base cases:

[\![ a ]\!]=\langle \{a^+\}^{\bot\bot},\{a^-\}^{\bot\bot} \rangle=\langle X_\mp,\bot_\mathfrak{B}\rangle \qquad [\![ b ]\!]=\langle \{b^+\}^{\bot\bot},\{b^-\}^{\bot\bot} \rangle=\langle X_\pm,X_\mp \rangle

We can use the formula for semantic consequence to show that \Gamma \vdash \Delta \iff [\![ \Gamma ]\!]\vDash [\![ \Delta ]\!].

We can compute syntactically that {a,b\vdash a\wedge b} from {a,b\vdash a} and {a,b\vdash b} and {a,b\vdash a,b} in \mathcal{X}.

We also observe on the right that {[\![ a ]\!],[\![ b ]\!]\vdash [\![ a\wedge b ]\!]}.

\begin{align*} \pi_1([\![ a ]\!]) \otimes \pi_1([\![ b ]\!]) \otimes \pi_2([\![ a \wedge b ]\!]) &\subseteq \bot_\mathfrak{B}\\ \pi_1([\![ a ]\!]) \otimes \pi_1([\![ b ]\!]) \otimes\qquad \qquad\qquad\qquad\quad& \\ \pi_2([\![ a ]\!]) \vee \pi_2([\![ b ]\!]) \vee (\pi_2([\![ a ]\!]) \otimes \pi_2([\![ b ]\!])) &\subseteq \bot_\mathfrak{B}\\ X_\mp \otimes X_\pm \otimes (\bot_\mathfrak{B}\vee X_\mp \vee (\bot_\mathfrak{B}\otimes X_\mp)) &\subseteq \bot_\mathfrak{B}\\ X_\mp \otimes X_\pm \otimes X_b &\subseteq \bot_\mathfrak{B}\\ X_\bot &\subseteq \bot_\mathfrak{B}\\ \end{align*}

\mathcal{X} satisfies “containment” (\Gamma\vdash \Delta \in \bot whenever \Gamma and \Delta overlap), so \vDash is supraclassical.

However \vDash is not monotone: we have \ \vDash [\![ b ]\!] and [\![ a ]\!]\nvDash [\![ b ]\!].

This fact about the property of the frame satisfying containment therefore the induced semantic consequence relation being supraclassical is one example of many in R4L of how traditional logics (although restricted to propositional logics, such as K3, LP, TS, ST) can all be viewed as the implication spaces of implication frames satisfying such-and-such properties.

Non-idempotent example

Let X=\{\bullet\}. Multisets of X can be identified with natural numbers, thus the set of positions is \mathbb{N}^2. Define \bot_\mathfrak{B}=\{01,12\}\cup R, i.e. 0\vdash 1 and 1\vdash 2, and n\vdash n. ¹

\begin{array}{||c||c|c|c||} \hline\hline (-)^{\bot\bot} & 0 & 1 & 2 \\ \hline\hline 0 & \{00\} & \{01\} & \{02\} \\ \hline 1 & \{10\} & \{00, 11\} &\{01, 12\} \\ \hline\hline \end{array}

To check whether linear modus ponens is valid, we test [\![ \bullet ]\!],[\![ \bullet\multimap\bullet ]\!]\vDash[\![ \bullet ]\!], which holds because \{10\}\otimes \varnothing \otimes \{01\} = \varnothing\subseteq \bot_\mathfrak{B}. This also follows from a theorem that reflexivity of \mathcal{X} implies \vDash is supralinear.

However, note \vDash is not transitive, as \vDash [\![ \bullet ]\!] and [\![ \bullet ]\!]\vDash [\![ \bullet ]\!],[\![ \bullet ]\!], yet we also have \nvDash [\![ \bullet ]\!],[\![ \bullet ]\!].

This is example is much more abbreviated, the details are in an appendix of the paper.

We also define an overline operation on \mathbb{N}^2 sending pairs to the smallest pair that has the same difference.

Our implication frame of interest is then \mathcal{X}:=(X,\bot_\mathfrak{B}). That is, \mathcal{X} endorses the implications 0\vdash 1 and 1 \vdash 2 as well as all implications which have the same number of \bullet on each side.

1. For brevity we’ll denote pairs with concatenation, e.g. (m,n) is written mn. Let R:=\{ii\ |\ i \in \mathbb{N}\} and mnR:=\{(m+i,n+i)\ |\ (m,n)\in\mathbb{N}^2\}. Also \overline{nm}=(n',m') where m'=n-m and n'=0 if m\leq n, otherwise n'=m-n and m'=0.

Outline

What is “logical expressivism”?

• Motivation: representationalism vs inferentialism
• As presented in (Hlobil and Brandom 2025)
• Examples: with and without contraction

Free Girard quantales

• from incompatibility frames

• from implication frames
• from reflexive implication frames
• from ‘containment’ implication frames

Relation to traditional / categorical logic

Phase spaces and Girard quantales

Phase spaces

A (commutative) phase space is a commutative monoid equipped with a distinguished subset. A phase space (X,+,0,\bot\subseteq X) has a natural (-)^\bot operation on its elements, a^\bot := \{x\ |\ x + a \in \bot\}, as well as on subsets of its elements {A^\bot := \bigcap_{a \in A} a^\bot = \{x\ | \forall a \in A\colon x+a \in \bot\}}. This also leads to an ordering on X given by a \leq b := a^\bot \supseteq b^\bot.

Challenge: what is an appropriate notion of morphism of phase spaces?

Category of Girard quantales

(Commutative, unital) Girard quantales are thin, *-autonomous categories.

\mathsf{GQ} has these as objects. Morphisms are quantale morphisms which weakly preserve \bot, i.e. f(\bot_\mathcal{X}) \leq \bot_\mathcal{Y}.

Phase spaces can be recovered as a full subcategory of the comma category of F^\vee (the free quantale of a monoidal preorder) and U^\bot (forgetting the dualizing structure of a Girard quantale).

In this talk, all monoids and quantales should be assumed to be commutative and unital.

Categories of phase spaces and unsigned frames

Category of phase spaces

\mathsf{PS} includes only the (P,Q,\phi) such that {\phi\colon F^\vee(P)\twoheadrightarrow U^\bot(Q)} is surjective and {\tilde \phi\colon P\rightarrowtail U^\vee U^\bot(Q)} is an embedding.

A morphism {(\mathcal{P},\mathcal{Q},\phi)\rightarrow (\mathcal{P}',\mathcal{Q}',\phi')} in \mathsf{PS} is a preordered monoid morphism {f\colon \mathcal{P}\rightarrow \mathcal{P}'} and a Girard quantale morphism {g\colon \mathcal{Q}\rightarrow \mathcal{Q}'} such that the square commutes.

• \phi and \phi' are surjective, g is fully determined by f.
• g is monotone and weakly preserves \bot, thus f(\bot)\subseteq \bot'.
• Continuity: or every lower set A \subseteq P, we need f(A)^{\bot'\bot'} = f(A^{\bot\bot})^{\bot'\bot'}.

Category of (unsigned) incompatibility frames

\mathsf{IF} has objects (X,\bot\subseteq \mathbb{N}[X]) and morphisms which are continuous functions which preserve \bot.

\mathsf{IF} is close to what we want, but it can only represent a language where assertions can be made, not assertions and denials. E.g. a,b,c\vdash and a,a\vdash.

We can think of morphisms just as those preordered monoid maps \mathcal{P}\rightarrow \mathcal{P}' that induce a quantale morphism F^\vee(\mathcal{P})^{\bot\bot}\rightarrow F^\vee(\mathcal{P}')^{\bot\bot}.

So thinking of phase spaces as monoids with a distinguished (arbitrary) subset, morphisms are continuous monoid homomorphisms which preserve the subset.

We call this condition continuity because, in topological terms, maps for which the image of the closure is contained in the closure of the image are called continuous maps.

Workhorse lemmas for constructing adjunctions

Let F\colon \mathsf{A \rightarrow C} and G\colon \mathsf{B\rightarrow C} be functors such that {F\dashv U} (with unit \eta and counit \varepsilon).

Lemma: Adjoints from pullbacks

If G has cartesian lifts \overline{\varepsilon_c} for all \varepsilon_c\colon FU(c)\to c,¹ then the pullback projection \pi_\mathsf{B}\colon \mathsf{A \times_C B\rightarrow B} has a right adjoint R\colon \mathsf{B\rightarrow A \times_C B} given by R(b)\mapsto (UG(b), \operatorname{dom}(\overline{\varepsilon_{G(b)}})).

Lemma: Adjoints from comma categories

The comma category F \downarrow G has a reflective subcategory R\colon \mathsf{B\rightarrowtail (F\downarrow G)} whose reflector is the projection \pi_\mathsf{B}\colon (F\downarrow G) \rightarrow \mathsf{B}. Concretely, R sends b\mapsto (UG(b),b,\varepsilon_{G(b)}) and f to (UGf,f).

1. Notation: the cartesian lift of f is denoted \overline{f}.

Free Girard quantales from unsigned frames

Two adjunctions, {F^\otimes\dashv U^\otimes} and {F^\oplus\dashv U^\oplus}, which can be composed.

There need not exist a function \otimes\colon \mathbb{N}[X]\rightarrow X such that {\forall \Gamma \in \mathbb{N}[X]:\otimes(\Gamma)^\bot = \Gamma^\bot}.

I.e. multisets represented by atoms. \eta^\otimes freely adds this structure.

\boxed{\begin{align*} \Gamma, a, b &\vdash \\ \hline \Gamma, a \otimes b &\vdash \end{align*}}\\

There need not exist a function \oplus\colon \mathcal{P}[\mathbb{N}[X]]\rightarrow X such that {\forall \{\Gamma_1,...,\Gamma_n\} \subseteq \mathbb{N}[X]: \oplus(\{\Gamma_1,...,\Gamma_n\})^\bot = \{\Gamma_1,...,\Gamma_n\}^\bot}.

I.e. sets of multisets are represented by atoms. \eta^\oplus adds this structure.

\boxed{\begin{align*} \Gamma, a \vdash \qquad & \Gamma, b \vdash \\ \hline \Gamma, a \oplus b &\vdash \end{align*}}\\

Conservativity in logic

Introducing new logical connectives does not change the goodness of inferences in sequents which do not feature the new vocabulary. If we think of \eta^{\otimes\oplus}\colon (X,\bot)\to (X',\bot') as the addition of new logically-complex, formal combinations of elements of X to yield X', then conservativity means \Gamma \in \bot \iff \eta(\Gamma) \in \bot'.

Proposition: \eta^{\otimes\oplus} is conservative

Outline

What is “logical expressivism”?

• Motivation: representationalism vs inferentialism
• As presented in (Hlobil and Brandom 2025)
• Examples: with and without contraction

Free Girard quantales

• from incompatibility frames
• from implication frames
• from reflexive implication frames
• from ‘containment’ implication frames

Relation to traditional / categorical logic

Free Girard quantales from implication frames

• use the free involutive commutative monoid on a set
• use the free quantale from an involutive preordered monoid.

Compose three adjunctions to obtain F^{\otimes\neg\oplus}\dashv U^{\otimes\neg\oplus}. The unit \eta^{\otimes\neg\oplus}\colon (X,\bot)\to (X',\bot') sends an implication frame to its implication space, thought of as an implication frame (\bot' now codifies \vDash). {X' = \mathbb{C}=\R^2}, and \eta^{\otimes\neg\oplus}(a)=[\![ a ]\!]={\langle \{a^+\}^{\bot\bot},\ \{a^-\}^{\bot\bot}\rangle}.

Challenge: where do the semantic clauses for \neg,\otimes,\wedge, etc. come from?

The difference, in a nutshell: (read two points)

Reflexive implication frames

Reflexive implication frames

An element x \in X of an implication frame (X,\ \bot\subseteq \mathbb{N}[X+X]) is reflexive if (x,x)\in \bot. If all elements are reflexive, we say the frame itself is reflexive. Let \iota^{\rm r}\colon \mathsf{IF}_\pm^{\rm r} \rightarrowtail \mathsf{IF}_\pm be the full subcategory of reflexive frames.

We can restrict the F^{\otimes\neg\oplus}\dashv U^{\otimes\neg\oplus} adjunction to F_\pm\dashv U_\pm between \mathsf{IF^r_\pm} and \mathsf{GQ}.

How the codomain differs for \eta_\pm\colon (X,\bot)\to (X',\bot'):

• Elements of X' are not the full \R^2, but the subset for which a \vDash a, i.e. a_+\otimes a_- \leq \bot.
• Let \mathcal{Q}:=F_\pm(\mathcal{X}). The Girard quantale operations of \mathcal{Q}\times \mathcal{Q}^{\rm op} are closed on this subset.

\begin{align*} &[\![ A \otimes B ]\!]&:=\langle \texttt{a}_+\otimes \texttt{b}_+,\texttt{a}_-\text{⅋ } \texttt{b}_-\rangle&& &[\![ A \oplus B ]\!]&:=\langle \texttt{a}_+\vee \texttt{b}_+,\texttt{a}_-\wedge \texttt{b}_-\rangle\\ &[\![ A \text{⅋} B ]\!]&:=\langle \texttt{a}_+\text{⅋ } \texttt{b}_+,\texttt{a}_-\otimes \texttt{b}_-\rangle&& &[\![ A \& B ]\!]&:=\langle \texttt{a}_+\wedge \texttt{b}_+,\texttt{a}_-\vee \texttt{b}_-\rangle \end{align*}

These are precisely the semantic clauses we wanted to recover.

Supralinearity

Prop: These clauses validate the logical rules of \rm MALL

\boxed{ \begin{array}{c} \Gamma \vdash A, \Delta\\ \hline\hline \Gamma,\neg A \vdash \Delta \end{array} }

\begin{align*} &&\pi_2(\langle \texttt{a}_+,\texttt{a}_-\rangle) &\subseteq (\Gamma_+\Delta_-)^\bot \\ {\scriptscriptstyle \iff\hspace{-3mm}}&& \texttt{a}_- &\subseteq (\Gamma_+\Delta_-)^\bot \\ {\scriptscriptstyle \iff\hspace{-3mm}}&& \pi_1(\langle \texttt{a}_-,\texttt{a}_+\rangle) &\subseteq (\Gamma_+\Delta_-)^\bot \\ \end{align*}

\boxed{ \begin{array}{c} \Gamma, A\vdash \Delta\\ \hline \hline \Gamma \vdash \neg A,\Delta \end{array} }

\begin{align*} &&\pi_1(\langle \texttt{a}_+,\texttt{a}_-\rangle) &\subseteq (\Gamma_+\Delta_-)^\bot \\ {\scriptscriptstyle \iff\hspace{-3mm}}&& \texttt{a}_+ &\subseteq (\Gamma_+\Delta_-)^\bot \\ {\scriptscriptstyle \iff\hspace{-3mm}}&& \pi_2(\langle \texttt{a}_-,\texttt{a}_+\rangle) &\subseteq (\Gamma_+\Delta_-)^\bot \\ \end{align*}

\boxed{ \begin{array}{c} \Gamma,A,B \vdash \Delta \\ \hline\hline \Gamma, A \otimes B\vdash \Delta \end{array} }

\begin{align*} \texttt{a}_+ \texttt{b}_+ &\subseteq \Gamma_+\Delta_-^\bot \\ \text{ (Holds }&\text{by defn)}\\ \end{align*}

\boxed{ \begin{array}{c} \Gamma \vdash A,\Delta \quad \Theta \vdash B,\Omega \\ \hline \Gamma, \Theta \vdash A \otimes B, \Delta,\Omega \end{array} }

\begin{align*} (\Gamma_+\Delta_-\subseteq \texttt{a}_-^\bot) &\wedge (\Theta_+\Omega_- \subseteq \texttt{b}_-^\bot) \\ {\scriptscriptstyle \implies} \Gamma_+\Delta_- \Theta_+&\Omega_- \subseteq \texttt{a}_-^\bot \texttt{b}_-^\bot\\ {\scriptscriptstyle \iff} \Gamma_+\Delta_- \Theta_+&\Omega_- \subseteq (\texttt{a}_- \mathop{\mathrm{⅋}}\texttt{b}_- )^\bot\\ \end{align*}

\boxed{ \begin{array}{c} \Gamma, A \vdash \Delta \quad \Gamma, B \vdash \Delta\\ \hline\hline \Gamma, A \oplus B \vdash \Delta \end{array} }

\begin{align*} (\Gamma_+\Delta_- \subseteq a_+^\bot)&\wedge (\Gamma_+\Delta_- \subseteq \texttt{b}_+^\bot) \\ {\scriptscriptstyle \iff} \Gamma_+\Delta_- &\subseteq \texttt{a}_+^\bot \wedge \texttt{b}_+^\bot \\ {\scriptscriptstyle \iff} \Gamma_+\Delta_- &\subseteq (\texttt{a}_+ \vee \texttt{b}_+)^\bot \\ \end{align*}

\boxed{ \begin{array}{c} \Gamma \vdash A,\Delta\\ \hline \Gamma \vdash A \oplus B,\Delta \end{array} }

\begin{align*} \Gamma_+\Delta_- &\subseteq \texttt{a}_-^\bot \\ {\scriptscriptstyle \implies} \Gamma_+\Delta_- &\subseteq \texttt{a}_-^\bot \vee \texttt{b}_-^\bot \\ {\scriptscriptstyle \iff} \Gamma_+\Delta_- &\subseteq (\texttt{a}_- \wedge \texttt{b}_-)^\bot \\ \end{align*}

Prop: the consequence relation \vDash from \eta_\pm is supralinear.

Suppose \Gamma \vdash_{\rm MALL}\Delta. By cut-elimination for MALL, \Gamma \vdash_{\rm MALL}\Delta has a cut-free proof. The base case is that the proof is a single identity rule, which holds in \mathcal{X}' in virtue of being a reflexive implication frame. Each remaining step in the proof is a logical rule of MALL, which holds in \mathcal{X}'. Therefore \Gamma \vDash \Delta. That the valid atomic sequents are precisely \bot is a restatement that \eta_\pm is conservative.

The quantale operation \otimes in the validations to the right of each rule is represented via concatenation for space. Invertible steps are denoted with \iff, whereas implications are denoted with \implies.

\mathop{\mathrm{⅋}} and \& need not be checked, as they are definable as De Morgan duals of \otimes and \oplus, i.e. [\![ A\mathop{\mathrm{⅋}}B ]\!]:=[\![ \neg(\neg A \otimes \neg B) ]\!] and [\![ A\& B ]\!]:=[\![ \neg(\neg A \oplus \neg B) ]\!].

Containment + idempotent implication frames

Containment implication frames

An element x of an implication frame (X,\bot\subseteq \mathbb{N}[X+X]) satisfies:

• idempotence:¹ (\{x\},\{\})^\bot=(\{x,x\},\{\})^\bot and (\{\},\{x\})^\bot=(\{\},\{x,x\})^\bot.
• containment: (\{x\},\{x\})^\bot=\mathcal{P}[X+X]

Let a containment implication frame be one where all elements satisfy idempotence and containment. Let \iota^{\rm c}\colon \mathsf{IF_\pm^{c}\rightarrowtail IF^{\rm r}_\pm} be the full subcategory of containment frames.

Join-idempotent Girard quantales

Let \mathsf{GQ^{ji}} be the full subcategory of \mathsf{GQ} where we restrict to join-idempotent Girard quantales, which are those for which every element q \in \mathcal{Q} can be expressed as some join of idempotent elements of \mathcal{Q}.

Idempotent restriction of a quantale

Any quantale \mathcal{Q} can be restricted to its idempotent elements \mathcal{Q}_\otimes. This has the same \otimes operation and a \overset{\sim}{\vee} b := a \vee b \vee a \otimes b.

Caveat: if \mathcal{Q} is a Girard quantale, \mathcal{Q}_\otimes need not be a Girard quantale.

Reflexive frames could be restricted to those for which all elements are idempotent and, separately, to those for which all elements satisfy containment. However, we focus on the subcategory where both properties obtain.

1. If all elements are idempotent, subsets of \mathbb{N}[X+X] are in bijection with subsets of \mathcal{P}[X+X].

Free Girard quantales from containment frames

We can restrict the F_\pm\dashv U_\pm adjunction to F_\pm^{\rm c}\dashv U_\pm^{\rm c} between \mathsf{IF^c_\pm} and \mathsf{GQ^{ji}}. For \eta_\pm^{\rm c}\colon (X,\bot)\to (X',\bot'):

• Let \mathcal{Q}:=F_\pm(\mathcal{X}). Elements of X'\subseteq \mathcal{Q}_\otimes^2 are those for which a_+\otimes a_- = 0.¹
• The quantale operations of \mathcal{Q}_\otimes \times \mathcal{Q}_\otimes^{\vee} are closed on this subset.

[\![ A \wedge B ]\!]:=\langle \texttt{a}_+\otimes \texttt{b}_+,\ \texttt{a}_-\vee \texttt{b}_- \vee \texttt{a}_-\otimes \texttt{b}_-\rangle \quad [\![ A \vee B ]\!]:=[\![ \neg(\neg A \wedge \neg B) ]\!]

These are precisely the semantic clauses we wanted to recover.

Prop: the consequence relation \vDash from \eta^{\rm c}_\pm is supraclassical.

Shown by proving Lindenbaum algebra of \vDash is a Boolean algebra.

\neg and \vee satisfy the Robbins equation (McCune 1997): {\neg(A \vee B) \vee \neg (A \vee \neg B) = \neg A}

1. We denote the bottom element of the lattice with 0.

Outline

What is “logical expressivism”?

• Motivation: representationalism vs inferentialism
• As presented in (Hlobil and Brandom 2025)
• Examples: with and without contraction

Free Girard quantales

• from incompatibility frames
• from implication frames
• from reflexive implication frames
• from ‘containment’ implication frames

Relation to traditional / categorical logic

Connections to category theory

Functorial semantics: functors \mathsf{C}\to\mathsf{D} are thought of representationally: the objects of \mathsf{C} are represented by objects of \mathsf{D}. The semantic category (e.g. \mathsf{Set}) usually has nice structure (e.g. cocompleteness).

However, even without some supplied semantic category \mathsf{D}, we have a canonical interpretation of \mathsf{C} into a nice category as \widehat{\mathsf{C}}=[\mathsf{C,Set}], the free cocompletion.

So our embedding \eta^{\otimes\neg\oplus} has some similarities to the Yoneda embedding, in particular because it arises from a free cocompletion.

We aren’t forced to pick exclusively between understanding \mathsf{C} ‘internally’ vs representationally.

Future work

• Connective clauses were shown to emerge from a relatively simple quantale built from \mathcal{Q} being closed on the relevant subset of

Despite Frege’s context principle, something needs to be said about subsentential structure. How to generalize to predicate logic?

• Enriched category theory: from \bot\colon \mathbb{N}[X]^2\to \mathbb{B} to \bot\colon \mathbb{N}[X]^2\to \mathsf{Set}.

Can we make use of structure in the category \mathsf{IF}_\pm?

• (Co)-limits? Closed structure?
• Work in Kleisli category of the monad of F^{\otimes\neg\oplus}\dashv U^{\otimes\neg\oplus}?

Computational implementation in Julia: ROLE.jl - Naive implementation

References

Fodor, Jerry A, and Ernest Lepore. 1993. “Why Meaning (Probably) Isn’t Conceptual Role.” Philosophical Issues 3: 15–35.

Girard, Jean-Yves. 1995. “Linear Logic: Its Syntax and Semantics.” Proceedings of the Workshop on Advances in Linear Logic, 1–42.

Hlobil, Ulf, and Robert B Brandom. 2025. Reasons for Logic, Logic for Reasons: Pragmatics, Semantics, and Conceptual Roles. Routledge.

McCune, William. 1997. “Solution of the Robbins Problem.” Journal of Automated Reasoning 19 (3): 263–76. https://doi.org/10.1023/A:1005843212881.

Restall, Greg. 2005. “Multiple Conclusions.” In Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress, 189–205.