A Category-theoretic Reconstruction of Logical Expressivism

Kris Brown

Press s for speaker notes, q to reveal hidden slides

7/8/26

Outline

$$

$$

What is “logical expressivism”?

• (Hlobil and Brandom 2025)
• Representationalism vs inferentialism
• Logical and non-logical consequence relations
• NMMS sequent calculus
• Implication space semantics

Free Girard quantales

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

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. I’ll talk about their solution, which isn’t at all ad hoc from a philosophical point of view, yet it can be supported by a mathematical perspective of how its numerous definitions are actually quite natural. To summarize how it works: we view their core data structure as an object of a category. Then, their framework shakes out as the free addition of some structure via the unit of an adjunction.

Background: representationalism vs inferentialism

Important concepts in the philosophy of language:

Understanding

Concept

Meaning

Truth

Syntax

Reason

Logic

Reference

(Language) use

Vocabulary

Semantics

Rationality

Norm

Inference

Definition

Model

Pragmatics

Justification

Philosophers of language are concerned with the relations between concepts like these.

What you think about the above heavily influences what you make of the following inferences:

“x is to the east of y” \vdash “y is to the west of x”

“t is a bird” \vdash “t can fly”

“t is a penguin”, “t can fly” \vdash

“t is a bird”, “t is a penguin” \nvdash

“I’ll strike a match, m” \vdash “m will light”

“I’ll strike a match, m”, “m is wet” \nvdash “m will light”

“n is a triangle” \vdash “n has 180^\circ (internal angle sum)”

“n is a triangle”, “n is noneuclidean” \nvdash “n has 180^\circ”

What you think about the above heavily influences what you make of the inferences at the bottom.

As a caveat, these inferences below are things that I’m personally claiming are good inferences (for our community, here in this room), though you may personally disagree or want to dive deeper into what it really means to be a good inference.

[Read the LHS]

Background: representationalism vs inferentialism

Core philosophical question: relation between mind and world language and world.

• What/why is it that “x is a dog” \vdash “x has four legs”?
• What/why is it that “x is to the east of y” \vdash “y is to the west of x”?

Here are some examples of ordinary language inferences.

The philosophical question is ‘what makes them good or bad?’

There are two major lines of response to that question.

Representationalism: meaning grounded by reference to the world (atomistic)

One we call representationalism. It’s an atomistic approach: words name objects or relations which get combined logically, giving you some overall meaning with a reference in the world, and the world determines which things really follow from what. One reason this is nice is that it explains the compositionality of language: the meaning of its syntactically complex exprs (very often) is a fn of their syntactic structures + meanings of their syntactic constituents.

Compositionality in turn explains how we’re able to understand each other despite the fact that almost every spoken sentence has never been (literally) spoken before.

Inferentialism: meaning is determined by inferential role within a language (holistic)

• Frege’s context principle: Only in the context of a sentence does a word have meaning.¹
• Open question: What about logic?²

The opposing intuition is a more holistic view of language. Inferentialists are more interested in what inferences obtain between sentences than what conditions make a sentence true. From that perspective, the meaning of ‘x is a dog’ is to be read off of all of the inferences involving that sentence, and the meaning of ‘dog’ the word must be read off of this web of inferences as well.

The intuition here is the context principle: only in the context of a sentence does a word have meaning (the proposition is the smallest unit of pragmatic force). This approach has some nice advantages and also invites some questions, and the one I’ll focus on is bolded here: what does the inferentialist have to say about recursive semantic formulas for logical connectives?

1. The proposition is the smallest unit of pragmatic force.

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

The key data structure

Implication frame

Implication frame: a set L equipped with a subset {\bot\subseteq \mathbb{N}^{L+L}}.

Idempotent implication frame: a set L with a subset {\bot\subseteq \mathbb{B}^{L+L}}.

\bot is the subset of good implications, according to the frame.

One way to model the situation we’re in (when I say ‘here are some good implications, here are some bad ones’) is an implication frame. This is a set equipped with a subset of the possible sequents you build (either multisets or sets of the atoms on either side). The most striking thing about this definition is how radically unconstrained the good sequents have to be.

Example frame

If our underlying set of propositional atoms is X=\{a,b\}, these are the possible sequents:

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

Even though the definition is simple, I just wanted to show a simple implication frame concretely so that you have a feel for one. This one has two possible things that can be said and, of the 16 possible sequents that can be built out of those atoms,

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

\begin{array}{||c||c|c|c|c||} \hline\hline & 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}

E.g. a\vdash a and \quad \vdash a and b\nvdash a in this frame.

In this example frame, 11 of the 16 are endorsed by the frame. It showcases an example of nonmonotonicity: ‘a’ is a theorem but ‘b’ does not imply ‘a’.

Isn’t it just probabilistic inference?

• Radically-substructural frames have a very rich algebraic structure.
  • Going for probability is just projecting away all of this rich structure onto the real line.
  • One can reconstruct a lot of the algebraic structure you started with, remarkably. But why do that? You should only if you have to.
• Rational choice (game theoretic or decision): relation between outcomes and options.
  • We reconstruct a notion of rationality in terms of maximizing expected utility
  • You can do that if you already have semantically determined outcomes and options
  • You’ve presupposed the semantic content of those things, so your consequence relation is not trying to do the same thing as these (one that is derivative/parasitic of a more basic notion of reason relation).
• Hyperintentionality can’t be reconstructed via probability theory (Kit Fine)
  • probability theory still has a lot of structure.

Logical consequence relations

Logical consequence relations are a source of \ \vdash\ relations. Common assumptions:

• monotonicity: weakening, portability of reasoning
• transitivity¹: (mixed) cut, composability of reasoning

We’re used to seeing consequence relations arising from logic, where they come from provability or truth preservation. This framing is going to come with certain expectations of properties that it must satisfy, such as monotonicity or transitivity. (In particular, reflexivity and transitivity are essential assumptions for consequence relations modeled by categorical logic.)

It is desirable to have such relations because of the portability and composability of reasoning. And it’s easy to slip from this desire to actually thinking any notion of ‘follows from’ must satisfy these.

Conceptions of consequence that admit radical substructurality:

Truthmaker semantics: (Fine 2017)

Bilateralism: (Restall 2005)

Truthmaker semantics is a generalization of possible worlds semantics designed to be hyperintentional, meaning it can distinguish properties which are true in all possibile worlds. Bilateralism is a way of understanding the turnstile to codify a norm which regulates proper and improper speech acts in a community.

Ordinary language transitivity violation:

“bird(x)” \textcolor{red}{\vdash} “flies(x)”    |||   “penguin(x)”, “flies(x)” \vdash    |||   “bird(x)”, “penguin(x)” \nvdash

Here are some examples of what could be thought of as good inferences. This can’t be represented by a logical consequence relation.

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

More substructural consequence relations

Transitivity violation from distinguishing explicit contradictions from implicit ones:

       C \wedge \neg C\vdash       |||      C \wedge \neg C\vdash A \wedge \neg A       |||      B\vdash A\wedge \neg A       |||       B \textcolor{red}\nvdash

Ordinary language monotonicity violation:

“I’ll strike a match, x” \textcolor{red}\vdash “x will light”    |||    “I’ll strike a match, x”, “x is wet” \nvdash “x will light”

[For sake of time, I’ll skip over some other examples.]

One may say “good reasoning” ought satisfy these because, look how successful mathematics and scientific communities have been! If we’re taken in by such a suggestion, we’ll then revisit the earlier implications we wrote and then conclude we must have been wrong.

But this gets the priority backwards modeling reason relations and the reason relations themselves: if the logician’s toolset is insufficiently rich to account for representing the “follows from” relation of a social practice, then it’s not the social practice that is wrong.

Consequence relation interpretations

Two ways to interpret the turnstile beyond provability or truth preservation:

Truthmaker semantics: (Fine 2017)

There is a lattice of possible world states, and a notion of some being impossible. Worldly propositions denote parts of the world that make them true vs make them false. This induces a frame.

\underline{\color{blue}{\text{Truthmakers}}}\qquad \underline{\color{black}{\text{Impossibility}}}\qquad \underline{\color{red}{\text{Falsemakers}}} \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” = “Any fusion of verifiers of \color{blue}\Gamma and falsifiers of \color{red}\Delta is an impossible state.”

Bilateralism: (Restall 2005)

There is a set of things one can claim, two possible speech acts per claim, and a notion of some move combinations being incompatible. This is (directly) the data of an implication frame.

\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”

When we look at the turnstile as provability or truth preservation, it’s hard to not expect the bot subset to satisfy certain properties.

Below I have two conceptions of logical consequence which do make it perfectly natural to allow it to be an arbitrary subset. These are quite different, on the left is a very semantic interpretation, a generalization of possible world semantics called truthmaker semantics. Bilateralism takes two kinds of speech acts as primitive and derives a notion of implication.

There is a set of things one can claim, two possible speech acts per claim, and a notion of some move combinations being incompatible. This is (directly) the data of an implication frame.

For the purposes of the talk, I’ll just adopt the social-pragmatic vocabulary for talking about frames, even though there is no need to commit to that interpretation of the turnstile.

Logical Elaboration + NMMS calculus

Logical expressivism

Logic ought express features of some antecedent, prelogical system of implications. Express means to make explicit (internalize) features of \bot that, were originally only describable in a metavocabulary.

So what can an inferentialist say about logic? One can view it as something one adds to a preexisting frame with a pre-logical notion of good inference. We’re going to add brand-new sentences to frame which internalize certain features of the frame that we might have stated in a metavocabulary.

The logical elaboration of a frame freely extends its atoms with connectives.

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

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

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

One can formally define a procedure for extending the set of atoms with logical connectives.

The mechanical process is specified by the sequent calculus NMMS (Nonmonotonic, multisuccedent logic) a fragment of which I show here.

There might not have been an atom in the frame which was good (on the left) in every and only context in which A was good on the right.

NMMS rules are bidirectional:

• Therefore the logical consequence relation is fully determined by the prelogical one.

Bidirectionality means have a mechanical procedure for taking a logically complex sequent and evaluting its goodness by looking at sequents with fewer connectives.

Full NMMS calculus

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

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

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

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

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

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

Implication space semantics

Implication space semantics extends this syntactic logical story with a semantic one.

Given a frame (X,\bot), the implication space semantics constructs:

• A space of semantic values, \mathbb{C}:=\mathbb{R}^2
• Some binary operations \sqcup, \sqcap\colon \mathbb{R}\times \mathbb{R}\to \mathbb{R}
• A consequence relation on pairs of (multi)sets of such semantic values
• An interpretation function [\![ - ]\!]\colon X\to \mathbb{C}
• Semantic clauses for MALL connectives (\otimes,\text{⅋},\oplus,\&)
• Semantic clauses for classical logic (\neg,\wedge, \vee)
  • Sound and complete for NMMS

Our goal

Recover these definitions, providing a mathematical perspective on where they come from.

Defining a category of implication frames

Let F' be the composition of a sequence of left adjoints:

F'(X) = (\mathcal{P}[\mathbb{N}[X+X]],\subseteq, \otimes)

We define implication frames with two building blocks. The first is a way to obtain quantales from sets via a series of functors. In particular, this sends a set to the power set of the set of pairs of multisets.

Pointed quantales (Q,\leq,\otimes,\bot\in Q)

Quantales with a distinguished element weakly-preserved by morphisms: f(\bot_X)\leq_Y \bot_Y

The second building block comes from a category of pointed quantales where the point is weakly preserved.

Implication frames (X,(\mathcal{P}(\mathbb{N}[X+X]), \subseteq, \bot\in \mathcal{P}(\mathbb{N}[X+X])))

Let \widehat{f}:=\mathcal{P}[\mathbb{N}[f+f]], lifting f to act on sets of pairs of multisets.

Morphisms: functions f\colon X\to Y such that \widehat{f}(\bot_X)\leq_Y \bot_Y.

We combine these building blocks by taking a pullback, using the obvious forgetful functor from dialogue quantales to quantales. So, formally, a frame would be a pair of a set and a dialogue quantale, though everything about the dialogue quantale is determined by the set except for its distinguished element, so we can recover our earlier definition of a set equipped an element of the power set of the set of pairs of multisets.

Defining a category of implication frames

We can characterize what the category of implication frames is by expressing it as a big pullback. Dashed arrows are projection functors from pullbacks in Cat. From this perspective, a frame is a set along with a dialogue quantale whose underlying quantale is the power set of N[X+X]. Because dialogue quantales are simply a quantale with a distinguished object, the only degree of freedom is to pick an element of that set, i.e. a bot which is a subset of N[X+X]. Note that dialogue quantales are a decategorification of cocomplete dialogue categories (which are called as such due to a game-theoretic interpretation).

This lemma lets us use the fact that the bottom functors are all left adjoints and the fact the vertical functors are all fibration-like in order to automatically generate right adjoints for the top row of pullback projections.

Free dialogue category implication frames

Lemma: Adjoints from pullbacks

If G has cartesian lifts \overline{\varepsilon_c} for all {\varepsilon_c\colon FU(c)\to c}, 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)}})).

Here is a helpful lemma which lets us lift left adjoints when functor we are pulling back to be enough of a fibration.

The pullback projection map \mathsf{IF_\pm\to PQ} is a left adjoint, as U^\bot is a fibration.

So we apply this to our left adjoint from Set to Quant to lift it into the setting where we also carry around the bot data. We need functor we are pulling back to be enough of a fibration: the resulting adjunction is the same on the underlying set.

U_bot is a fibration: the lift of f:X->Y and Y’ = (Y,bot) is (X,bot’), where bot’:=f_*(bot) = V{x | f(x)≤ y} … the largest element of X which allows weak preservation to Y.

Free Girard quantales

Girard quantales (Q,\leq,\otimes,\bot\in Q)

\bot is a dualizing element: a=a^{\bot\bot} where {a \otimes b \leq \bot} \iff a \leq b^\bot.

Dialogue quantales are almost what we want. Girard quantales further stipulate that the distinguished element is dualizing, as in a *-autonomous category.

These form a full subcategory of dialogue quantales, but the natural way to obtain a functor from dialogue quantales to Girard quantales is not functorial.

If we restrict the morphisms of both \mathsf{IF}_\pm and \mathsf{PQ} to satisfy continuity,¹ we freely add the structure of a Girard quantale. Let this adjunction be F \vdash U.

We have to restrict the class of morphisms we allow (for dialogue quantales, and then correspondingly in implication frames) in order to view GQ as a reflective subcategory. We call this condition continuity: image of the closure is contained in the closure of the image.

1. A^{\bot_X}\subseteq B^{\bot_X} {\implies} \widehat{f}(A)^{\bot_Y}\subseteq \widehat{f}(B)^{\bot_Y}. Tantamount to \widehat{f}(A^{\bot_X\bot_X}) \subseteq \widehat{f}(A)^{\bot_Y\bot_Y} and to \widehat{f}(A)^{\bot_Y\bot_Y} = \widehat{f}(A^{\bot_X\bot_X})^{\bot_Y\bot_Y}.

Adjunction unit

Let’s interpret \eta_\mathcal{X}\colon \mathcal{X}\to\widehat{\mathcal{X}}=(X',\bot') as sending a base frame to its semantic frame with a semantic consequence relation, \ \vDash.

I’ve given an abstract argument for why there is an adjunction with Girard quantales, but let me be a bit concrete about what the unit of it looks like.

X' elements are pairs of \mathcal{Q}:=F(\mathcal{X}) elements: (-)^{\bot\bot}-closed subsets of \mathbb{N}[X+X].

Given premises [\langle \Gamma_1^+,\Gamma_1^-\rangle,...] and conclusions [\langle \Delta_1^+,\Delta_1^-\rangle,...], we have:

\langle \Gamma_1^+,\Gamma_1^-\rangle,...\langle \Gamma_n^+,\Gamma_n^-\rangle \vDash \langle \Delta_1^+,\Delta_1^-\rangle,...\langle \Delta_m^+,\Delta_m^-\rangle := \bigotimes_i \Gamma_i^+ \otimes \bigotimes_j \Delta_j^- \leq \bot

\eta_\mathcal{X} sends x \in X to \langle (\{x\},\{\})^{\bot\bot}, (\{\},\{x\})^{\bot\bot}\rangle.

Progress so far

Progress towards recovering implication space semantics

We’ve now recovered:

• What \mathbb{R} is: elements of the free Girard quantale \mathcal{Q}= F(\mathcal{X})
• the space of semantic values (\mathbb{C}=\mathbb{R}^2=X')
• the notion of semantic consequence (\bot')
• the interpretation [\![ - ]\!] = \eta_\mathcal{X}\colon X \to X'.
• semantic clause for negation: it’s swap! [\![ \neg A ]\!]:=\langle a_-,\ a_+\rangle

Believe it or not, we’ve already done most of the work, we’ve recovered the space of semantic values (elements of X’), the notion of semantic consequence (bot’), and a canonical interpretation for atomic propositions into this space (eta itself).

Now, what about the MALL semantic clauses? We know how to interpret \otimes,\text{⅋},\oplus,\& in a GQ.

The semantic values are pairs of elements of \mathcal{Q}, but this doesn’t have a unique Girard quantale structure.

There are two strong candidates for Girard quantale structures whose underlying set is Q^2. Either Q x Q or Q x Q^op, since every Girard quantale has an opposite Girard quantale. How do we resolve this?

Reflexive implication frames

Reflexive implication frames

An implication frame (X,\ \bot\subseteq \mathbb{N}[X+X]) is reflexive if \forall x \in X\colon (x,x)\in \bot.

Let \iota^{\rm r}\colon \mathsf{IF}_\pm^{\rm r} \rightarrowtail \mathsf{IF}^{\rm cont}_\pm be the full subcategory of reflexive frames.

We can restrict our previous adjunction to F_\pm\dashv U_\pm between \mathsf{IF^r_\pm} and \mathsf{GQ}.

We can consider the full subcategory of frames which do satisfy reflexivity and restrict our adjunction to this subcategory.

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

• Elements of X' are not the full \mathbb{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*}

Once we restrict to the reflexive subset of pairs of quantale elements, it becomes clear the natural Girard quantale structure is Q x Q^op. The tensor of that quantale becomes the interpretation of tensor in the space of semantic values, etc.

Supralinearity

\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 MALL semantic clauses we wanted to recover.

What is some evidence that these are good definitions?

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

E.g.

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

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

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*}

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

parr and & need not be checked, as they are definable as De Morgan duals of and .

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.

Conservativity in logic means that the addition of new vocabulary does not change the goodness of inferences between sentences that only use the old vocabulary. This is requiring that morphisms not merely preserve but also embed it.

Analogous story for classical logic

Containment implication frames

A containment implication frame (X,\bot\subseteq \mathbb{N}[X+X]) has each element x satisfying:

• idempotence: \Gamma, x \vdash \Delta {\iff} \Gamma,x,x\vdash \Delta and \Gamma \vdash x,\Delta {\iff} \Gamma\vdash x,x,\Delta.
• containment: all \Gamma, x \vdash x, \Delta are in \bot

There is a similar story one can do for propositional logic: one restricts the type of frames one is considering, then there is a natural algebraic structure that arises on the elements in the frame which the adjunction sends a frame to. In the case for classical propositional logic, we restrict attention to frames which are idempotent and satisfy containment, which means that reflexive sequents are not just good but indefeasibly good.

When we restrict the adjunction to these frames, \eta^{\rm c} sends a frame to subset of \mathcal{Q}^2 with a natural quantale structure whose natural \otimes operation is:

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

I’ll note that the multiset story (with linear logic clauses) and the set story (with classical propositional logic clauses) are told as two disjoint (though spiritually similar) stories in the book, so by showing they are both restrictions of the same adjunction we have unified their story in addition to showing its naturality. In particular, it was always a pain to justify the third clause (a v b v (a x b)) in NMMS and the semantics, but now we see it is forced on us as the natural choice of join in the idempotent subquantale.

These are precisely the propositional logic semantic clauses¹ we wanted to recover. We’re done!

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

When the only good implications are the ones which are forced by containment, then the consequence relation is exactly the classical logic consequence relation.

1. Define disjunction as the De Morgan dual: [\![ A \vee B ]\!]:=[\![ \neg(\neg A \wedge \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.

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 propositional logic semantic clauses we wanted to recover. We’re done!

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}

Here I again restrict the adjunction, look at what structure matches the subset of pairs of quantale elements that survive the filtering, and use that to define semantic clauses which can be proven to be supraclassical.

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

Implication space semantics

The following are all parameterized by a background implication frame (X,\mathbb{I}).

Range of subjunctive robustness operation

The range of subjunctive robustness function, (-)^*\colon {\mathcal{P}[\mathbb{N}[X+X]]\to\mathcal{P}[\mathbb{N}[X+X]]}, 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 \mathbb{I}\}}.

Implicational roles and conceptual contents

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

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

Symjunction and adjunction of roles A,B \in \mathbb{R}

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

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

Semantic consequence + base interpretation

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

For all atoms a \in X:

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

Now we’re able to make sense of the implication space semantics formalism!

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 \mathbb{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.

The debate from a CT perspective

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 \eta 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.

I think there should be more to say at a technical level for how this relates to category theory.

Here compositionality of reasoning is a property rather than structure. ACT is concerned with open systems, and these are (pre)logical systems which are radically open.

Next steps

\mathsf{IF_\pm} is bicomplete + closed, has analogues which are \mathcal{V}-enriched

IF_^cont is a terrible category (as initial/terminal object, but no (co)products).

Computational implementation in Julia: ROLE.jl - Naive implementation

Frege’s context principle: we need to understand subsentential structure from the starting point of inferences between sentences.

“Nobody is hurting me!”

How to deduce the logical form from \bot?

Whether or not we ought parse this sentence as a negative existential or as predicating an individual who happens to be named ‘Nobody’ is determined by the inferences that we ought draw from it. So just like there were inferential roles associated with the propositional connectives, there are also roles associated with quantifiers, so we want to make this explicit as well.

Conclusions

• Logical consequence relations => supra-logical consequence relations
  • Precise control over goodness of atomic sequents
  • Can represent richly-structured concepts / systems

Thank you for listening

And even more thanks to:

      Kevin Carlson

      David Jaz Myers

      Evan Patterson

      Lucy Horowitz

And the Research on Logical Expressivism (ROLE) group:

• Robert Brandom, Ulf Hlobil, Ryan Simonelli, Rea Golan, Shuhei Shimamura, and others.

Future work

Ordered implication frame

An ordered implication frame is a preorder (X,\leq) equipped with a monotone map \mathbb{N}[X^{\rm op}+X]\to 2

Enriched implication frame

An enriched implication frame is a \mathcal{V}-category \mathcal{A} equipped with a \mathcal{V}-presheaf in \widehat{S[\mathcal{A}^{\rm op}+\mathcal{A}]}, where S(-) denotes the free symmetric monoidal \mathcal{V}-category.

It’s not clear how \mathcal{V}-enrichment leads to subsentential structure.

• In \mathsf{Set}-enriched setting, a frame a set of substitutions between any two claimables.

• There is also a set of reasons why \Gamma \vdash \Delta.

Another idea inspired by hyperdoctrines: fix a category of contexts \mathsf{C} and decompose one’s frame to a give a functor \mathsf{C \to OIF} (this requires work, there isn’t a canonical way to do it).

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 preorder¹

• U^\bot: forget the dualizing structure of a Girard quantale

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

1. Free join completion, \otimes of the result given by Day convolution

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.

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

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}=\mathbb{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)

References

Fine, Kit. 2017. “A Theory of Truthmaker Content i: Conjunction, Disjunction and Negation.” Journal of Philosophical Logic 46 (6): 625–74.

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.” Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress, 189–205.