Deriving semantics from pragmatics

Kris Brown

Press s for speaker notes, q to reveal hidden slides

9/3/26

Background (1/6)

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            (2025)

Robert Brandom

          Ulf Hlobil

Project Timeline

• 2024: Formal pragmatics \to semantics construction for propositional logics

Recently there was a book Reasons for Logic Logic for Reasons that made a big step forward in understanding the relationship between pragmatics and semantics.

Background (2/6): semantics

Competing strategies for understanding the meanings of our ordinary language:

The semantic attitude

Bits of language have meaning by referring to the world.

Let me explain how I’m going to use those terms. You might say they are competing strategies for understanding the meanings in our ordinary language. Let’s start with the semantic attitude which draws from techniques for understanding meaning in artificial languages.

One possibility:

• particular nouns pick out objects (from some set of ‘real-world objects’)
• adjectives pick out subsets of objects (i.e. predicates)
• sentences pick out subsets of possible worlds (the worlds in which they’re true)
• and picks out the intersection of such subsets

The semantic attitude might take it that for example, our particular nouns pick out objects in some domain of real-world objects, that our adjectives pick out subsets of these, that our sentences pick out subsets of possible worlds and our logical conjunction intersects these sets, and so on. To handle more richness of language, people throw in other gizmos like accessibility relations to handle modality.

Background (3/6): semantic analysis example

[\![ \text{Amy is poor and honest} ]\!] \\ [\![ \text{Amy is poor} ]\!] [\![ \text{and} ]\!] [\![ \text{Amy is honest} ]\!] \\ \underset{\rm Predicate}{\underbrace{{[\![ \text{Poor} ]\!]}}}(\underset{\rm Object}{\underbrace{{[\![ {\text{Amy}} ]\!]}}}) \underset{\rm Connective}{\underbrace{\raisebox{-1mm}{$\cap$}}} \underset{\rm Predicate}{\underbrace{{[\![ \text{Honest} ]\!]}}}(\underset{\rm Object}{\underbrace{{[\![ {\text{Amy}} ]\!]}}})\\

Let’s consider an example sentence. We could potentially analyze ‘Amy is poor and honest’ to have this covert or implicit logical structure, involving objects, predicates, and a propositional connective.

\text{As conjunctions: }[\![ \text{and} ]\!]=[\![ \text{but} ]\!] = \cap [\![ \text{Amy is poor and honest} ]\!] = [\![ \text{Amy is poor but honest} ]\!] [\![ \text{Amy is poor and honest} ]\!] \vDash [\![ \phi ]\!] \quad \text{ iff }\quad [\![ \text{Amy is poor but honest} ]\!]\vDash [\![ \phi ]\!]

Now, let’s assume that, as conjunctions, ‘and’ and ‘but’ are taken to have the same ‘literal’ meaning (that of intersecting scenarios), this means that ‘Amy is poor and honest’ and ‘Amy is poor but honest’ have the same semantic value (literal meaning)! Therefore any sentence phi which semantically is a consequence of one (note the double turnstile) must be likewise for the other.

Upshot: semantic attitude accounts for compositionality of natural language

As a final comment on the semantic attitude, I’ll highlight that this has the upshot that it neatly explains how language is compositional: the meaning of a complex is often the meaning of its parts combined in some algorithmic way.

Background (4/6): pragmatics

The pragmatic attitude

To say something is to do something, a speech act.

The meanings of speech acts are to be read off of their consequences.

By contrast, the pragmatic attitude starts with observing that we do a lot of things with language which are not best understood as referring to or describing the world.

Meanings of speech acts are to be understood in a more holistic way, the overall consequence of making the speech act.

• E.g. “Court is in session” not a description of the court
• E.g. propriety of a chess move not justified by reference.
  • Some particular knight movement allowed because we have agreed on the valid moves.

For example a judge declaring court is in session isn’t describing it as such but making it the case. People like to talk about ‘language games’ in this space: the analogy is good because games are a setting where there are genuine rules that regulate proper and improper behavior, and proper moves are not justified by reference (the knight moves the way it does because we have agreed that’s the rule, not because actual horses move that way, for example).

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A notion of consequence from pragmatic raw materials (Restall 2005)

• Assume we have at least two specific kinds of speech acts: assertion and denial.
• Assume a normative status ‘impropriety’, e.g. {\rm Improper}(A^+) means it’s improper to assert A.
• {\rm Improper}(A^+,\ B^-) \ \ \equiv\ \ A \vdash B\ \ \equiv\ \ B\text{ follows from }A \ \ \equiv \ \ A \text{ is a reason for }B

The last thing I’ll say about pragmatics: just because we have a more squishy sense of what meaning is doesn’t prevent us from having a notion of consequence. We can get a notion of consequence with just some very basic ingredients. Two very particular speech acts (assertion and denial) and a notion of a speech act being `improper’. This model, which is called bilateralism, interprets the norm which which forbids simultaneously asserting A and denying B, as a norm in which A is a good reason for B which we denote with the turnstile.

Note that improper is not analyzable in terms of something simpler (like in the chess example).

Background (5/6): semantics versus pragmatics

Competing strategies for understanding the meanings of our ordinary language:

The semantic attitude

Language has meaning by referring to the world.

 

The pragmatic attitude

To say something is to do something (speech act).

Meanings are to be read off of the consequences.

So where might these two attitudes come in contact with each other?

---

\text{When all goes well: }\quad [\![ A ]\!]\vDash [\![ B ]\!] \quad \text{ iff } \quad A \vdash B

They both have a sense of consequence, so we expect these to agree. The problem is: what if they don’t? This is similar to the problem of a formal model which disagrees with your starting assumptions / intuitions / data. Did you make a mistake in your definitions? Did you make a computational error? Or were your starting intuitions wrong? Or did you have bad data?

\text{However: }\quad [\![ \text{Amy is poor but honest} ]\!]\nvDash [\![ \text{Wealth+honesty are related} ]\!]

\quad \ \text{... yet: } \quad [\![ \text{Amy is poor but honest} ]\!]\vdash [\![ \text{Wealth+honesty are related} ]\!]

Which side of this biconditional has explanatory priority? At the core of this debate is whether it is more enlightening to first understand the use of language or to understand the meanings of the components of our language. Which do we have a better grip on?

Background (6/6): project timeline

• 2024: Formal pragmatics \to semantics construction for propositional logics

(5 min) There has been a lot of ink spilled both ways in this debate (e.g. “Why Meaning (Probably) Isn’t Conceptual Role” (Fodor and Lepore 1993)). This book, Reasons for Logic, starts with a notion of pragmatic rationality (in terms of assertions and denials) and derives a semantic space, semantic consequence relation, and recursive semantic clauses. It shows how depending on the structure of the input pragmatic reason relation, when you turn the semantic crank you might end up with any variety of substructural (propositional) logics.

Problems

1. Concise presentation of the technical formalism, showing design decisions as ‘natural’
2. Generalization to syntax with subsentential structure (e.g. predicate logic)

However, there were two issues from my perspective!

1. it was not easy to concisely explain the work and justify all of its choices as particularly natural.
2. the wide scope was limited syntactically to logics that had no subsentential structure. E.g. no FOL.

Background (6/6): project timeline

Project Timeline

• 2024: Formal pragmatics \to semantics construction for propositional logics

• 2025: Described as the \eta of an adjunction involving simple, commonplace categories.

The first issue was addressed in work I did in 2025, which I presented at ACT 2026. It turns out, in some sense, the technical apparatus was “just” the unit of a particular adjunction, the free addition of Girard quantale structure to their model of a pragmatic system of norms.

• 2026: Generalized internally to any \mathcal{E} (some fixed topos with a natural numbers object)

• 2026: Instantiate \mathcal{E}\mapsto \mathsf{Nom}, use particularities of nominal sets to address predicate logic.

The second issue has really made a lot of progress after a comment of David Jaz Myers who encouraged me to look into nominal sets. In order to take that advice, it was important to formulate the above in terms of an arbitrary topos.

Technical Outline

1. Internal (pointed, Girard) quantales
   
   
2. The ‘free sequent-set’ functor
   
   
3. Implication frames + the logical completion
   
   
4. Interpreting MALL and classical logic in a frame
   
   
5. Work in progress: Nominal sets to model subsentential structure
   
   

(8 min) Here is the mathematical story. I’ll first have to tell you the propositional story at a topos-agnostic level before telling you about nominal sets.

I’ll talk about pointed and Girard quantales internal to one’s topos, and then we’ll introduce a functor “the free sequent-set functor” and then talk and putting those together will allow us to define an “implication frame” which is the core data structure that the philosophers care about for representing pragmatics.

Quantales internal to a topos (1/4)

A (commutative, unital) quantale is a complete lattice with a (commutative, unital) multiplication \otimes such that: a \otimes \bigvee_i b_i = \bigvee_i a\otimes b_i.

In this talk I’ll mention quantales a lot. I’ll only be interested in ones which are commutative and unital. Ordinarily, these are just lattices equipped with a multiplication which distributes over joins. This turns out to be a quantale internal to the topos of sets.

Internal suplattices in a topos

An internal suplattice in a topos \mathcal{E} is an object X equipped with a join operation \bigvee\colon \mathcal{P}X \to X.¹

Now we’re going to generalize this by working internal to a topos. To start, we can define internal suplattices by axiomatizing when a morphism from a power object its underlying object plays the role of join.

Internal quantale in a topos (1/2)

A (commutative, unital) internal quantale is an internal commutative monoid (X, \mu, e) and internal suplattice (X, \bigvee) structure, satisfying:

Then, if that object also happens to be an internal (commutative) monoid, then we can ask if the multiplication distributes across the join, which in the topos setting corresponds to this diagram.

We also have a category of internal quantales being morphisms on the underlying objects which commute with all of the extra structure operations.

Note t_{X,X}\colon\mathcal{P}X\times X\to\mathcal{P}(X\times X) sends (A, b)\mapsto \{(a,b)\ |\ a \in A\}.

1. Moreover satisfying unitality (\{-\}\cdot \bigvee = 1_X) and associativity (\bigcup \cdot \bigvee = \mathcal{P}(\bigvee)\cdot \bigvee as maps \mathcal{P}\mathcal{P}X\to X).

Quantales internal to a topos (2/4)

Internal quantale in a topos (1/2)

A (commutative, unital) internal quantale is an internal commutative monoid (X, \mu, e) and internal suplattice (X, \bigvee) structure, satisfying:

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Example: lifting monoid structure on X to quantale product on \mathcal{P}X

If (X,+,0), then (\mathcal{P}X,\bigcup, \otimes_{\rm Mink},\{0\}) is an internal quantale.

A \otimes_{\rm Mink} B = \{a+b\ |\ a \in A, b\in B\}

Rather than worrying about quantales in full generality, the only kind we’ll see in this talk are those where the quantale is a power object and its multiplication comes from some monoid structure on the object underlying power object. The Minkowski product shown here is a monoid operation on the power object. It is compatible with the join operation. This is the form of all the quantales we’ll be interested in.

Quantales internal to a topos (3/4)

Residuation morphism

Suppose we have a distinguished point: I\colon 1\to Q.

There is a residuation morphism (-)^*\colon \mathcal{Q}\to \mathcal{Q} given by q \mapsto \bigvee \{x\ |\ x\otimes q \leq I\}.¹

So quantales have an internal hom operation called residuation, and when we do that in our topos we get a morphism from the quantale object to itself (note this is NOT a quantale morphism, it’s just a sup-lattice morphism from Q to Q^op).

These residuation morphisms have nice properties.

Closure operation

(-)^{**} is a \mathsf{Quant}_\mathcal{E} endomorphism and closure (idempotent, increasing) operation.

The most important thing we’ll do with a distinguished point is apply the associated closure. It’s going to be important.

1. Internally, this is \chi_R \cdot \bigvee where R\rightarrowtail Q \times Q is given by \langle q_1,q_2\rangle\mapsto q_1\otimes q_2\leq_\mathcal{Q}I

Quantales internal to a topos (4/4)

A sequence of forgetful functors:

Here is a sequence of forgetful functors. We start with the category of quantales internal to our topos. There is a forgetful functor from pointed quantales. We then could define a wide subcategory including just the morphisms compatible with the closure operation.

Then it turns out that the full subcategory where the closure operation is the identity is the category of internal of Girard quantales (these are thin, cocomplete, star-autonomous categories).

Girard quantales are a reflective subcategory

Girard quantales are a reflective subcategory of pointed quantales with continuous maps

(I can explain more with a hidden slide, but let’s move on for now).

Extra: Pointed quantales and residuation

\begin{align*} (-)^*\in\ &\mathsf{Sup}_\mathcal{E}(\mathcal{Q},\mathcal{Q}^{\rm op}) :=& a\mapsto \bigvee\{b\ |\ a\otimes b\leq I\}\\ j(-) \in\ &\mathsf{Quant}_\mathcal{E}(Q,Q) := &(-)^{**} \end{align*}

Given any pointed quantale in our topos, we can define a residuation (or internal hom) which is an internal Sup-lattice morphism to the opposite suplattice. If we do this twice, however, we get a quantale morphism.

When we epi-mono factorize this, we get a quantale that has some interesting properties.

Dashed lines are supp-lattice morphisms, solid lines are quantale homomorphisms

The free sequent-set functor

F^\vdash \colon \mathcal{E}\to \mathsf{Quant}_\mathcal{E}

E.g. in \mathsf{Set}, it sends a set X to the set of assignments of X-sequents¹ as good or bad.

So hold that thought on the various kinds of quantales. Our other main ingredient is this functor which sends an object of a topos to the power object of its object of sequents, which are pairs of (internal) multisets on that object.

So, in Set, this sends a set of propositional atoms to subsets of the set of subsets all propositional sequents you could build out those elements. So, all possible ways you could partition the space of sequents into good ones and bad ones.

F^\vdash a left adjoint

All three composite functors of the free sequent-set functor are left adjoints, so the overall functor is one too. Obviously the free commutative monoid and free join completion are, but so is the doubling functor (which is left adjoint to itself).

1. We’ve baked in the exchange structural rule by using commutative monoids.

Extra: The free sequent-set functor … is a left adjoint

F^\vdash \colon \mathcal{E}\to \mathsf{Quant}_\mathcal{E}\qquad U^\vdash \colon \mathsf{Quant}_\mathcal{E}\to \mathcal{E}

All three composite functors of the free sequent-set functor are left adjoints, so the overall functor is one too. Obviously the free commutative monoid and free join completion are, but so is the doubling functor (which is left adjoint to itself).

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The (co)unit might seem kind of messy at first…

\begin{align*} \eta^\vdash_X(x) &= \langle \{([x],0)\}, \{(0,[x])\} \rangle\\ \varepsilon^\vdash_X(S) &= \bigvee\big\{\textstyle\bigotimes_j a^+_j\otimes\bigotimes_k b^-_k \ \big|\ \langle\sum_j(a^+_j,a^-_j),\sum_k(b^+_k,b^-_k)\rangle\in S\big\} \end{align*}

I think these (co)unit formulas are quite intimidating and maybe even suspicious at first glance. In particular, if “S” is a set of pairs of multisets of pairs of elements, it seems to be discarding about half of its data (the negative “a” stuff and the positive “b” stuff).

Extra: The free sequent-set functor … is a left adjoint

F^\vdash \colon \mathcal{E}\to \mathsf{Quant}_\mathcal{E}\qquad U^\vdash \colon \mathsf{Quant}_\mathcal{E}\to \mathcal{E}

The (co)unit might seem kind of messy at first…

\begin{align*} \eta^\vdash_X(x) &= \langle \{([x],0)\}, \{(0,[x])\} \rangle\\ \varepsilon^\vdash_X(S) &= \bigvee\big\{\textstyle\bigotimes_j a^+_j\otimes\bigotimes_k b^-_k \ \big|\ \langle\sum_j(a^+_j,a^-_j),\sum_k(b^+_k,b^-_k)\rangle\in S\big\} \end{align*}

… but the oddness comes from the doubling self-adjunction:

\mathsf{CMon}(X^2,Y)\cong \mathsf{CMon}(X,Y)^2\cong \mathsf{CMon}(X,Y^2)

\begin{align*} \eta^{\rm Dbl}_X(&x) &=&\ \langle (x,0), (0,x) \rangle\\ \varepsilon^{\rm Dbl}_X(\langle (a,b)&,(c,d)\rangle) &=&\ a+d \end{align*}

But this is just because the doubling functor adjunction isn’t very familiar to us. We’re used to free monoids and join completions, but the the doubling functor happens to have these units and counits.

Internal implication frames (1/4): definition

An object of \mathsf{IF}^{\rm cont}_\mathcal{E} is a pair (X,I)

• An object X \in \operatorname{Ob}\mathcal{E}
• A sequent sub-object I \rightarrowtail \mathbb{N}[X]^2\quad (i.e. 1\to F^\vdash X)

Now we can take a pullback of two of the functors we’ve seen so far to define the category of something called an implication frames internal to our topos. An implication frame (what the philosophers called it) is an object equipped with a distinguished subobject of its object of sequents. So in Set this is a set (of propositional atoms) equipped with a subset of all the possible pairs of multisets one could build.

This is precisely the toy model of pragmatic reason relations: there is a norm governing the assertions and denials of various claimables, but no structure is antecedently presumed about this pragmatic consequence relation.

Internal implication frames (2/4): example frame in \mathsf{Set}

Some assumptions about a frame (X,I) to make it finitely representable:

X=\{a,b\}

Multiplicity doesn’t matter for whether or not a sequent is good or not.

\begin{array}{||c||c|c|c|c||} \hline\hline \mathcal{P}(X)^2 & 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,

\begin{array}{||c||c|c|c|c||} \hline\hline I & 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. in this frame:

a\vdash a

\quad \vdash a

b\nvdash a

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’.

Extra: Why represent radically substructural relations?

Logical consequence relations are a source of $ $ relations. Common assumptions:

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

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”

Ordinary language transitivity violation:

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

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

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.

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

Internal implication frames (3/4): free pointed quantale

The pullback of a left adjoint along a fibration is itself a left adjoint, so \pi_{\mathsf{PQ}} is a left adjoint!

So we’ll use the fact that being a left adjoint is preserved by a pullback with a fibration.

This means that our pullback projection pi-PQ is a left adjoint as well.

Internal implication frames (4/4): free Girard quantale

An object of \mathsf{IF}^{\rm cont}_\mathcal{E} is a pair (X,I)

• An object X \in \operatorname{Ob}\mathcal{E}
• A sequent sub-object I \rightarrowtail \mathbb{N}[X]^2\quad (i.e. 1\to F^\vdash X)

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Compose free pointed quantale with \mathsf{GQ} reflective subcategory adjunction:

The pullback projection to pointed quantales is also a left adjoint, so we can compose the adjunction with the Girard quantale reflective subcategory to get the overall adjunction which I claim is the mathematical core of the philosopher’s pragmatics-to-semantics pipeline, which they had uncovered via different means.

Semantic consequence relation

The unit is \eta_\mathcal{X}\colon \mathcal{X}\to \hat{\mathcal{X}} where \hat{\mathcal{X}}:=(\hat{X},\hat{I}) is the corresponding semantic frame.

\hat X = \{(A,B)\ |\ A,B \in F^\vdash X, A=A^{**}, B=B^{**}\}

The key thing is the unit of the adjunction. We think of the unit as sending a frame to its implicit semantic counterpart.

The elements of the semantic frame are pairs of elements of the free sequent-set object, which has elements which are subobjects of pairs of multisets. Importantly these are the closed such sequent-sets. The philosophers call these “implicational roles”. It’s a kind of equivalence class which identifies things which can be substituted for each other without changing the goodness of an inference in any context.

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Semantic values are pairs of sub-sequent objects (premisory role and conclusory role)

• Only the former is relevant to consequence when the value is a premise
• Only the latter is relevant to consequence when the value is a conclusion

Example semantic consequence between two arbitrary premises + conclusions

\langle a_+,a_-\rangle,\langle b_+,b_-\rangle \vDash \langle c_+,c_-\rangle,\langle d_+,d_-\rangle \iff a_+\otimes b_+\otimes c_-\otimes d_- \subseteq I

Why pairs of values? We see from the formula for semantic consequence that only the premisory role matters when the semantic value is a premise and only the conclusory role matters when it is a conclusion.

Semantic consequence relation: conservativity

The unit is \eta_\mathcal{X}\colon \mathcal{X}\to \hat{\mathcal{X}} where \hat{\mathcal{X}}:=(\hat{X},\hat{I}) is the corresponding semantic frame.

\hat X = \{(A,B)\ |\ A,B \in F^\vdash X, A=A^{**}, B=B^{**}\}

---

The unit is a \mathcal{E} morphism, thought of as assigning semantic values.

Property of conservativity

\Gamma \vdash \Delta \iff [\![ \Gamma ]\!]\vDash [\![ \Delta ]\!]

We’ve faithfully embedded our pragmatics into a ‘semantic’ space which is quite hard to do since you saw how wildly unstructured the starting implication frame is allowed to be.

Defining operations on the semantic space

For any semantic frame (\hat X, \hat I) we can intelligibly apply the following operations:

Formula	Interpretation	Formula	Interpretation
[\![ \neg A ]\!]	\langle \texttt{a}_-,\texttt{a}_+\rangle	[\![ A\wedge B ]\!]	\left\langle \begin{matrix} \texttt{a}_+ \otimes \texttt{b}_+, \\ \texttt{a}_-\wedge \texttt{b}_- \wedge (\texttt{a}_-\otimes \texttt{b}_-) \end{matrix} \right\rangle
[\![ A\otimes B ]\!]	\langle \texttt{a}_+\otimes \texttt{b}_+,\texttt{a}_+\mathop{\mathrm{\raisebox{-0.2ex}{⅋}}}\texttt{b}_+\rangle	[\![ A\oplus B ]\!]	\langle \texttt{a}_+ \vee \texttt{b}_+,\, \texttt{a}_- \wedge \texttt{b}_- \rangle

For any implication frame, here are some things you can do with the elements. Firstly, you can always swap the premisory and conclusory role (an operation you might wish to call negation).

Since we have all sorts of operations we can do on the elements of the free Girard quantale, we have lots of other kooky things we concoct do with pairs of Girard quantale elements.

Yet these ones aren’t arbitrary. One thing I’ll forgo is an explanation of how to ‘naturally’ derive these semantic clause formulas. There is a story there though, as I describe in my ACT26 paper and I think that is one of the main upshots of reconstructing the formalism from this lens.

We obtain other operations by DeMorgan duals:

• classical disjunction \quad \quad [\![ A\vee B ]\!]=[\![ \neg (\neg A \wedge \neg B) ]\!]
• multiplicative disjunction \ [\![ A\mathop{\mathrm{\raisebox{-0.2ex}{⅋}}}B ]\!]=[\![ \neg (\neg A \otimes \neg B) ]\!]
• additive conjunction \quad \quad [\![ A\& B ]\!]=[\![ \neg (\neg A \oplus \neg B) ]\!]

Moreover, we can obtain more operations via De Morgan duals.

Special kinds of implication frames

The semantic space is often rich enough to interpret flavors of propositional logic.

So we were pinned on one end by wanting to faithfully represent the radically unstructured input frame, but on the other end we want something that is nicely structured enough in order to interpret a variety of flavors of propositional logic. On this slide I’ll show we can do just that.

Reflexive implication frames

• contain all sequents x \vdash x
• Semantic consequence is supralinear

Formula	Interpretation
[\![ A\otimes B ]\!]	\langle \texttt{a}_+\otimes \texttt{b}_+,\texttt{a}_+\mathop{\mathrm{\raisebox{-0.2ex}{⅋}}}\texttt{b}_+\rangle
[\![ A\oplus B ]\!]	\langle \texttt{a}_+\vee \texttt{b}_+,\texttt{a}_+\wedge \texttt{b}_+\rangle

For example, when we restrict our attention to frames where it’s always the case that x |- x, we see that (with the interpretations here) that consequence is supralinear. And it is exactly MALL when reflexive sequents are the only good sequents.

Indefeasibly-reflexive + contractive frames

• contain all sequents \Gamma, x \vdash x, \Delta
• \Gamma, x \vdash \Delta\iff \Gamma,x,x\vdash \Delta
• Semantic consequence is supraclassical

Formula	Interpretation
[\![ A\wedge B ]\!]	\left\langle \begin{matrix} \texttt{a}_+ \otimes \texttt{b}_+, \\ \texttt{a}_-\wedge \texttt{b}_- \wedge (\texttt{a}_-\otimes \texttt{b}_-) \end{matrix} \right\rangle

Note: these supra-linear/classical frames need not satisfy monotonicity or even transitivity!

For example, when we restrict our attention to frames where it’s indefeasibly the case that x |- x and multiplicity doesn’t matter, we see that (with the interpretations here) that consequence is supraclassical. And it is exactly the classical logic consequence relation when reflexive sequents are the only good sequents.

Takeaway: look at the structure of your domain you’re trying to elucidate with semantic analysis. Depending on its features, some logics may be fit/unfit for purpose.

This is nice because we have a test for taking our domain of interest and picking an appropriate logic for modeling it. It’s kind of obvious, but if you ever have any inferences whose goodness relies on multiplicity somewhere, obviously classical logic is not suited for what you’re trying to do and you should reach for something closer to linear logic.

The book talks more about all sorts of substructural logics (e.g. K3, ST) which likewise can be characterized in terms of features of the implication frame. One thing that is interesting is that linear logic is appropriate just in virtue of satisfying reflexivity: you don’t need cut!

There are a wide variety of supralinear consequence relations which linear logic is suitable for describing - many of these have cut violations because the sentences have genuine non-logical inferences that can be drawn from, not merely the formal inferences.

Extra: 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_-)^* \\ {\scriptscriptstyle \iff\hspace{-3mm}}&& \texttt{a}_- &\subseteq (\Gamma_+\Delta_-)^* \\ {\scriptscriptstyle \iff\hspace{-3mm}}&& \pi_1(\langle \texttt{a}_-,\texttt{a}_+\rangle) &\subseteq (\Gamma_+\Delta_-)^* \\ \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_-)^* \\ {\scriptscriptstyle \iff\hspace{-3mm}}&& \texttt{a}_+ &\subseteq (\Gamma_+\Delta_-)^* \\ {\scriptscriptstyle \iff\hspace{-3mm}}&& \pi_2(\langle \texttt{a}_-,\texttt{a}_+\rangle) &\subseteq (\Gamma_+\Delta_-)^* \\ \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{\raisebox{-0.2ex}{⅋}}}\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.

Extra: Idempotent Example (1/3)

Let \mathcal{X}:=(X,I) where X=\{a,b\} and I 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 I & 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 (-)^* computation

\begin{array}{||c||c|c|c|c||} \hline\hline \{a^+\}* & 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 singletons (\{-\})^*:

\begin{array}{||c||c|c|c|c||} \hline\hline (\{-\})^* & 0 & a^- & b^- & a^-b^- \\ \hline\hline 0 & I & 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]

(DON’T ACTUALLY DO THE EXAMPLE, JUST SHOW IT EXISTS)

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.

Extra: Idempotent Example (2/3)

Now we can derive more inferential roles in \mathfrak{G} 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 (-)^{**} & 0 & a^- & b^- & a^-b^- \\ \hline\hline 0 & X_b & I & X_\mp & X_\bot \\ \hline a^+ & X_\mp & I & 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 & I & 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 & I & X_\pm & X_\mp & \top \\ \hline % 2 I & X_b & I & I & 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 & I & X_\pm & X_\mp & \top \\ \hline\hline X_b & X_b & X_\bot & I & X_\pm & X_\mp & \top \\ \hline % 1 X_\bot & X_\bot & X_\bot & X_\bot & X_\bot & X_\bot & X_\bot \\ \hline % 2 I & I & X_\bot & I & 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.

Extra: Idempotent Example (3/3)

We have base cases:

[\![ a ]\!]=\langle \{a^+\}^{**},\{a^-\}^{**} \rangle=\langle X_\mp,I \rangle \qquad [\![ b ]\!]=\langle \{b^+\}^{**},\{b^-\}^{**} \rangle=\langle X_\pm,X_\mp \rangle

We can use the formula for semantic consequence to show that: {[\![ a ]\!],[\![ b ]\!]\vDash [\![ a\wedge b ]\!]}

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

\mathcal{X} is indefeasibly reflective+contractive, so \vDash is supraclassical (but not monotonic).

This fact about the property of the frame being indefeasibly reflexive 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.

Extra: 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}

When restricted to indefeasibly reflexive, contractive frames, the semantics is sound and complete for NMMS (Non-monotonic, multisuccedent logic).

Computer implementation

These calculations can be hairy but can be mechanized: ROLE.jl

"""
a = 'Zazzles the cat has four legs', 
b = 'Zazzles the cat lost a leg'

|     |   | a | b | a,b |
|-----|---|---|---|-----|
|     | ✓ | ✓ | × |   ✓ |
|   a | × | ✓ | × |   ✓ |
|   b | × | × | ✓ |   ✓ |
| a,b | ✓ | ✓ | ✓ |   ✓ |
"""
C = ImpFrame([[]=>[:a], []=>[:a,:b], [:a,:b]=>[]], [:a,:b]; containment=true)
𝕒, 𝕓 = contents(C)
∅ = typeof(𝕒)[]               # empty list of contents
@test ∅ ⊩ (((𝕒 → 𝕓) → 𝕒) → 𝕒) # pierce's law
@test ∅ ⊮ ((𝕒 → 𝕓) → 𝕒)       # not pierce's law

Especially when considering contractive frames (finite set of candidate implications).

This is the same frame now in a computer implementation of implication frames. You can compute semantic values and then ask logically-rich queries within the corresponding semantic space. Recall that, because our example frame met the criteria for supraclassicality, theorems like pierce’s law will automatically be theorems in the semantic frame.

Technical Outline

1. Internal (pointed, Girard) quantales

 (Hidden: deriving reflector of Girard quantale reflective subcategory)

2. The ‘free sequent-set’ functor

 (Hidden: unit and counit formulas)

3. Implication frames and the logical completion

 (Hidden: why model radically-substructural reason relations?)

4. Interpreting MALL and classical logic in a frame

 (Hidden: Supralinearity proof)

 (Hidden: Nonmonotonic Multisuccedent Sequent calculus)

5. Nominal sets to model subsentential structure

(23:30)

I would describe everything up to this point as “applied category theory of the first kind” where you take something somebody has already done and redescribe it in the language of category theory. This can be elucidating and helpful and various ways (e.g. we can now summarize the pragmatics to semantics pipeline in a pithy sentence, we can now justify their choice of the semantic values (of pairs of sets of pairs of multisets), a semantic consequence relation which seems to ignore half of its input, and semantic clauses which seem scary at first for something as simple as classical conjunction.

However, ACT of the second kind, you solve a problem that the domain expert knew about before they met you.

The category of nominal sets

Two ways of looking at \mathsf{Nom} the category of nominal sets:

Nominal sets as actions

Let \mathbb{A}=\{a,b,c,...\} be a countably infinite set.

A nominal set is a finitely-supported \rm Perm\ \mathbb{A} action.

• Each element x \in X has a finite support, the variables it depends on.
• If the support is fixed by some permutation \pi, then \pi’s action fixes x

There are two different / equivalent ways to view the category of nominal sets. The first considers them as a group action from permutations of a countably infinite set A of names. The constraint is that each element is unchanged by permutations that fix some finite set of names, thus we have a ‘support’ function which tells us, for any element, which names it ‘actually’ depends on.

This is not just a category but a topos. We’ll see later what the power objects look like.

Nominal sets as copresheaves

Let \mathbb{I} be the category of finite sets and injective maps.

A nominal set is a pullback-preserving functor \mathbb{I}\to\mathsf{Set}.

We can also treat nominal sets as Set-valued functors from the category of finite sets and injections, ones that happen to preserve pullbacks.

\Delta \mapsto \{\text{subset of $X$ which is supported by $\Delta$}\}

To connect this to the previous representation, we think of the functor as sending a context ∆ to the elements of X which are supported by ∆.

Also, if you’re familiar with combinatorial species: the objects of Nom are essentially in bijection with combinatorial species; it’s just that the morphisms are more flexible.

Extra: Elements of nominal sets

Let X \in \mathsf{Nom} and Q_{ab} \in X be an element with support \{a,b\}.

• this is not necessarily the same thing as a syntax term application Q(a,b)
  • \sigma_{ab}\bullet Q_{ab} \ \ \ \ \ \ = Q_{ba}\ \ \ \ \ \overset{?}{=}Q_{ab} (in general, may or may not be equal)
  • \sigma_{ab}\bullet Q(a,b)= Q(b,a)\ne Q(a,b) (terms are free nominal set elements)¹

This is more and less expressive than standard predicate logic syntax:

• Can represent unordered predicates
• Cannot represent Q(a,a) — this might as well be an element Q'(a) with support \{a\}
• Substitution only defined for free variables (or permutations)

So the elements of nominal sets have a set of names they depend on. It’s easy to think of an element of a nominal set as a predicate applied to some variables, as in FOL; however, this can be misleading. Rather it’s always a quotient of such term: nonidentity permutation of the variables always gives a distinct term. This lets us be still quite abstract about what these elements actually are, beyond the fact that they depend on variables and that variables can be permuted.

1. Terms with distinct arguments.

What is an implication frame internal to \mathsf{Nom}

It is a nominal set X equipped with a sub-nominal I\rightarrowtail \mathbb{N}[X]^2.

• “Claimables” are now allowed to depend on parameters: e.g. \psi, P_a,P_b,Q_{ab}

• Sequent components can share variables: e.g. P_a \vdash Q_{ab} and Q_{ba},P_a \nvdash Q_{ab}

So now let’s instantiate our general definition of an implication frame in this specific topos.

A nominal implication frame now has extra structure on its set of propositional atoms. We know what (finite set of) parameters each atom depends on, and we know how non-merging substitutions act on the set of atoms. The sequents you can build out of these atoms can share variables, and the frame picks which sequents are the good sequents.

I still picks out the subset of good sequents, but the choice must be equivariant:

• label names cannot matter: P_a \vdash Q_{ab} \iff P_b \vdash Q_{ba} \iff P_a \vdash Q_{ac}

• notation to help with this: P_- \vdash Q_{-=}

However, now the category Nom imposes a guard rail: our choice of the good sequents can’t depend on the choice of variable names.

What is the free Girard quantale in \mathsf{Nom}

Definition: finitely-supported subset of a nominal set

A subset of nominal set is supported by \Gamma if it is unchanged by permutations that fix \Gamma, applied pointwise to the elements of the subset.

• Any nominal subset supported by \varnothing because it’s equivariant.
• The subset \{P_a,P_b\} is not equivariant but it is finitely supported by \{a,b\}.
• The infinite subset \{P_a,P_c,P_d,...\} is not finitely supported.
  

In order to talk about the elements of a power object in Nom, we need to define what it means for a subset to be finitely supported. (read def) These are precisely the elements of a power object in Nom.

---

Let \mathfrak{G} be the free internal Girard quantale for (X,I) \in \mathsf{IF}_{\mathsf{Nom}}.

• These are the (-)^{**}-closed f.s. subsets of \mathbb{N}[X]^2.

So G has closed, finitely-supported subsets as its elements and this has Girard quantale structure. We could just ignore the support structure and treat these nominal sets as sets, and that would be enough to recover the propositional story we covered earlier.

We need more than just Girard structure on \mathfrak{G} in order to define [\![ \forall a\colon \Phi(\Gamma,a) ]\!] on \mathfrak{G}^2 elements.

MALL Hyperdoctrine

If we view \mathfrak{G} from the indexed perspective, we have a functor \mathfrak{G}\colon \mathbb{I}\to \mathsf{GQ}.

We do have more structure which is transparent when viewing the nominal set G as a copresheaf.

---

This is the data of a MALL hyperdoctrine if some other properties obtain:

• Left/right adjoints for each \Gamma \rightarrowtail \Delta

• Frobenius reciprocity

• Beck-Chevalley

We almost have the structure of a hyperdoctrine, though the Beck-Chevalley condition is not guaranteed by default (I think of this in analogy to how reflexivity was required for the semantic clauses of MALL to behave as expected). Let’s approach these components one by one.

MALL Hyperdoctrine: adjoints

Let S \in \mathfrak{G}(\Gamma+\{a\}) and let \iota\colon \Gamma\hookrightarrow\Gamma+\{a\}.

So you can go from a role in a large context to a small context. This particular role which depends on Gamma and a has a symmetric copy for every substitution of a (with a variable outside of Gamma). So we can consider the meet of these and their join. On right this is taking the largest thing in the sublattice below and the smallest thing in the sublattice above.

Formula for right adjoint to inclusion of Girard quantales: \forall^{a}_\Gamma(S)=\bigcap_{b \notin \Gamma} S[a:=b]

The formula for the right adjoint is here. It’s just intersection which is the meet in the overall Girard quantale lattice. Likewise the left adjoint is just the join.

Extra: MALL Hyperdoctrine: adjoints

Let S \in \mathfrak{G}(\Gamma+\Delta) with \iota\colon \Gamma\hookrightarrow\Delta.

\forall^\Delta_\Gamma(S)=\bigcap_{b \notin \Gamma} S[a:=b] \qquad \text{ and } \qquad \exists^\Delta_\Gamma(S) =(\bigcup_{b \notin \Gamma} S[a:=b])^{\bot\bot}

\mathfrak{G}(\iota) \dashv \forall^\Delta_\Gamma      i.e.      y \leq_{\mathfrak{G}(\Gamma)} \forall^\Delta_\Gamma S \iff \mathfrak{G}(\iota)(y) \leq_{\mathfrak{G}(\Gamma+\Delta)} S

\small \begin{align*} y &\leq_\Gamma \forall^\Delta_\Gamma(S) && \\ &\iff y \leq \textstyle\bigcap_{\pi \in G_\Gamma} \pi S && \text{definition}\\ &\iff \forall \pi:\ y \leq \pi S && \text{universal property of }\textstyle\bigcap\\ &\iff \forall \pi:\ \pi^{-1}y \leq S && \phi_{\pi^{-1}}\text{ order-auto.},\ \phi_{\pi^{-1}}(\pi S)=S\\ &\iff \forall \pi:\ y \leq S && y\in \mathfrak{G}(\Gamma),\ \text{so }\pi^{-1}y = y\\ &\iff y \leq S && \text{independent of }\pi,\ \operatorname{id}\in G_\Gamma\\ &\iff \mathfrak{G}_\mathcal{Q}(\iota)(y) \leq_{\Gamma+\Delta} S && \text{weakening is the inclusion} \end{align*}

I’ve included a proof.

MALL Hyperdoctrine: Beck-Chevalley

Let \iota\colon \Gamma\rightarrowtail \Delta and S \in \mathfrak{G}(\Gamma+\{a\}).

The Beck-Chevalley condition¹ is to the right:

Frobenius reciprocity holds automatically, where things get interesting is Beck Chevalley.

This property holds if the subobject I in (X,I\rightarrowtail \mathbb{N}[X]^2) is substitution-equivariant.

Although nominal sets don’t naturally support arbitrary substitution, we have something close when there is an internal monoid structure on the set.

1. In contexts with arbitrary substitution, B.C. can be described as: (\forall y\colon \phi(x,y))[x{:=}t] = \forall y\colon (\phi(x,y)[x{:=}t]). This is for \forall; there is an analogous condition for the \exists.

Substitution-equivariance (1/2)

Substitution equivariance of a monoid subobject in \mathsf{Nom}

A \rightarrowtail M, is substitution-equivariant iff closed under identifying names in the following sense. Suppose \{a,b\}\subseteq \Gamma with let x_a \in M(\Gamma\setminus\{b\}) and y_b \in M(\Gamma\setminus\{a\}) and x_ay_b \in A(\Gamma). Substitution-equivariance means that x_ay_a \in A(\Gamma\setminus\{b\}), where y_a:=y_b[b{:=}a].

Let me define what it means for a subobject to be closed under arbitrary substitutions. This only makes sense when we’re talking about a subobject of a monoid.

Extra: Substitution-equivariance (1.5/2)

Substitution equivariance of a monoid subobject in \mathsf{Nom}

A \rightarrowtail M, is substitution-equivariant iff closed under identifying names in the following sense. Suppose \{a,b\}\subseteq \Gamma with let x_a \in M(\Gamma\setminus\{b\}) and y_b \in M(\Gamma\setminus\{a\}) and x_ay_b \in A(\Gamma). Substitution-equivariance means that x_ay_a \in A(\Gamma\setminus\{b\}), where y_a:=y_b[b{:=}a].

---

Substitution equivariance example and nonexample

Consider \mathbb{A} as a nominal set. Then \mathbb{N}[\mathbb{A}] is the nominal set of multisets of names.

• Let A \rightarrowtail \mathbb{N}[\mathbb{A}] pick out all multisets which mention some a\in\mathbb{A} at least twice.

         ✅: \{c^2,a,b\}=\{c,a\}+\{c,b\} \in A(\{a,b,c\}) and \{c,a\}+\{c,a\} \in A(\{a,c\})

• Let B \rightarrowtail \mathbb{N}[\mathbb{A}] be the complement of A.

        ❌: \{a,b\} = \{a\} + \{b\} \in B(\{a,b\}) and \{a\}+\{a\} \notin A(\{a\}).

Here’s a simple example:

Substitution-equivariance (2/2)

Substitution equivariance of a monoid subobject in \mathsf{Nom}

A \rightarrowtail M, is substitution-equivariant iff closed under identifying names in the following sense. Suppose \{a,b\}\subseteq \Gamma with let x_a \in M(\Gamma\setminus\{b\}) and y_b \in M(\Gamma\setminus\{a\}) and x_ay_b \in A(\Gamma). Substitution-equivariance means that x_ay_a \in A(\Gamma\setminus\{b\}), where y_a:=y_b[b{:=}a].

---

Substitution equivariance nonexample in implication frames

Consider Roberts Rules of Order, which states a motion must be seconded for the motion to be considered. We encode this norm in an implication frame.

{\rm Motion}(a,m),{\rm Second}(b,m)\vdash {\rm Considered}(m)

This is an element of I. But we do not want the following in I, which is required by sub. equivariance:

{\rm Motion}(a,m),{\rm Second}(a,m)\vdash {\rm Considered}(m)

Therefore we should allow the possibility of frames which are not substitutionally-equivariant.

Here’s an example from of an implication frame whose subobject I of good implications is not substitutionally equivariant. First, let’s make sure we understand the example that its choice of good and bad implications makes sense (READ EXAMPLE).

Now, we can see that it’s not substitutionally invariant by noting that when we apply a permutation to some subpart of the overall sequent, we can take ourselves from a good sequent to a bad one.

So, insofar as we want to model a language where inequality of parameters can be significant to the meanings of things, then we cannot bake this in. We must be content to have an ‘almost’ hyperdoctrine in general.

Interpretations of quantifiers¹

For the semantic clause, let [\![ \phi(\Gamma;a) ]\!] have \texttt{p} as a premisory role and \texttt{c} as a conclusory role.

[\![ \forall a\colon \phi(\Gamma;a) ]\!]:=\langle \forall^a_\Gamma(\texttt{p}),\ \exists^a_\Gamma(\texttt{c}) \rangle \quad \equiv\quad [\![ \mathop{\&}\limits_{b \notin \Gamma} \phi(\Gamma;a)[a:=b] ]\!]

So, it doesn’t matter whether or not we have a hyperdoctrine. The upshot of that analysis was that we do have adjoint operations on the elements of the free Girard quantale, and we can use those as building blocks for semantic clauses for the universal and existential quantifiers.

The following semantic clause for \forall a\colon \phi(\Gamma;a) would be tantamount to (if such infinite conjunctions were legitimate formulas). As the semantic clause for & has meet of premisory roles on the left and join of conclusory roles on the right.

Now we can logically say what it means for a candidate implication to be in I:

a_1,...,a_n\vdash_\Gamma b_1,...,b_m \quad \iff\quad \vDash \forall \Gamma\colon (a_1 \otimes ... \otimes a_n)\multimap (b_1 \mathop{\mathrm{\raisebox{-0.2ex}{⅋}}}... \mathop{\mathrm{\raisebox{-0.2ex}{⅋}}}b_m)

Now we can see what is the logical sentence in the semantic frame which is true iff a particular candidate implication is pragmatically a good inference. If you feel you have more of a grip on the semantics, then this helps you understand what you are doing by a pragmatic commitment, though the explanation could run the other direction too.

The following rules depend on Beck Chevalley:

One of the upshots in the propositional story was that we could show that our choice of the logical connectives validated the inference rules for a wide variety of logics (provided the frames meet such and such structural conditions). What can we say about the sequent rules that these quantifiers obey? This is something I’ve yet to really unpack. In the case where we have this substitutional equivariance, we obey state these rules, but it’s certainly a different kind of rule from the typical sequent rules for the quantifiers, namely because we don’t have a general notion of substitution (in nominal sets you can only substitute in general for free variables).

I know there is literature about “freshness quantifiers” which I’ll need to look into and potentially the connection is better drawn there.

1. We let the semantic interpretation of \exists be determined by De Morgan duality.

Next steps

How to handle substitution more broadly?

• “Renaming sets” (subcategory of [\mathbb{F},\mathsf{Set}]) (Gabbay and Hofmann 2008)

Ok that’s my present understanding of this new generalization to quantifiers.

This issue of arbitrary substitutions is a big next step.

How to draw more connections to existing techniques in categorical logic.

It’s still not obvious how to connect this work to traditional categorical logic. We have for example, a supraclassical consequence relation under certain circumstances, but even then we don’t have a boolean algebra.

Computational implementation: substructural knowledge bases

• Work with finite presentations of implication frames
• Use the implicit logical structure to build / query them
• Use the category of them to collaboratively / compositionally manipulate them

There is work in progress towards generalizing the computational aspects of this to the nominal case, which involves lots of computational group theory but seems quite tractable overall, at least for small frames. Leveraging the compositionality of these things (the category of frames) will likely be significant in getting good performance.

A vision for neurosymbolic AI

We want AI systems to have some kind of intelligible, interrogatable model.

At the same time, the lack of imposing structure (radical flexibility of present AI architectures) is practically useful.

Compromise: imp. frames have the right balance of unstructuredness and (latent) structure.

Here is a vision for simultaneously making implication frames useable as a knowledge base backend AND for making interpretable (or auditable) AI.

Extra: Future work

Def: 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

The order codifies a kind of substitutional license: A \leq B means that conclusions can be weakened A \mapsto B and premises can be weakened B \mapsto A.

Def: 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 what \mathcal{V}-enrichment leads to:

• 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.

Conclusions: logical expressivism

We can model a norm where some claimables can be asserted and denied, and some combinations of assertions and denials are in-bounds or not.

• This is the data of a radically-substructural consequence relation.
• This is a model of the thing that we care about when we do formal modeling, including the assignment of informal concepts to formal constructs:
  • when we design formal calculi, a criterion of adequacy is that they reproduce inferences we already take ourselves to be entitled to make!

Even if we assume nothing else about the domain, we can introduce new claimables to be asserted or denied, built out of connectives of the old ones.

• This does not change the goodness of inference between the original claimables (conservative extension).

• When the norm meets some basic criteria, these connectives recover the inferential properties we expect from well known logics (MALL, classical logic).

By generalizing predicate logic, we can compactly represent an infinitude of judgments into a single sequent.

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.

References

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

Gabbay, Murdoch J, and Martin Hofmann. 2008. “Nominal Renaming Sets.” International Conference on Logic for Programming Artificial Intelligence and Reasoning, 158–73.

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.

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