Inferentialism and applied category theory

Kris Brown - August 17, 2024

Logic, Methodology of Science & its Applications

8/17/24

Two related problems: scientific communication

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What framework for cooperation doesn’t trap us in our present misconceptions?

• Need some sort of fixed structure / common protocol in order to communicate
• However: fixed structures may bake in unknown assumptions we’ll want to change

Alice and Bob study a very basic form of chemistry, and find it productive to communicate their theories and discoveries to each other via messages like “\(A\rightarrow B\), \(B\rightarrow C\), \(C\rightarrow A\)”. If everyone were to buy into this protocol, one could write software which checks things like “Is anyone else studying reaction networks with \(A\) and \(C\)?” or “Is there a pathway that gets me from \(A\) to \(D\)?”

So this can be a technical problem: whenever there is communication, there is some fixed protocol, database schema, controlled vocabulary. There are all sorts of affordances for cooperation once one obtains when one has something like this. But we’re immediately pulled in two directions: the more we hold fixed, the it seems like the more powerful things we can coordinate on. However, invariably some aspects of the things we fix we realize were mistakes, or we’ve omitted something. Then something that was liberating now becomes restrictive, constraining. Let me demonstrate this with a toy example.

(…) One particular affordance I’ll be focusing on (relevant to practical scientific computing) is that, when many people coordinate around some shared structure, one can write software to perform high-level reasoning at scale.

Two related problems: scientific communication

What framework for cooperation doesn’t trap us in our present misconceptions?

Chris also studies chemistry. He’d like to communicate with her too via messages; however, what she calls ‘A’ he calls ‘B’, and he thinks what she calls ‘B’ and ‘C’ are actually both the same thing, ‘D’. They end up not collaborating.

Alice and Bob later discover that they find it much more productive to analyze some phenomena as “multi-reactions”, where one can have multiple inputs and outputs. They could modify their code to add a new data structure: “A + B \(\Rightarrow\) C + D, A \(\rightarrow\) B”, but this will be a lot of work (so for the time being, they continue to use singular reactions).

Alternatively they could consider their old notion of ‘reaction’ to be inadequate: that they thought were reactions are ‘multi-reactions’. But the same practical problem remains.

Can we find a synthesis of structure and flexibility?

So everything is going well so far, let’s see what happens next. (…)

(…) Whether or not we think of the old framework as incorrect vs omitting something makes no practical difference.

(…) Ok, so we have an unhappy ending.

(…) The takeaway is that we need some middle way, a synthesis of these two competing intuitions: that we need both structure and flexibility.

Two related problems: semantics

How do we think about semantics in a way that’s amenable to redescription?

Compositionality: understanding each component’s meaning allows one to determine the overall statement’s meaning.

There is a related problem in semantics. I’ll talk about redescription in a second, but first let me tell you about the naive, status quo, which I’ll call representationalist semantics. The content of a proposition is determined by the contents of its components, which represent the world in a certain way.

(…) Take the example of: “The green frog is on the log.” This gets translated into a statement in some logic, often first-order logic, where we interpret certain atomic predicates as tuples of objects.

This is a powerful idea and a lot of intellectual effort has been put into making it address things such as “how do you represent ‘possibility’?” (possible-worlds semantics)

Two related problems: semantics

How do we think about semantics in a way that’s amenable to redescription?

• In order to qualify as a semantics, we need compositionality. (E.g. truth conditions)
• Semantics traditionally casts meanings in stone (fixed meanings determine proper use), whereas redecription is a process in the opposite direction.

If I can’t work, then I don’t deserve food … unless food is a human right.

If it’s a cat, then it’s a mammal…unless future scientists discover cats are actually aliens.

The electron is at position \(p\) with velocity \(v\)… but then we learn Heisenberg uncertainty.

The triangle has more than 180 degrees… but the triangle is not a planar triangle.

Can we obtain compositionality without meaning constituted by fixed truth conditions?

Let me show by example what I mean by redescription. For example, we may take the concept of ‘deserving’ to be essentially linked to a notion of exchange/markets, but later on we learn that, in fact, ‘deserving’ instead has something to do with rights. A notion of meaning as definition would say we mean something different by ‘deserves’ now, but we really we’re talking about the same thing.

(…) And it might be essential to something being a cat that it is a mammal, but if we learn from future scientists that everything-that-we-call-cats are in fact not cats. Clearly we were wrong about something, but were we wrong b/c “there never were any cats”? Or, more plausibly, we were wrong about the necessary conditions of something being a cat?

(…) Science has these Kuhnian paradigm shifts, too: e.g. redescription of mass, energy, space (etc.) from Newtonian to Einsteinian physics

• some things contingently true become essential
• some things true become approximately true (true under special circumstances)

We don’t want to say that no one in the 19th century ever actually referred to electrons when they said “electron” due to their misconceptions.

(…) But this drive to redescribe isn’t necessarily tied to some empirical experience we didn’t anticipate. Even in mathematics, we could think of a triangle as essentially a polygon with 180 internal degrees and accidentally three points connected by segments (it seems to not matter which we take to be the truth conditions) yet future mathematicians find it much easier to generalize when the latter is the definition.

(…) These examples should remind you of our first problem: we may or may not want to change the definition of a ‘reaction’ as we learn more. If we don’t have the ability to redescribe, then

• we become trapped in backwards ways of thinking
• we’re not prepared for the fact that cats are aliens
• our scientific vocabulary can’t refer to the real world.
• Mathematics becomes ossified

Again we need a middle way. Clearly we do address this problem, somehow, in practice, but we want a semantic theory that is compatible with this fact.

Outline

Two approaches to a common kind of problem

• Work in categorical logic at the Topos Institute:
  • Broaden notions of logic and semantics
  • Make shifting foundations ergonomic
  • Pluralist computational science

• Work in infentialist semantics: logical expressivism

  • Rethink the relationship between logic and reasoning
  • Open reason relations

I’m going to argue that category theory and recent work in inferentialist semantics address these common problems in different ways. I hope to gesture towards it being interesting to try to synthesize these approaches, but for this talk I’m just going to try to give cursory explanations of the problems addressed and the proposed solutions.

Background: Category Theory

Focuses on relationships between things without talking about the things themselves.

CT studies certain shapes of combinations of arrows.

• These can be local shapes, e.g. a span

• These can be global, e.g. an initial object:

• Functors \(F: \mathsf{C}\rightarrow \mathsf{D}\) are assignments objects and arrows of \(\mathsf{C}\) to those in \(\mathsf{D}\) that respect the local structure of \(\mathsf{C}\).

• Power of CT to abstract comes from the fact its definitions can be applied in any category.

• Its definitions don’t require that its subject matter be anything in particular, rather they make pre-existing structure explicit.

In a context like this, for a general audience, there isn’t enough time to do justice to the full definitions involved, but I hope to provide intuition and keyword for you to search later if you’re interested. So for present purposes, category theory is a branch of math that focuses on relationships between objects rather than objects themselves. A category has a collection of objects and morphisms (or arrows).

(…)

(…) Here I’ve drawn where the functor \(F\) sends the objects of \(C\). I’ll note CT was invented in the 1940’s to connect different branches of math. I.e. CT is ACT, and original application of CT was to math itself. E.g. we could imagine \(C\) here being algebra, and algebraic objects are systematically related to geometric objects such that we can use algebra to study geometry.

(…)

ACT and scientific pluralism: functorial semantics

In this diagram, theories \(T_1,T_2,T_3\) are categories. We specify the semantics of a theory by giving a functor from the theory to a semantics category \(S\). Let \(S = \mathbf{Set}\) for now.

• No need to be confined to a single theory (nor reducible to a single theory)
• No need to be confined to a fixed semantics¹
• A meta-theory \(\mathbb{T}\) (or doctrine) defines a space of theories
  • Higher category theory allows us to iterate this procedure as needed.

The general story for how ACT addresses the opening problem is that we understand theories as living in a space of possible theories. Let’s suppose theory is itself a category, and the kind of semantics we wish to give models of the theory is also a category. A common choice for this is the category of sets and functions. Giving a model of the theory then is taking each object of one’s theory, assigning it a set, and taking each morphism of one’s theory and assigning it a function.

This space of theories is itself a category, where the arrows are coherent ways of interpretating one theory in another. This enables us to fluidly move between different theories in a rational, non-arbitrary way.

Here \(\mathbb{T}\) is a doctrine, aka a 2-theory. We can move higher up as needed if we feel our space of theories is too restrictive; we can understand it not as a fixed thing but rather as a choice among others in a category.

This is a kind of structure without foundations in a sense. Alternatively you could call it “version control” for mathematical theories.

Now at the risk of boring you with details, I want to dig into this picture and give some concrete examples of what these theories and models can look like.

1. Such as the set of ‘real things’ / set of arrangements of subatomic particles

Example scientific model: Petri nets

Petri nets consist of species and transitions, connected by input and output arcs.

We can of these as a syntax for chemical reaction networks:

• 🔵 The species are chemical species
• 🟧 The transitions are chemical reactions
• The inputs/outputs tell you how many of each molecule is a reactant/product

The category of all Petri nets is the category of models of a particular theory.

Returning back to our original scenario, here is a mathematical specification of chemical reaction networks.

Theories for graphs and Petri nets

Let’s return to our first example. We want to redescribe our earlier notion of reaction as a one-in-one-out relationship to a many-in-many-out relationship between species.

Reaction networks in the former paradigm are graphs, which have two sets and two functions.

Petri nets describe the new paradigm of reaction networks. Petri nets consist in four sets and four functions.

Remember that our semantics category \(S\) was Set. So the collection of models of a given theory is going to be choosing sets for each object of the theory and functions for each arrow in the theory. Let’s start with graphs, as they are simpler.

(…)

(…)

Example: data migration of graphs to Petri nets

What we want is a way of systematically turning our graphs into Petri nets

• I.e. a functor from category of graphs to category of Petri nets
• However, involves a massive amount of data.

In order for computers to store things and for human beings to comprehend them, they must be finitely presentable. How do we represent this relationship between the (infinitely large) category of graphs and the category of Petri nets?

Functors are more than mere isomorphisms, they are required to preserve the compositional structure.

Example: data migration of graphs to Petri nets

What we want is a way of systematically turning our graphs into Petri nets

• I.e. a functor from category of graphs to category of Petri nets
• However, involves a massive amount of data.

Conceptually-legible way to packaging this: merely need a functor between the two schemas.

Functorial data migration redescribes graph-vocabulary as Petri net vocabulary.

Functorial data migration connected to Kan extensions.

Example: open Petri nets

The category of Petri nets can be given 2-dimensions of composition¹

ACT can systemically construct ‘open’ versions of models and assign functorial semantics.

Closed systems are a special case of a system with an empty boundary.

One integrates more (previously unknown) information by composing one’s model.

Open systems express constraints on future behavior but do not generally determine future behavior (once you have an empty interface, behavior could be determinate).

Closed system are unable to compose with other models of the environment we may come to discover.

1. Baez, John C., and Jade Master. “Open petri nets.” Mathematical Structures in Computer Science 30.3 (2020): 314-341.

ACT: key points

• CT works with external structure rather than internal structure
• Categories use universal properties to make a category’s pre-existing structure explicit
  • Logic is interpreted within any category if it has enough structure for it (pluralism)
• Emphasis on open systems rather than closed systems
• One can reason within a theory while retaining flexibility
• Syntax and semantics are both categories, rather than two different kinds of things
• Mathematically-justified notions of redescription, e.g. functorial data migration

There are various of ways CT can make sense of redescription as a rational process, but one particular one I showed was functorial data migration.

Inferentialism

Inferentialist theories of meaning prioritize inferential role over reference in the order of explanation.

An ongoing practice of giving and asking for reasons is the starting point, rather than than the end point.

What and means is captured entirely by its circumstances of application and consequences of application. No account of reference needed.

Can we do this with our ordinary concepts, too?

I’m going to now switch to the second topic. ‘Inferentialism’ is a broad class of theories that generally try to understand reference in terms of a conceptually prior notion of inferential role. This isn’t in direct conflict with the earlier picture of “the frog on the log” as representational: we still can and ought think of it that way. However, in terms of an overarching view of propositional content, inferentialists will understand representational relations in terms of, e.g., the roles ‘frog’ and ‘log’ and observational reports in general play. Rather than an atomistic picture of the concepts having some reference, which determines what sentences mean, and that determines which inferences are good, the order is reversed: given a notion of good judgment we understand what it is we’re talking about.

The paradigm case of a concept being understood in terms of inferential roles is logical vocabulary, such as “and”. Here are introduction and elimination rules for “and”. We understand “and” iff we understand these rules.

One particular strand of this comes from a book by Ulf Hlobil and Robert Brandom that has recently come out. I can’t do justice to it here, but I can try to highlight some observations that there are some similarities with the story we just heard, even though the technical details are very different.

Reason relations, implication frames

Suppose we have a base lexicon of claimables \(L_B = \{a,b,c,d,...\}\).

Reason relations among these claimables look like “\(a,b \mid\hspace{-0.1em}\sim c,d\)” (implication) and “\(a,b \mid\hspace{-0.1em}\sim\ \)” (incompatibility). The space of possible implications/incompatibilities is \(S_B = \mathscr{P}(L_B)^2\). An implication frame (or ‘vocabulary’), \((L_B, \mathbb{I}_B)\), endorses a subset \(\mathbb{I}\subseteq S_B\) of these reason relations, which Restall suggests we read as a claim of incoherence:

\[(\Gamma,\Delta) \in \mathbb{I} \text{ means rejecting all of } \Delta \text{ and accepting all of } \Gamma \text{ is incoherent}\]

The notion of a vocabulary here is very broad:

• \(a =\) “It is pious” and \(b =\) “It is loved by the gods” \(a\mid\hspace{-0.1em}\sim_{TL} b\) (some kind of theological vocabulary)

• \(x =\) “Pittsburgh is to the west of NY” and \(y =\) “NY is to the east of Pittsburgh”, then \(x \mid\hspace{-0.1em}\sim_{OL} y\) (ordinary language / material inference)

• \(a =\) “\(\forall m,n, eastOf(m,n) \iff westOf(n,m)\)”, \(b =\) “\(westof(P,NY)\)” and \(c =\) “\(eastof(NY,P)\)”, then \(a,b\mid\hspace{-0.1em}\sim_{FOL} c\) (first-order logic)

So we care about inferences, inferences between what? We start with a lexicon, a set of claimables. We can construct ‘positions’ such as “a,b \(\vdash\) c,d” out of these.

We call this a vocabulary,a language in use - there is a set of things that can be said and norms governing what are good reasons for what and what are good reasons against what.

(…) Let me showcase how broad of a concept this is: it’s part of what ‘pious’ and ‘God’, mean in the context of theological vocabulary, that this follows.

(…) Rather than this niche kind of discourse, we can also talk about our ordinary pre-theoretical notion of a good reason, what Sellars calls a ‘material inference’.

(…) Note that, from the implication frame’s perspective, these logical expressions in quotes are simply elements of the lexicon with no internal structure.

Logical expressivism

Claim: logic makes reason relations explicit: \(\Gamma \mid\hspace{-0.1em}\sim a \rightarrow b\) says that \(\Gamma, a \mid\hspace{-0.1em}\sim b\).

We can extend any base lexicon to include logically-complex claimables, e.g. \(a \land b\): what then of the reason relations between these new claimables, or implications that mix base claimables and logical ones?

A framework for talking about reason relations shouldn’t impose restrictions on what is a ‘good’ implication frame: chemists, theological scholars, relevance or paraconsistent logicians, etc. all have their own expertise. Yet many of these, while still be rational, fail to obey monotonicity and transitivity. Classical logic is not harmonious in these cases.

Example of non-harmonious rules of inference:

Logic should be LX: elaborated from a base vocabulary and expressive of it. Being universally LX means this is the case for any base vocabulary.

A motivating idea of logical expressivism is that logic makes reason relations explicit.

Now we consider logical elabrations of base lexicon, and it’s important these logical symbols are quite different from the ones on the previous slide, which were invisible to the perspective of reason relations which just deals with claimables. So, given a lexicon (which can have whatever internal structure it wants), we can add logically complex bearers like so, and we have to ask what are the good reason relations that involve these new bearers?

The sequent rules for classical logic have this tonk-like property when the base reason relation has nonmonotonicity or nontransivity. It is still worthy of being called a logic because it does play an LX role for certain kinds of reason relations; however, the next move Reasons for Logic makes is to introduce a universally LX logic.

Logicism vs logical expressivism

	Logicism	Logical expressivism
Logic ought study relations between	sentences with only logical vocabulary	sentences with logical vocabulary and sentences in a base vocabulary
Reason relations and logical vocabulary	Reason relations should be reducible to logical vocabulary	Reason relations should be expressible by logical vocabulary
Genuine nonlogical content, material inferences	Not possible (it’s just incompletely-specified)	Possible (fever, cruel, birds can fly, torture is wrong)
Search for generality	Exhibit all (doxastic) reasons as covertly logical reasons	Find a logic which is universally LX

Here I re-summarize logical expressivism by contrasting it with the converse theory, which we can call logicism.

NonMonotonic, MultiSuccedent logic

NMMS are Gentzen-style sequent calculi which conservatively extend implication frames.

This sequent calculus tells us precisely which sequents involving the new logically-extended claimables are good ones. It satisfies a number of nice properties that we expect the connectives to satisfy, it’s a conservative extension of the base reason relation (which may be non-monotonic), and if we look at sequents involving purely the logically-elaborated claimables, we recover classical propositional logic.

This isn’t a technical talk, so I won’t work through all of these, but, for example:

• if \(\Gamma \vdash \Delta\) is a reason against \(A\), then it’s a reason for \(\neg A\). In general, negation makes explicit flipping sides of the turnstile, i.e. swapping reasons for and against.
• The ‘and’ operator makes explicit the comma on the left hand side of the turnstile as a premise.
• A reason against \(A\) and against \(B\) (and their disjunction) is a reason against the conjunction.

A powerful representation theorem for NM-MS is proven which shows that this enables us to have all sequents we need in order to describe the underlying base vocabulary.

Syntax and semantics on the same playing field

Rather than arguing for truth-taking (pragmatic) to be conceptually prior to truth-making (semantic), they are shown to have a remarkable common structure:

Truth-maker theory, as developed by Kit Fine:

• generalizes possible-worlds semantics by being hyperintentional: can distinguish meanings that are still true in all possible worlds, such as \(1+1=2\) and \(isPrime(3)\).

• It does this via notions of worldly states, combining states, and impossible states.

• One can understand the meaning of \(\Gamma \vdash \Delta\) as “there isn’t any possible state that makes every sentence in \(\Gamma\) true and every sentence in \(\Delta\) false”

Every consequence relation that can be codified in an NM-MS system over a material semantic frame can also be codified in truth-maker theory, and vice versa.

You can be an inferentialist about anything as long as you have notion of compatibility.

Open reason relations

Traditionally, logical consequence is closed:

\[Cn(\Gamma) = Cn(Cn(\Gamma))\ \ \ \ \ \ \ \Gamma\subseteq\Delta \implies Cn(\Gamma)\subseteq Cn(\Delta)\]

This notion forces, from the outset, for nothing to change if we had \(\Gamma \mid\hspace{-0.1em}\sim \alpha\) and decided to add \(\alpha\) back in as a premise: if \(\Gamma, \alpha \mid\hspace{-0.1em}\sim \beta\) then it was already a conclusion all along \(\Gamma \mid\hspace{-0.1em}\sim \beta\)

But this is essential to rationality: we derive consequences which, if we accept them as our new premises, allow us to derive new things, which may invalidate earlier premises or rectify earlier contradictions. Explosion is implausible for non-logical conceptual contents.

Rational hysteresis in science: we use theories to extract consequences from data, though this data can make us change our theories (yield consequences incompatible with our premises) and then the new theory extracts even newer consequences.

A fixed implication frame can capture the steps of this evolutionary process as rational steps.

\[\Gamma, a, \mid\hspace{-0.1em}\sim b,c\ \ \ \ \ \Gamma,a,b,c \mid\hspace{-0.1em}\sim\ \ \ \ \ \ \Gamma,b \mid\hspace{-0.1em}\sim c \ \ \ \ \ \ \Gamma,c \mid\hspace{-0.1em}\sim b\]

Corfield writes about Friedman saying our practice has retrospective rationality but not prospctive (rational incommensurability is wrong).

Logical expressivism mathematizes prospective rationality by opening up space between logical and rational. ‘Logically following from’ can be a good reason for something but is not the only way to be a good reason, and in fact should be seen as parasitic on other reasons (logic makes rationality explicit, rather than constituting it itself).

Strong and weak compositionality

Traditional semantics obeys a strong compositionality property

• “Knowing every mentioned small sentence in \(\varphi\) will enable knowing complex sentence \(\varphi\).”

Inferentialism satisfies a weaker (but nontrivial) compositionality property

• “Knowing all small sentences lets you know all complex sentences.”

Definitions in implication space are recursive but not strongly compositional.

\[\mathsf{R}(\Gamma\mid\hspace{-0.1em}\sim\Delta) := \{(X,Y) \in S\ |\ \Gamma,X\mid\hspace{-0.1em}\sim Y,\Delta \in \mathbb{I}\}\]

\[ ⟦(\neg a)^+⟧ := ⟦a^-⟧ \ \ \ \ ⟦(\neg a)^-⟧ := ⟦a^+⟧\]

\[ ⟦(a \rightarrow b)^-⟧ := ⟦a^+⟧\sqcup ⟦b^-⟧\]

\[ ⟦(a \rightarrow b)^+⟧ := ⟦a^-⟧\sqcap⟦b^+⟧\sqcap (⟦a^-⟧\sqcup ⟦b^+⟧)\]

Our motivating problem was to allow redescription while preserving compositionality. Do we still have compositionality? In some sense, we do. We can give recursive definitions of the inferential role of logically complex sentences in terms of the inferential roles of the simpler ones.

The definition of the implicational role of a particular implication is the set of other implications which we can add it to in order to make it a good implication.

However this is not strongly compositional because inferential roles are defined holistically (among the base vocabulary).

Conclusions

Remarkable similarities between ACT and logical expressivism

• Focus on making pre-existing structure explicit
  • Content in terms of external structure rather than internal structure
  • Logical structure is special but not guaranteed. Pluralism about the ‘right’ logic.
  • Traditional thinking about logic can be good/useful, but not as a foundation
• Emphasis on open systems / reason relations
• Theoretically ground norms that are binding without fixed foundations
• Syntax and semantics on a par, rather than a gulf between fundamentally different things.
• A capacity to make sense of redescription

Thanks for listening!

And special thanks to:

• Kevin Carlson, David Spivak, Owen Lynch (Topos Institute)

• Ryan Simonelli (U Chicago), Ulf Hlobil (Concordia), Bob Brandom (U Pittsburgh)

To summarize, I didn’t try to defend these two approaches, rather in my limited time I hope I could show the shape of the moves they make and their similarities.

Speculative parallels: the two-language problem

	Universality: impractical and restrictive norms	Contingency: norms profilerate, fail to legitimately bind us
Protocols	Semantic web: universal ontology but illegible, inflexible, clunky	Domain-specific languages: fragmented, no interoperability, only local legibility
Semantics	Representionalism: mind as mirror of nature	Poststructuralism/regularism: language games all the way down, no reason for cooperation

Solution is to contextualize the locally-authorative norms, allow them to talk to each other.

Can only be done from a vantage point that does not presuppose the content of what it’s talking about.

ACT solves something my colleague Evan Patterson calls “the two language problem”. The Semantic Web aims to offer a universal ontology. There has been massive effort to formalize science based on Semantic Web (e.g. ontologies in biomedicine) but it has largely failed.

Conversely, domain specific languages promote legibility only at small scales and do not interoperate. There’s no reason for anyone new to the party to get in line.

I think there’s a potential parallel on the semantic side, where the universality of the idea that we are all responsible to this external world forces us to share a set of norms. But representationalism is useless to someone who wants to practically make decisions: who cares if after 50 layers of interpretation, saying “I like pie” is a statement about arrangements of subatomic particles? The level at which statements become monotonic is too low-level for practical use.