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Towards an Inferentialist Account of Information Through Proof-theoretic Semantics

This paper proposes an inferentialist account of information grounded in proof-theoretic semantics, replacing traditional truth with inferability to develop a mathematical-logical framework for the "inferon" that enables rigorous reasoning about information flow in distributed systems.

Original authors: Matthew Collins, Timo Eckhardt, David Pym

Published 2026-05-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matthew Collins, Timo Eckhardt, David Pym

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: What is Information, Really?

Imagine you are trying to build a house. You have bricks, wood, and nails, but you don't have a blueprint. You know how to stack bricks, but you don't know why they hold up or how they fit together to make a room.

The authors of this paper argue that in our modern world, we talk about "information" all the time (in computers, news, science, and daily life), but we don't really have a solid, mathematical blueprint for what information is. Without this blueprint, it's hard to understand how complex systems (like the internet or global supply chains) actually work.

Their goal is to build a new kind of blueprint. Instead of defining information by whether it is "true" in a static sense (like a fact written in a book), they want to define it by how it is used to make a deduction. They call this an "inferentialist" approach.

The Old Way vs. The New Way

The Old Way (The "Truth" Model):
Imagine a librarian who only accepts books that are factually correct. If a book says "The sky is blue," the librarian checks a giant, perfect map of the universe to see if the sky is actually blue. If it is, the book is "information." If the book says "The sky is green," it's rejected as false.

The Problem: This requires a perfect, external map of the universe (a "model") that everyone agrees on. It's hard to build that map for complex, changing systems.

The New Way (The "Inference" Model):
The authors propose a different librarian. This librarian doesn't care about a giant map of the universe. Instead, they care about rules.

Imagine you have a set of rules: "If it is raining, then the ground is wet."
If you tell the librarian, "It is raining," they can immediately infer, "Therefore, the ground is wet."
In this view, information isn't about matching a fact to reality; it's about what you can logically conclude based on the rules you have.

The authors call this new unit of information an "inferon" (short for "inferential unit"). Think of an inferon not as a fact, but as a move in a game of logic. It's a piece of information that says, "Given these rules, I can deduce this next step."

The Three Pillars of the Paper

The paper builds this new theory on three main legs:

1. The Philosophy (The "Why")

The authors look at a famous philosopher named Dretske, who said information must be about something, must be true, and must be transmissible.

The Twist: The authors keep "about something" and "transmissible," but they swap "truth" for "inferability."
Analogy: Instead of asking, "Is this statement true in the real world?" they ask, "Can I prove this statement using the rules I have right now?" If you can prove it, it counts as information. This makes information dependent on the person (or computer) doing the thinking and the rules they are using.

2. The Logic (The "How")

To make this work mathematically, they use a tool called Proof-Theoretic Semantics.

The Analogy: Imagine a giant, infinite notebook of rules called a "Base."
- Some rules are simple: "If you have a key, you can open the door."
- Some rules are complex: "If you have a key AND the door is unlocked, you can enter."
In their system, an "inferon" is a piece of information that is supported by a specific set of rules in this notebook. If you have the right rules (the Base), the information "holds up." If you change the rules, the information might change.
This is different from the old way (Situation Theory), which treats information like a snapshot of a scene in a movie. The new way treats information like a step in a recipe.

3. The Systems (The "Where")

Finally, they show how this works in real-world systems, like a network of computers or a security checkpoint.

The Analogy: Think of a Flashlight. A flashlight has a switch and a bulb.
- In the old view, you might say, "The switch is ON, so the bulb is ON."
- In their new view, they define a "channel" (a connection) between the switch and the bulb. They use rules to say: "If the switch is ON (in this specific context), then the bulb must be lit."
They apply this to a complex example: Airport Security.
- A passenger goes through check-in, baggage screening, and passport control.
- At each step, different "bases" (sets of rules) are used. The check-in agent has rules about passports; the baggage agent has rules about luggage.
- The paper shows how information flows from one station to the next not just as data, but as logical deductions. For example, "Because the passport was checked at step 1, and the rules say a checked passport allows boarding, the passenger can board at step 5."

Key Takeaways

Information is a Process: It's not a static object sitting in a database; it's an active process of reasoning.
Context Matters: What counts as information depends on the "rules" (the Base) you are currently using. A fact that is information to a doctor might not be information to a layperson because they have different rulebooks.
Agents Matter: The paper emphasizes that "agents" (people or computers) have their own specific abilities to infer. A system works when these agents can pass information to each other using shared rules.

Summary in One Sentence

The paper proposes that we stop thinking of information as "facts about the world" and start thinking of it as "steps in a logical argument," where the value of the information comes from what you can deduce from it using a specific set of rules.

Technical Summary: Towards an Inferentialist Account of Information Through Proof-Theoretic Semantics

1. Problem Statement

The paper addresses the lack of wholly convincing logical or mathematical foundations for the concept of "information," despite its centrality to modern society and informatics. Current approaches, particularly those rooted in model-theoretic semantics (such as Situation Theory), rely on truth conditions derived from external models. The authors argue that this reliance on truth (specifically the "Veridicality Thesis") and external models limits the ability to reason about complex, distributed information ecosystems. They posit a need for a framework where the meaning of information is determined by its use in inference rather than its correspondence to a model.

2. Methodology

The authors propose an inferentialist semantic theory of information grounded in Proof-Theoretic Semantics (P-tS), specifically utilizing Base-Extension Semantics (B-eS). The methodology involves three interacting components:

Conceptual Analysis (Metaphysics): The authors reinterpret Fred Dretske's metaphysics of information (Intentionality, Truth, Transmissibility). They replace the requirement of Truth with Inferability. Information is defined not as a fact that is true in a model, but as a unit derivable from a set of atomic rules (a "base").
Logic (Formalization): The paper employs P-tS to construct a mathematical-logical theory of a primitive unit of information called the inferon. This approach counterpoints the model-theoretic infon of Situation Theory (Barwise, Seligman, et al.).
- Inferons: Defined as triples $\langle\langle P, \mathcal{P}, b \rangle\rangle$ , where $P$ is an atomic proposition or predicate, $\mathcal{P}$ is a local inferonic base (a set of atomic rules), and $b$ is a polarity (assertion or denial).
- Meaning: The meaning of an inferon is derived from a global base of atomic rules and the inferon's local base, rather than a first-order model.
- Logic: The authors develop a logic of inferons based on intuitionistic logic, extending Sandqvist's B-eS for Intuitionistic Propositional Logic (IPL) and Gheorghiu's extension to first-order logic.
Systems (Application): The developed P-tS tools are applied to model distributed systems. The authors adapt the "Logic of Distributed Systems" (LDS) framework (Barwise & Seligman), replacing classifications and infomorphisms with inferonic bases and inferomorphisms.

3. Key Contributions

A. The Inferon and Inferentialist Semantics

The core contribution is the definition of the inferon as an inferentialist counterpart to the Situation Theory infon.

Structure: An inferon $\iota = \langle\langle P, \mathcal{P}, b \rangle\rangle$ consists of a proposition $P$ , a local base $\mathcal{P}$ , and a polarity $b$ .
Support Relation: Validity is defined via a support relation $\Vdash_B$ relative to a base $B$ . An atomic inferon is supported if it is derivable in the union of the global base and the local base: $\Vdash_B \langle\langle P, \mathcal{P}, b \rangle\rangle$ iff $\vdash_{B \cup \mathcal{P}} \langle\langle P, b \rangle\rangle$ .
Agency: The framework introduces agent modalities ( $[a]$ and $\langle a \rangle$ ) to represent inferential capacities. An agent's ability to support an inferon depends on the union of the ambient base and the agent's specific set of bases.

B. Logical Results

The paper establishes the formal properties of the logic of inferons:

Soundness and Completeness: The authors prove Theorem 11 (Soundness) and Theorem 12 (Completeness) for the propositional logic of inferons, adapting Sandqvist's proofs. These results are extended to the first-order case (Theorem 11 and 12 extended).
The (Inferon) Axiom: A specific axiom is introduced to handle the interaction between local bases and the global support relation, ensuring coherence between the natural deduction calculus and the base-extension semantics.
Compound Inferons: The framework allows for logical connectives within inferons, with support relations defined recursively (Figure 9).

C. Information Flow and Inferomorphisms

The paper develops an inferentialist account of information flow in distributed systems:

Inferomorphisms: Analogous to infomorphisms in LDS, an inferomorphism is a pair of functions ( $f^\vee$ on terms, $f^\wedge$ on predicates) that preserves the derivability relation across different bases.
Channels: A channel is defined as a family of inferomorphisms sharing a common core (a central base).
Information Flow: Information flow is characterized by constraints between inferons across channels. If a constraint holds in the core, it transfers information from one component to another.
Sites and Contexts: The authors introduce sites (finite sets of inferonic atoms) and a contextual support relation $\Vdash^P_B$ . This allows for modeling information flow where constraints are internalized via implication and the "Inf" rule, rather than being external semantic conditions.

4. Results and Examples

The paper demonstrates the utility of the framework through several examples:

No Smoke Without Fire: Formalizes the "attunement" to constraints as inference rules linking inferonic bases. It shows how an agent can derive information about a remote situation (fire) based on local perception (smoke) and a shared constraint, without direct access to the remote facts.
The Wise Men Puzzle: Models the epistemic puzzle using fixed inferential capabilities. Agents reason about the inability of other agents to infer, demonstrating how the framework handles reasoning about inference itself without requiring unbounded state-space exploration.
Access Control: Models a system where a password and token lead to access, illustrating how constraints can be internalized in the logic via implication.
Airport Security: A detailed case study (Figure 11) modeling the flow of information (passports, tickets, baggage) through a distributed system. It demonstrates how sites (information available at a location) and bases (reasoning capabilities) evolve and interact via channels to ensure system integrity (e.g., matching passengers to baggage).

5. Significance and Claims

The paper claims to provide a conceptually rigorous mathematical-logical account of information grounded in inference and reasoning. Its significance lies in:

Shifting the Foundation: Moving from a model-theoretic view (truth in a model) to a proof-theoretic view (derivability in a base). This aligns with the inferentialist philosophy that meaning is determined by use.
Addressing Van Benthem's Categories: The authors argue their approach primarily addresses Information-as-Correlation (via inferons and situations) but also offers pathways to address Information-as-Range (via ranges of bases) and Information-as-Code (via the inferential content of messages).
Modelling Distributed Systems: It provides a reasoning-based theory of information flow that is applicable to the organization of information processing systems, offering a tool for understanding complex ecosystems without relying on potentially unmanageable global models.
Modesty: The authors characterize this work as a "first step" towards a full inferentialist account. They acknowledge that deeper discussions on veridicality, agency, and the integration with other logical dynamics (e.g., van Benthem's work) are left for future work. They do not claim to have solved all foundational problems but have established a viable logical framework for further development.