New perspectives
on theoretical equivalence
SPEAKERS
ABOUT
Philosophers and logicians have long inquired under which conditions two theories should count as theoretically equivalent, proposing a plethora of syntactic and model-theoretic criteria (e.g., definitional equivalence, bi-interpretability, categorial equivalence and so on). In recent years, this topic has gained renewed popularity thanks to various debates across philosophy. Among these, some notable examples are the equivalence between potentialist and actualist frameworks, and the lifting of type distinctions in higher-order metaphysics. This workshop aims to take stock of these developments, clarify the landscape of competing notions of theoretical equivalence, and explore their implications for contemporary philosophy of science and mathematics. The event will bring together the team of C-FORS members actively working on the topic and three external leading experts in the field, whose contributions have significantly shaped the debate over the past decade.
TOPICS
Topics may include:
- How much can we rely on formal criteria to assess theoretical equivalence without taking into account how linguistic expressions are interpreted?
- Do formal criteria match an informal and pre-theoretical notion of theoretical equivalence (e.g., Church-Turing Thesis)?
- How should we handle theoretical equivalence outside classical first-order theories?
- Should we be monists or pluralist on theoretical equivalence?
- How shall we assess the equivalence of modal and non-modal formal theories?
- How shall we assess the equivalence of higher-order and first-order theories?
SCHEDULE
Monday 22nd (GMH 652)
09.00 – 09.30: Coffee and Welcome
09.30 – 11.00: Hans Halvorson (Princeton University), Equivalence: A Program
Abstract
Over the past two decades, a substantial body of technical work has accumulated on the question of when two physical theories are equivalent. Meanwhile, critics have questioned whether this work has any philosophical relevance whatsoever. I argue that the critics have it backwards: the formal study of equivalence is not a technical sideshow to philosophy of science but one of its central instruments — the means by which questions about realism, theoretical content, and interpretation become tractable at all. Rather than defending past work, I set out the questions that the next generation of philosophers and logicians is positioned to answer, and the conceptual tools they will need to answer them.
11.00 – 11.30: Break
11.30 – 13.00: Alessandro Giglia (University of Oslo), Against the Equivalence of Potentialist and Actualist Set Theories
Abstract
The Equivalence Thesis, championed by Tim Button in Level Theory Part 2, holds that potentialist and actualist conceptions of set do not genuinely disagree but are equivalent ways of expressing the same content. In this talk I argue against this thesis in two stages.
First, I present a strategy for assessing the thesis by discussing actualist and potentialist theories within the same language. The strategy relies on embedding actualist theories in the modal logic triv—the logic obtained by extending FOL with the axiom `□φ ↔ φ’. When doing so, the comparison can proceed via logical equivalence, and the equivalence thesis fails for a straightforward reason: the logic of the potentialist theories is a proper fragment of the logic of the actualist theories.
Second, I argue against Button’s argument for the thesis, which rests on a battery of near-synonymy results between potentialist and actualist theories. I focus on two considerations. First, several of the theories Button discusses may not faithfully capture the conceptions they are meant to represent. Second, and more importantly, near-synonymy is too weak to entail theoretical equivalence, as it is not an equivalence relation and does not preserve important features of the theories.
13.00 – 14.00: Lunch
14.00 – 15.30: Øystein Linnebo (University of Oslo), Theoretical equivalence: the Case of Potentialism
Abstract
What is the philosophical significance of various formal results about theoretical equivalence? I begin with some general remarks about what such formal results do and do not establish. Then, I focus on the philosophical significance of some theoretical equivalence results concerning potentialism, both Aristotle’s and my own. I argue that each version of potentialism has the resources to deny—each in its own way—that the relevant formal results about theoretical equivalence establish a philosophical equivalence of each version of potentialism with its actualist competitor.
15.30 – 16.00: Break
16.00 – 17.00: Discussion/Round Table
19.30: Dinner at Der Peppern Gror (Aker Brygge)
Tuesday 23rd (GMH 652)
09.00 – 09.30: Coffee and Welcome
09.30 – 11.00: Tim Button (University College London), When are theories equivalent? A case study concerning weak categori(c)al theories
Abstract
Let’s regard cardinals as collections of otherwise featureless points. A famous theory of cardinals is Lawvere’s ETCS. There are some very nice discussions of ETCS’s axioms, but I have not seen anyone attempt to present the ‘intended model’ of ETCS, or indeed of any theory of cardinals. (By contrast, we are often told that the intended model of ZFC is the cumulative hierarchy of pure sets.)
So: in this talk, I will introduce you to a new theory of cardinals (Stagewise Relational Cardinals, or SRC for short). This axiomatizes the idea that cardinals are found stage-by-stage. It has an obvious intended model.
We can go further. Osius (1974) showed that an extension of ETCS is equivalent (in a precise sense) to an extension of Zermelo’s theory. Using similar proof ideas: RSC is equivalent (in that same precise sense) to a weak but quasi-categorical theory of ‘the cumulative hierarchy of pure sets’. (Indeed, the quasi-categoricity is key to the proof.)
Should we conclude that theories of (stagewise) cardinals and theories of (cumulative) sets are just “notational variants”? No! The theories are not (and must not be) bi-interpretable. Mathematical content is lost if we move from sets to cardinals. This undermines the idea (not uncommon among category theorists) that theories of cardinals and theories of [material] sets are equivalent, but that the former are somehow more faithful to mathematical practice.
11.00 – 11.30: Break
11.30 – 13.00: Toby Meadows (University of California Irvine), Theoretical Equivalence & Mathematics: insufficient, unnecessary and yet?
Abstract
Some people say that if two theories are definitionally equivalent, then they’re “really” the same theory. Other people say that mathematical relations like definitional equivalence have no bearing on the questions of when two theories are “really” the same. I find both views baffling.
My goal in this talk is to air some dirty laundry and then offer some new logical appliances to wash it. My target notion will be definitional equivalence. I aim to show that this is neither a sufficient nor necessary condition for naive theoretical equivalence. And yet it is an invaluable element of our deliberative apparatus for such problems. I will suggest that some of us have been asking the “wrong” questions. I’ll then pose the “right” questions and offer some groundwork toward answering them.
13.00 – 14.00: Lunch
14.00 – 15.30: Davide Sutto (University of Oslo), Iterating Critical Pluralities
Abstract
In this talk I outline an iterative conception for the process of plural-to-set abstraction recently proposed by Florio and Linnebo on the grounds of their critical approach to plural logic, according to which every plurality collapses into a set. To frame this view in iterative terms I adapt the tools of level theories, as recently sharpened by Tim Button, to the plural context. The result is a plural theory of levels where every plurality is level-bounded and hence collapses into a set. This theory is then used to assess the role of critical plural logic as a kind of two-sorted first-order theory. In particular, by concatenating a series of synonymy results, I use the theory to assess the role of critical plural logic as a non-modal route to potentialism as recently argued by Linnebo.
ATTENDANCE
Attendance is free, but in person only.
To register please fill the form below.
Deadline for registration: 30 April 2026.
VENUE
The conference takes place at the University of Oslo, Department of Philosophy, Classics, History of Art and Ideas, situated in Blindern Campus, building: Geor Morgenstiernes hus.
The conference room is GMH 652. This is the conference room situated at the 6th floor of the department, at the end of the corridor.
How to get to Blindern (Link 1/Link 2)
- Tram: line 17 and 18 towards Rikshospitalet, Stop at Universitetet Blindern (or John Colletts Plass in case of road works).
- Metro (T Bane): line 4 (Vestli via Storo), 5 (Ringen via Storo) 5 (Sognsvann), Stop at Blindern.
ORGANIZERS: Alessandro Giglia, Øystein Linnebo, Davide Sutto.
FOR MORE INFORMATION, CONTACT