Lugano Workshop on Theoretical Equivalence and Related Topics

8th April 2025

Rooms SI004 and SI006, Palazzo Nero

Lugano

Tentative Schedule

Morning session, Room SI004

9:00 AM – 10:00 AM: Hans Halvorson, "Equivalence - State of Play"

10:00 AM – 11:00 AM: Øystein Linnebo, “What is potentialism? A case study concerning theoretical equivalence”

11:00 AM – 11:15 AM: Coffee break

11:15 PM – 12:15 PM: Tim Button, “Nature's joints and artefacts of representation” (online talk)

12:30 PM – 2:00 PM: Lunch

Afternoon session, Room SI006

2:15 PM – 3:15 PM: Joshua Babic and Lorenzo Lorenzetti, “Saving functionalism from inconsistency”

3:15 PM – 4:15 PM: Neil Dewar, "What conventionalism demands"

4:15 PM – 4:30 PM: Coffee break

4:30 PM – 5:30 PM: Léon Probst, "Interpretation and meaning questions in metamathematics"

5:30 PM – 6:30 PM: Andrea Salvador, "Identicals by logic"

Abstracts

• Hans Halvorson, Equivalence - State of Play 

In this talk, I give an overview of definitions, results, and open questions about theoretical equivalence. In particular, I consider a spectrum of notions of equivalence from extremely liberal (coarse grained) to extremely conservative (fine grained). I then go on to discuss the metaphysical upshot of the "formal" theory of equivalence, paying special attention to questions raised by Babic, Calosi, and Teitel.

• Neil Dewar, What conventionalism demands

Conventionalism can seem puzzling, because it appears to be appropriate only in circumstances where (a) there is a choice to be made between two options, and (b) that choice is somehow “empty”, since choosing between the options is a matter of mere convention. In this paper, I propose a straightforward account of the circumstances under which (a) and (b) are true, and hence in which conventionalism might be appropriate: when we are faced with pairs of theories that are inconsistent but equivalent. The inconsistency gives rise to the need for choice (as per (a)), and the equivalence gives rise to the choice’s emptiness (as per (b)). I illustrate this with various examples, discuss how this distinguishes the conventionalist from their opponents, and compare my treatment to other recent discussions in the literature.

• Øystein Linnebo, Potentialism demodalized

Potentialism is the view that certain objects are successively generated and that this generation cannot be completed. It is natural to analyze this view modally. So-called mirroring theorems connect the resulting modal analysis of potentialism with the non-modal languages of ordinary mathematics. This article proves a mirroring theorem for plural logic, which determines the correct plural logic for non-modal reasoning about a merely potential domain. Even better, a result about definitional equivalence is proved, which enables a new and entirely non-modal explication of potentialism. Some advantages of this non-modal explication are discussed. It is simpler and more user-friendly than the extant modal analysis, as an application to potentialist set theory illustrates. The explication also enables potentialists to sidestep the question of which mathematical objects are actual.

• Tim Button, Nature's joints and artefacts of representation

Kant asked: how is metaphysics so much as possible? Quine answered: because metaphysics is broadly continuous with science. But Quine's answer gives us no reason to think that "joint-carving" metaphysics is possible. Joint-carvers want our theoretical primitives to keep track of what is metaphysically primitive (or fundamental, as opposed to derivative). But we have no reason to think that such a thing is possible. To explain why, I'll offer some general considerations, a particular case study (about space), and a logical argument. The upshot of all of these is that we (provably) end up with (arbitrary) artefacts of representational choices.

• Joshua Babic and Lorenzo Lorenzetti, Saving functionalism from inconsistency

Functionalism continues to thrive in philosophy and has gained significant traction in the philosophy of science in recent years, particularly as a valuable framework for interpreting theoretical terms like spacetime. However, a seemingly overlooked argument by George Bealer, recently revived by Hans Halvorson, claims to demonstrate that functionalism is inconsistent. This argument relies on Beth’s definability theorem, a fundamental result in model theory. We introduce a more streamlined version of the argument, showing its validity without invoking second-order logic and applying it directly to David Lewis’s first-order formal account of functionalism. We then propose a solution by challenging Bealer’s definition of functionalism, saving functionalism from the threat of inconsistency.  

• Léon Probst, Interpretation and meaning questions in metamathematics

The notion of (relative) interpretation plays a central role in metamathematics. For instance, it allows one to prove Gödel’s second incompleteness theorem (G2) for non-arithmetical theories, such as ZFC. The consistency of ZFC being expressed by an arithmetical statement (Con(ZFC)), the usual proof of G2 relies on translating such an arithmetical statement into the language of set theory (more precisely, it requires interpreting an arithmetical theory, e.g. Q, on some domain such as omega). In which sense, then, does ZFC fail to prove its own consistency? More generally, if a theory T cannot prove the relativisation of Con(T) to its language, under what conditions are we justified in concluding that T fails to prove its own consistency?

In this talk, we explore how the notion of interpretation interacts with these so-called meaning questions about consistency. We focus on whether Q fails to prove its own consistency and, more generally, how some generalisation theorems of G2 via interpretation interact with these meaning questions. Such considerations support a new understanding of the two standard approaches to meaning questions. Finally, we argue that a more fine-grained analysis of the choice of (the domain of the) interpretation is needed and discuss some potential approaches.

• Andrea Salvador, Identicals by logic

By asserting a statement, we say things are in a certain way. Also, different statements can say things are in the  same way, or describe the same facts. Similar remarks apply to theories, where two theories are equivalent when, intuitively, they describe the same facts. Following a recent custom, I call “generalised identities” (GIs) statements that claim that two statements describe the same facts. A question then naturally arises: what statements are identical, in this general sense, by logic alone? Dorr (2016), Correia (2016) and Elgin (2023) recently answered this question. However, I give reasons for believing that their answers are incorrect. I thus provide an exact truthmaker semantics for GIs which instead captures the correct logic of GI. Notably, and in contrast with the other proposals, the semantics provides exact verification and falsification clauses for GIs. This is done by introducing “identity-states”—states which consider certain propositions identical—and imposing suitable constraints on the models to ensure they provide adequate possible identity-states and enough identity-states. Furthermore, the same semantics can capture the interaction between GI and metaphysical necessity in a natural way.