2025-2029: Determinacy and Non-Classicality in Arithmetic

• Funder: FWF (Austrian Science Fund).
• 430.000€.
• Applicants: Julien Murzi and Nuno Maia
• PI: Julien Murzi.
• DOI: 10.55776/PAT2449425

In this project, we propose to study determinacy within arithmetic, and argue that, when viewed non-classically, traditional theorems and arguments lead to unexpected conclusions and, in some cases, actually go against the established consensus, supporting arithmetical indeterminacy and making arithmetic very much like set theory. In this respect, our results are in line with an early conjecture made by Joel Hamkins to the effect that the present confidence in arithmetical determinacy is contingent on current logical techniques, like forcing methods. As he writes: “[m]y long-term expectation is that technical developments will eventually arise that provide a forcing analogue for arithmetic, allowing us to modify diverse models of arithmetic in a fundamental and flexible way, just as we now modify models of set theory by forcing, and this development will challenge our confidence in the uniqueness of the natural number structure […]” (Hamkins, 2012, p. 428). Furthermore, we think that a non-classical treatment of some of those traditional theorems and arguments gives rise to mathematically interesting logical systems, and has the potential to shed light on many other philosophical questions (to give an example: we develop a paraconsistent logic which provides an entirely new perspective on how contradictions might arise within arithmetic).

2021-2026: Categoricity by Convention

• Funder: FWF (Austrian Science Fund).
• 430.000€.
• Applicants: Julien Murzi and Brett Topey
• PI: Julien Murzi.
• DOI: 10.55776/P33708

The project develops a moderate inferentialist view on which our open-ended rules for the higher-order quantifiers determine the full interpretation of second-order logic (and indeed of all logics of finite order), so that, via standard categoricity and quasi-categoricity results, our higher-order mathematical theories can be seen to be categorical or quasi-categorical (pace Skolem, Putnam etc.). The idea is of course old — Vann McGee among others has been arguing for much the same view — but the details are new. Among other things, we provide a new solution to Carnap’s categoricity problem for propositional logic, and strengthen and generalize to higher-order logic a recent result by Denis Bonnay and Dag Westerståhl concerning the categoricity of predicate logic. Our view yields a novel, largely syntactic criterion for logicality, a moderate form of pluralism, and an attractive epistemology of the a priori. Or so we hope!

2017-2021: The Liar and its Revenge in Context

• Funder: FWF (Austrian Science Fund).
• 390.000€.
• Applicants: Julien Murzi and Lorenzo Rossi.
• PI: Julien Murzi.
• DOI: 10.55776/P29716

The project has started in January 2017 and will last until July 2021. It involves (temporal parts of) two (awesome) post-doctoral researchers: Lorenzo Rossi and Brett Topey. Lorenzo is now an Assistant Professor in Munich; we are currently writing a book for OUP. Brett is still in Salzburg; we are currently working on issues surrounding conventionalism and the determinacy of mathematical language. You can visit the website of the project here.

2013-2015: Inference and Logic

• Funder: British Academy.
• £10.000.
• Applicants: Julien Murzi and Florian Steinberger.

The Inference and Logic project was funded by the and led by Julien Murzi and Florian Steinberger. It aimed at investigating inferentialist approaches to logic: approaches according to which (roughly) the meaning of a logical expression is determined by the way we use it, and to know the meaning of a logical expression is to know how to use it. Two controversial but, we think, fecund thoughts! Between April 2013 and July 2015, we wrote a bunch of papers and organised this conference.