Description

This project seeks to develop and defend Upper Logicism, a philosophy of mathematics inspired by Russell’s logicism according to which arithmetic can be interpreted in higher-order modal logic with Finitary Plenitude as an axiom. It is characterized by the following theses:

UL1: Predicates have a semantic role distinct from those of individual and plural terms, and quantification into predicate position is legitimate and irreducible to first-order singular or plural quantification.

UL2: Natural, rational and real numbers, and sets, are all higher-type entities (and so properties or relations) whose canonical applications are contained within their nature, and the primitives of their standard theories are all truly characterizable as higher type-entities definable in pure modal type theory.

UL3: The metaphysically necessary truths formulated in pure modal and type-theoretic languages are all logically true.

UL4: Finitary Plenitude, the thesis that any finite cardinality could have been instantiated, is necessarily true.

UL5: For every standard, true deductive theory T of arithmetic, rational or real analysis or set theory, there is a deductive system of pure modal type-theory which includes Finitary Plenitude as an axiom, has only metaphysically necessary truths as theorems, and includes as theorems translations of T’s theorems by interpreting T’s primitives according to UL2.

UL6: The truths of arithmetic, rational and real analysis, and set theory, are all metaphysically necessary truths expressible in a pure modal type theory by interpreting them according to UL2.

For a paper kickstarting the project, see Jacinto, B. (2024) ‘Finitary Upper Logicism’, The Review of Symbolic Logic.

OBJECTIVES

O1: To investigate the history of logicism – in particular, the work of the early logicists Frege and Russell – concerning both their own interpretability results and how they grappled with the philosophical presuppositions required for sustaining their logicist projects.

O2: To identify and defend the main philosophical presuppositions concerning higher-order metaphysics, modality, logic, mathematics, philosophy of language and epistemology required for seeing the interpretability results of (O3) as establishing Upper Logicism, and to show how Upper Logicism compares vis-à-vis recent logicist rivals.

O3: To establish novel technical results concerning the interpretability of arithmetic, rational and real analysis, and set theory by minimally demanding pure modal type theories incorporating the thesis that every finite cardinality could have been instantiated, and through equating natural, rational and real numbers, and sets, with properties of a logically definable kind.

O4: To root research in the philosophy of mathematics, and more specifically logicism, in Portugal, and more specifically Lisbon, through the training of undergraduate and master’s students.

Research Team

PRINCIPAL RESEARCHER: Bruno Jacinto (Lisbon)

TEAM MEMBERS: Joan Bertran-San-Millán (Complutense), Tabea Rohr (FSU Jena), António Zilhão (Lisbon), Sébastien Gandon (Clermont Auvergne), Nicholas Jones (Oxford), Agustín Rayo (MIT), Christopher Scambler (Oxford), Hannes Leitgeb (LMU Munich), Sonia Roca-Royes (Stirling), Francesca Boccuni (San Raffaele), Francisca Silva (Lisbon).

POSTDOCTORAL FELLOW: José Mestre (Lisbon)

MASTER’S STUDENT: New Hire

CONSULTANTS: Fernando Ferreira (Lisbon), Øystein Linnebo (Oslo), Timothy Williamson (Oxford), Crispin Wright (Stirling).

RESEARCH BRANCHES

HISTORICAL ROOTS: Joan Bertran-San-Millán, Sébastien Gannon, Tabea Rohr, António Zilhão, José Mestre.

PHILOSOPHICAL FOUNDATIONS: Francesca Boccuni, Agustin Rayo, Bruno Jacinto, Nicholas Jones, Hannes Leitgeb, Sonia Roca-Royes.

TECHNICAL DEVELOPMENTS: Francesca Boccuni, Bruno Jacinto, Christopher Scambler, Francisca Silva.

UPPER SEMINAR

The project’s regular seminar will be meeting weekly. Send an email to upperlogicism@gmail.com in case you would be interested in joining it.

Publications

CALENDAR

Events

OPENING WORKSHOP: 17, 18, 19 June 2026.

Speakers include Joan Bertran-San-Millán, Francesca Boccuni and Chris Scambler.

MASTERCLASS+WORKSHOP: June 2027.

FINAL CONFERENCE: June 2028.