Jeremy Avigad
Carnegie Mellon University
Stephen Mackereth
University of Pittsburgh
Neo-Fregean logicists claim that Hume's Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A longstanding problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck's Two-sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn't. In fact, 2FA is not conservative over $n$-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic.
Keywords Frege  logicism  neo-Fregeanism  neologicism  abstractionism  Hume's Principle  conservativeness  Field-conservativeness  second-order logic  second-order arithmetic  stipulative definition
Categories (categorize this paper)
Reprint years 2022
DOI 10.1017/s1755020322000156
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy

 PhilArchive page | Other versions
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

Fixing Frege.John P. Burgess - 2005 - Princeton University Press.
Subsystems of Second-Order Arithmetic.Stephen G. Simpson - 2004 - Studia Logica 77 (1):129-129.

View all 31 references / Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

Neo-Logicism and Its Logic.Panu Raatikainen - 2020 - History and Philosophy of Logic 41 (1):82-95.
Frege's Other Program.Aldo Antonelli & Robert May - 2005 - Notre Dame Journal of Formal Logic 46 (1):1-17.
Predicative Fragments of Frege Arithmetic.Øystein Linnebo - 2004 - Bulletin of Symbolic Logic 10 (2):153-174.
Finitude and Hume's Principle.Richard G. Heck Jr - 1997 - Journal of Philosophical Logic 26 (6):589 - 617.
Finitude and Hume’s Principle.Richard Heck - 1997 - Journal of Philosophical Logic 26 (6):589-617.
The Strength of Abstraction with Predicative Comprehension.Sean Walsh - 2016 - Bulletin of Symbolic Logic 22 (1):105–120.
Is Hume's Principle Analytic?Crispin Wright - 1999 - Notre Dame Journal of Formal Logic 40 (1):6-30.
Comparing Peano Arithmetic, Basic Law V, and Hume’s Principle.Sean Walsh - 2012 - Annals of Pure and Applied Logic 163 (11):1679-1709.


Added to PP index

Total views
87 ( #132,172 of 2,498,184 )

Recent downloads (6 months)
87 ( #8,411 of 2,498,184 )

How can I increase my downloads?


My notes