Frege meets Belnap: Basic Law V in a Relevant Logic

In Andrew Tedder, Shawn Standefer & Igor Sedlar (eds.), New Directions in Relevant Logic. Springer. pp. 381-404 (forthcoming)
  Copy   BIBTEX

Abstract

Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects $\S e_1, \S e_2$, are identical if, and only if, an equivalence relation $Eq_\S$ holds between the entities $e_1, e_2$) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? In this paper, we investigate the matter by deriving the axioms of second-order Peano Arithmetic from Frege's Basic Law V (the extension of $F$ is identical with the extension of $G$ if, and only if, $F$ and $G$ are extensionally equivalent) in the presence of a relevant higher-order logic. The results are interesting. Not only must we take on logic as true, and not only must we apply our logic to abstraction principles, but also we have to apply our theory of abstraction back to the logic in order to arrive at arithmetic. Thus, what Abstractionism gives us is not simply what we get from abstraction via logic, but also what we get from logic via abstraction.

Links

PhilArchive

External links

  • This entry has no external links. Add one.
Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Frege on Sense Identity, Basic Law V, and Analysis.Philip A. Ebert - 2016 - Philosophia Mathematica 24 (1):9-29.
Frege meets Brouwer.Stewart Shapiro & Øystein Linnebo - 2015 - Review of Symbolic Logic 8 (3):540-552.
Plural Frege Arithmetic.Francesca Boccuni - 2022 - Philosophia Scientiae:189-206.
Plural Frege Arithmetic.Francesca Boccuni - 2022 - Philosophia Scientiae 26:189-206.
Essays on Frege's Basic Laws of Arithmetic.Philip A. Ebert & Marcus Rossberg (eds.) - 2019 - Oxford: Oxford University Press.
Nothing but the Truth.Andreas Pietz & Umberto Rivieccio - 2013 - Journal of Philosophical Logic 42 (1):125-135.

Analytics

Added to PP
2024-02-06

Downloads
89 (#179,596)

6 months
89 (#42,046)

Historical graph of downloads
How can I increase my downloads?

Author Profiles

Francesca Boccuni
University Vita-Salute San Raffaele
Shay Logan
Kansas State University

Citations of this work

No citations found.

Add more citations

References found in this work

Relevant Logics and Their Rivals.Richard Routley, Val Plumwood, Robert K. Meyer & Ross T. Brady - 1982 - Ridgeview. Edited by Richard Sylvan & Ross Brady.
Anti-realism and logic: truth as eternal.Neil Tennant - 1987 - New York: Oxford University Press.
Frege’s Conception of Numbers as Objects.Crispin Wright - 1983 - Critical Philosophy 1 (1):97.
Paradoxes and Inconsistent Mathematics.Zach Weber - 2021 - New York, NY: Cambridge University Press.

View all 18 references / Add more references