# Abstraction Principles and the Classification of Second-Order Equivalence Relations

@article{EbelsDuggan2019AbstractionPA, title={Abstraction Principles and the Classification of Second-Order Equivalence Relations}, author={Sean Ebels-Duggan}, journal={Notre Dame J. Formal Log.}, year={2019}, volume={60}, pages={77-117} }

This paper improves two existing theorems of interest to neo-logicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation $E$ is defined without non-logical vocabulary, then the bicardinal slice of any equivalence class---those equinumerous elements of the equivalence class with equinumerous complements---can have one… Expand

#### Topics from this paper

#### 2 Citations

Identifying finite cardinal abstracts

- Computer Science
- 2020

A novel cross-sortal identity principle is proposed, based on embeddings of the induced models of abstracts developed by Walsh (2012), but offers interestingly different answers to the more controversial identifications made by ECIA2. Expand

#### References

SHOWING 1-10 OF 35 REFERENCES

Logicality and Invariance

- Computer Science, Mathematics
- Bulletin of Symbolic Logic
- 2008

The standard arguments in favor of invariance under permutation, which rely on the generality and the formality of logic, should be modified and shown to support an alternative to Tarski's criterion, according to which an operation is logical iff it is invariant under potential isomorphism. Expand

Notions of Invariance for Abstraction Principles

- Mathematics
- 2010

The logical status of abstraction principles, and especially Hume’s Principle, has been long debated, but the best currently availeble tool for explicating a notion’s logical character ‐ permutation… Expand

Reals by Abstractiont

- Mathematics
- 2000

ions like the Direction equivalence and Hume’s principle and that it is reasonable to regard it as one. We might bring EM into line with the characterisation of abstraction principles with which I… Expand

Finitude and Hume’s Principle

- Mathematics, Computer Science
- J. Philos. Log.
- 1997

‘Finite Hume’s Principle’ suffices for the derivation of axioms for arithmetic and is equivalent to a version of them, in the presence of Frege's definitions of the primitive expressions of the language of arithmetic. Expand

RELATIVE CATEGORICITY AND ABSTRACTION PRINCIPLES

- Computer Science, Mathematics
- The Review of Symbolic Logic
- 2015

It is shown that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. Expand

Logical Indefinites∗

- 2014

The best extant demarcation of logical constants, due to Tarski, classifies logical constants by invariance properties of their denotations. This classification is developed in a framework which… Expand

Comparing Peano arithmetic, Basic Law V, and Hume's Principle

- Mathematics, Computer Science
- Ann. Pure Appl. Log.
- 2012

There is a consistent extension of the hypear arithmetic fragment of Basic Law V which interprets the hyperarithmetic fragment of second-order Peano arithmetic, so that in this specific sense there is no predicative version of Frege’s Theorem. Expand

Logical operations

- Computer Science
- J. Philos. Log.
- 1996

This paper lends support to the Tarski-Mautner proposal to characterize the “logical” operations on a given domain as those invariant under arbitrary permutations. Expand

The Nuisance Principle in Infinite Settings

- Mathematics, Computer Science
- 2015

It is shown that logically this situation persists if one looks at joint (second-order) consistency rather than satisfiability: under a modest assumption about infinite concepts, NP is also inconsistent with HP. Expand