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Full Idea
The result of joining Hume's Principle to second-order logic is a consistent system which is a foundation for arithmetic, in the sense that all the fundamental laws of arithmetic are derivable within it as theorems. This seems a vindication of logicism.
Clarification
Hume's Principle involves one-to-one correlation
Gist of Idea
Neo-logicism founds arithmetic on Hume's Principle along with second-order logic
Source
B Hale / C Wright (Logicism in the 21st Century [2007], 1)
Book Ref
'Oxf Handbk of Philosophy of Maths and Logic', ed/tr. Shapiro,Stewart [OUP 2007], p.169
A Reaction
The controversial part seems to be second-order logic, which Quine (for example) vigorously challenged. The contention against most attempts to improve Frege's logicism is that they thereby cease to be properly logical.
| 8784 | Neo-logicism founds arithmetic on Hume's Principle along with second-order logic [Hale/Wright] |
| 8786 | One first-order abstraction principle is Frege's definition of 'direction' in terms of parallel lines [Hale/Wright] |
| 8783 | Logicism might also be revived with a quantificational approach, or an abstraction-free approach [Hale/Wright] |
| 8787 | The Julius Caesar problem asks for a criterion for the concept of a 'number' [Hale/Wright] |
| 8788 | Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology [Hale/Wright] |