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Full Idea
If one thought of second-order quantification as quantification over first-level Fregean concepts [note: one under which only objects fall], talk of domains might be regimented as talk of first-level concepts, which are not objects.
Gist of Idea
Perhaps second-order quantifications cover concepts of objects, rather than plain objects
Source
Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2)
Book Ref
'Absolute Generality', ed/tr. Rayo,A/Uzquiano,G [OUP 2006], p.8
A Reaction
That is (I take it), don't quantify over objects, but quantify over concepts, but only those under which known objects fall. One might thus achieve naïve comprehension without paradoxes. Sound like fun.
| 13449 | We could have unrestricted quantification without having an all-inclusive domain [Rayo/Uzquiano] |
| 13448 | The domain of an assertion is restricted by context, either semantically or pragmatically [Rayo/Uzquiano] |
| 13450 | Absolute generality is impossible, if there are indefinitely extensible concepts like sets and ordinals [Rayo/Uzquiano] |
| 13453 | Perhaps second-order quantifications cover concepts of objects, rather than plain objects [Rayo/Uzquiano] |
| 13451 | The two best understood conceptions of set are the Iterative and the Limitation of Size [Rayo/Uzquiano] |
| 13452 | Some set theories give up Separation in exchange for a universal set [Rayo/Uzquiano] |