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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.
| 10705 | Putting a predicate letter in a quantifier is to make it the name of an entity [Quine] |
| 9186 | First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett] |
| 13671 | Second-order quantifiers are just like plural quantifiers in ordinary language, with no extra ontology [Boolos, by Shapiro] |
| 10569 | If you ask what F the second-order quantifier quantifies over, you treat it as first-order [Fine,K] |
| 10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| 10978 | In second-order logic the higher-order variables range over all the properties of the objects [Read] |
| 5740 | Second-order logic needs second-order variables and quantification into predicate position [Melia] |
| 13453 | Perhaps second-order quantifications cover concepts of objects, rather than plain objects [Rayo/Uzquiano] |
| 18761 | Second-order variables need to range over more than collections of first-order objects [McGee] |
| 18763 | Basic variables in second-order logic are taken to range over subsets of the individuals [Anderson,CA] |