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Full Idea
The defining factor of second-order logic is that, while the domain of its individual variables may be arbitrary, the range of the first-order variables is all the properties of the objects in its domain (or, thinking extensionally, of the sets objects).
Gist of Idea
In second-order logic the higher-order variables range over all the properties of the objects
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Ref
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.47
A Reaction
The key point is that the domain is 'all' of the properties. How many properties does an object have. You need to decide whether you believe in sparse or abundant properties (I vote for very sparse indeed).
| 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] |