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Full Idea
In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets.
Gist of Idea
Three types of variable in second-order logic, for objects, functions, and predicates/sets
Source
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
A Reaction
It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types.
| 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] |