more on this theme | more from this thinker
Full Idea
First-order logic is distinguished by generalizations (quantification) only over objects: second-order logic admits generalizations or quantification over properties or kinds of objects, and over relations between them, and functions defined over them.
Gist of Idea
First-order logic concerns objects; second-order adds properties, kinds, relations and functions
Source
Michael Dummett (The Philosophy of Mathematics [1998], 3.1)
Book Ref
'Philosophy 2: further through the subject', ed/tr. Grayling,A.C. [OUP 1998], p.134
A Reaction
Second-order logic was introduced by Frege, but is (interestingly) rejected by Quine, because of the ontological commitments involved. I remain unconvinced that quantification entails ontological commitment, so I'm happy.
| 9186 | First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett] |
| 9187 | Logical truths and inference are characterized either syntactically or semantically [Dummett] |
| 9191 | Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett] |
| 9192 | The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett] |
| 9193 | ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett] |
| 9194 | The main alternative to ZF is one which includes looser classes as well as sets [Dummett] |
| 9195 | Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett] |