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Single Idea 9186

[filed under theme 5. Theory of Logic / G. Quantification / 5. Second-Order Quantification ]

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.

The 11 ideas with the same theme [quantifiyng over both objects, and features or sets of objects]:

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]