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Single Idea 8490

[filed under theme 5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic ]

Full Idea

Just as functions are fundamentally different from objects, so also functions whose arguments are and must be functions are fundamentally different from functions whose arguments are objects. The latter are first-level, the former second-level, functions.

Gist of Idea

First-level functions have objects as arguments; second-level functions take functions as arguments

Source

Gottlob Frege (Function and Concept [1891], p.38)

Book Ref

Frege,Gottlob: 'Translations from the Writings of Gottlob Frege', ed/tr. Geach,P/Black,M [Blackwell 1980], p.38

A Reaction

In 1884 he called it 'second-order'. This is the standard distinction between first- and second-order logic. The first quantifies over objects, the second over intensional entities such as properties and propositions.

The 13 ideas with the same theme [elements in logical systems to create new objects]:

8490 First-level functions have objects as arguments; second-level functions take functions as arguments [Frege]
21566 'Propositional functions' are ambiguous until the variable is given a value [Russell]
18961 We can identify functions with certain sets - or identify sets with certain functions [Putnam]
13812 A 'zero-place' function just has a single value, so it is a name [Bostock]
13811 A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock]
10074 A 'total function' maps every element to one element in another set [Smith,P]
10076 The 'range' of a function is the set of elements in the output set created by the function [Smith,P]
10075 A 'partial function' maps only some elements to another set [Smith,P]
10605 Two functions are the same if they have the same extension [Smith,P]
10612 An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P]
10209 A function is just an arbitrary correspondence between collections [Shapiro]
13700 A 'total' function must always produce an output for a given domain [Sider]
15105 F(x) walked into a bar. The barman said.. [Sommers,W]