| 8490 | First-level functions have objects as arguments; second-level functions take functions as arguments [Frege] |
| 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. | |
| From: Gottlob Frege (Function and Concept [1891], 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. |
| 21566 | 'Propositional functions' are ambiguous until the variable is given a value [Russell] |
| Full Idea: By a 'propositional function' I mean something which contains a variable x, and expresses a proposition as soon as a value is assigned to x. That is to say, it differs from a proposition solely by the fact that it is ambiguous. | |
| From: Bertrand Russell (The Theory of Logical Types [1910], p.216) | |
| A reaction: This is Frege's notion of a 'concept', as an assertion of a predicate which still lacks a subject. |
| 18961 | We can identify functions with certain sets - or identify sets with certain functions [Putnam] |
| Full Idea: Instead of identifying functions with certain sets, I might have identified sets with certain functions. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.9) |
| 13811 | A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock] |
| Full Idea: Usually we allow that a function is defined for arguments of a suitable kind (a 'partial' function), but we can say that each function has one value for any object whatever, from the whole domain that our quantifiers range over (a 'total' function). | |
| From: David Bostock (Intermediate Logic [1997], 8.2) | |
| A reaction: He points out (p.338) that 'the father of..' is a functional expression, but it wouldn't normally take stones as input, so seems to be a partial function. But then it doesn't even take all male humans either. It only takes fathers! |
| 13812 | A 'zero-place' function just has a single value, so it is a name [Bostock] |
| Full Idea: We can talk of a 'zero-place' function, which is a new-fangled name for a familiar item; it just has a single value, and so it has the same role as a name. | |
| From: David Bostock (Intermediate Logic [1997], 8.2) |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| Full Idea: The 'range' of a function is the set of elements in the output set that are values of the function for elements in the original set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
| A reaction: In other words, the range is the set of values that were created by the function. |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| Full Idea: A 'total function' is one which maps every element of a domain to exactly one corresponding value in another set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| Full Idea: A 'partial function' is one which maps only some elements of a domain to elements in another set. For example, the reciprocal function 1/x is not defined for x=0. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1 n1) |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| Full Idea: We count two functions as being the same if they have the same extension, i.e. if they pair up arguments with values in the same way. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 11.3) | |
| A reaction: So there's only one way to skin a cat in mathematical logic. |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| Full Idea: If a function f maps the argument a back to a itself, so that f(a) = a, then a is said to be a 'fixed point' for f. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 20.5) |
| 10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
| Full Idea: The modern extensional notion of function is just an arbitrary correspondence between collections. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 1) | |
| A reaction: Shapiro links this with the idea that a set is just an arbitrary collection. These minimalist concepts seem like a reaction to a general failure to come up with a more useful and common sense definition. |
| 13700 | A 'total' function must always produce an output for a given domain [Sider] |
| Full Idea: Calling a function a 'total' function 'over D' means that the function must have a well-defined output (which is a member of D) whenever it is given as inputs any n members of D. | |
| From: Theodore Sider (Logic for Philosophy [2010], 5.2) |
| 15105 | F(x) walked into a bar. The barman said.. [Sommers,W] |
| Full Idea: F(x) walked into a bar. The barman said, 'Sorry, we don't cater for functions'. | |
| From: Will Sommers (talk [2019]) |