Re: Function and Variable: fatal problem?

On Sun, 06 Jan 2008 03:27:07 +0100, G. Frege <nomail@invalid> wrote:


John loves Mary

Here, John and Mary are distinguished by the function 'loves'. 'Loves'
distinguishes John and Mary. [John - who may or may not love Mary.]

Again, as GF says, "loves" is not a function.

At least not if we deal with this sentence in a contemporary "standard"
system of logic, namely FOPL. There "loves" is translated with a
(binary) "predicate symbol", say "L", which is _not_ referring to a
function (but a relation).


Ironically, the real Frege would say that loves is a function.

I'm surprised to hear that! How so?

Right. Actually, he would say that /loves/ is a concept (Begriff).
(Hence "Conceptscript" ["Begriffsschrift"] for his system.)

But he (later) defined /concepts/ as functions from ... to to the set of
truth values.


Just found the following:

"In Frege's analysis, the verb phrase 'loves' signifies a binary
function of two variables: L(( ),( )). This function takes a pair of
arguments x and y and maps them to The True if x loves y and maps all
other pairs of arguments to The False. Although it is a descendent of
Frege's system, the modern predicate calculus analyzes /loves/ as a
two-place relation (Lxy) rather than a function; some objects stand in
the relation and others do not. The difference between Frege's
understanding of predication and the one manifest by the modern
predicate calculus is simply this: in the modern predicate calculus,
relations are taken as basic, and functions are [or can be --GF] defined
as a special case of relation, namely, those relations R such that for
any objects x, y, and z, if Rxy and Rxz, then y=z. By contrast, Frege
took functions to be more basic than relations. His logic is based on
functional application rather than predication, and so relations become
analyzed as a binary functions which map a pair of arguments to a
truth-value. Thus, a 3-place relation like /gives/ would be analyzed in
Frege's logic as a function that maps arguments x, y, and z to an
appropriate truth-value depending on whether x gives y to z; the 4-place
relation /buys/ would be analyzed as a function that maps the arguments
x, y, z, and u to an appropriate truth-value depending on whether x buys
y from z for amount u; etc."

(Edward N. Zalta, Gottlob Frege, 2005
http://plato.stanford.edu/entries/frege/)


F.

--

E-mail: info<at>simple-line<dot>de
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