Re: wittgensteins Tractarian logic any suggestions.?

On Wed, 21 Feb 2007 19:00:10 +0100, G. Frege <nomail@invalid> wrote:


TLP 5.52

"If § has as its values all the values of a function fx
for all values of x then
_
N(§) = ~(Ex).fx."

It seems that § refers to (certain) statements. Let's write

(a) § ==> P

if § refers just to the statement P ("has P as its value"). And

(b) § ==> P, Q, R

if § refers just to the statement P, Q, R ("has as its values P, Q,
R). Finally:

(c) § ==> fx for all x

if § "has as its values [refers to] all the values of a function fx
for all values of x".

Consider -for example- the function fx : "x is a man".

(If our universe of discourse would only consist of finitely many
objects, say, a, b, c, then this would amount to § ==> fa, fb, fc.)

Now Wittgenstein writes
_
N(§)

to express the "conjunction" (in the finite case) of the negations of
all the statements § refers to.

_
"5.502 N(§) is the negation of all the values of the [...]
variable §."


Hence in case of (a) we would have:
_
N(§) = ~P.

In case of (b) we would have:
_
N(§) = ~P . ~Q . ~R.

_
"5.51 If § has only one value, then N(§) = ~p (not p), if it
_
has two values then N(§) = ~p . ~q (neither p nor q).


And in case of (c) [following Wittgenstein] we would have:
_
N(§) = (x).~fx ,

or equivalently
_
N(§) = ~(Ex).fx.

(If our universe of discourse would only consist of finitely many
objects, say, a, b, c, then this would amount to
_
N(§) = ~fa . ~fb . ~fc ( = (x).~fx in this case) ,

or
_
N(§) = ~(fa v fb v fc) (= ~(Ex).fx in this case). )


Funny coincident (I just stumbled over). "Owen" wrote (at some other
place):


"N(p) = (p nor p) = ~p.
N(p,q) = (p nor q) = ~p & ~q.
N(p,q,r) = ~p & ~q & ~r.
N(Fa,Fb,Fc,...) = ~Fa & ~Fb & ~Fc & ... = ~ExFx.
etc."

This seems to "confirm" my own analysis. :-)


F.

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