|
5.4
Here it becomes clear that there are no such things as "logical
objects" or "logical constants" (in the sense of Frege and
Russell). |
|
|
5.41
For all those results of truth-operations on truth-functions are identical,
which are one and the same truth-function of elementary propositions. |
|
|
5. 42
That v,
 ,
etc., are not relations in the sense of right and left, etc., is obvious.
The possibility of
crosswise definition of the logical "primitive signs" of Frege
and Russell shows by itself that these are not primitive signs and that
they signify no relations.
And it is obvious
that the "
 "
which we define by means of "~" and "v" is identical
with that by which we define "v" with the help of "~",
and that this "v" is the same as the first, and so on. |
|
|
5.43 That
from a fact p an infinite number of others should follow,
namely, ~~p, ~~~~p, etc., is indeed hardly to be believed,
and it is no less wonderful that the infinite number of propositions of
logic (of mathematics) shold follow from half a dozen "primitive
propositions".
But the
propositions of logic say the same thing. That is, nothing |
|
|
5.44 Truth-functions
are not material functions.
If e.g. an
affirmation can be produced by repeated denial, is the denial -- in any
sense -- contained in the affirmation?
Does "~~p"
deny "~p", or does it affirm p; or both?
The proposition
"~~p" does not treat of denial as an object, but the
possibility of denial is already prejudged in affirmation.
And
if there was an object called "~", then "~~p"
would have to say something other than "p". For the one
proposition would then treat of ~, the other would not. |
|
|
5.45 If
there are logical primitive signs a correct logic must make clear their
position relative to one another and justify their existence. The
construction of logic out of its primitive signs must become clear.
|
|
|
5.46 When
we have rightly introduced the logical signs, the sense of all their
combinations has been already introduced with them: therefore not only
"p v q" but also "~(p v ~q)",
etc. etc. We should then already have introduced the effect of all
possible combinations of brackets; and it would then have become clear
that the proper general primitive signs are not "p v q",
"(
 x) . fx",
etc., but the most general form of their combinations.
|
|
|
5.47 It
is clear that everything which can be said beforehand about the
form of all propositions at all can be said on one occasion.
For all logical
operations are already contained in the elementary proposition. For "fa"
says the same as "(
 x) . fx . x=a".
Where there is
composition, there is argument and function, and where these are, all
logical constants already are.
One could say: the
one logical constant is that which all propositions, according to
their nature, have in common with one another.
That however is the
general form of proposition.
|
|
|
|
|
|
|
|
| . |
|
|