Op 21/12/2017 om 10:13 schreef Helmut Richter:
Ambiguous definitions are one source, ambiguous reference frames
another. I try to explain the latter:
In mathematics as well as in the 'real world', a falsifiable statement
is something (A) that is said, about something (B) else, in the world
'out there'. When I say 'The weather is nice' (A), I'm saying it about a
phenomenon, the weather (B) outside of my statement's frame; and it may
be verified independently. Similarly, when I say 'Parallel straight
curves don't intersect' (A), none of these objects and properties (B)
belong themselves to the frame of my statement A.
But when I say 'What I tell now is a lie' (A), the thing I'm talking
about (A) is not something outside there, but belongs itself to the
frame of my statement! There is no independent
falsification-verification method. It is a statement that apparently
respects linguistic and/or mathematical syntax. But it is a void
statement, nothing falsifyable.
The liars' paradoxes are in this vein of void statements. They are a
feature of self-referentiality (or mutual self-reference, eg: A says 'B
is telling the truth', B says 'A is lying').
An example of the other source of confusion, ambiguous definitions, is
in the barber paradox.
The barber shaves all men in the village who don't shave themselves.
Does the barber shave himself?
The solution is in realising that the definition is incomplete WRT the
barber himself. So, depending on the case, the statement has to be
finetuned to:
1) The barber shaves all men who don't shave themselves, AND himself. Or
2) The barber shaves all men who don't shave themselves, EXCEPT himself.
In mathematical set theory, avoiding (mutual) self-reference has to do
with the distinction between element and set. A set should be distinct
from its element(s). An element can't be identical to its defining set
and vice versa. A set mustn't be describing itself, but something 'out
there'.
And as for ambiguous definitions, the equivalence of the barber paradox
is here Bertrand Russell's set of 'All sets which don't contain
themselves as an element'. Does this set contain itself? (And my
solution is similar to the barber paradox's. But it goes against the
distinction element/set anyway :-).
--
guido wugi