[FOM] About Paradox Theory
Zuhair Abdul Ghafoor Al-Johar
zaljohar at yahoo.com
Tue Sep 13 16:56:09 EDT 2011
Dear FOMers,
I would like to know what would be the foundational
initial stand point on the following suggestion
of mine about establishing a comprehensive study
of logical paradoxes on their own.
A study that characterize those paradoxes in such
a manner that each category of paradoxes would bear
specific implications especially as regards machinery
of producing further paradoxes within each category.
A study that also depicts various inter-category issues,
i.e. the relation between each category to the other.
A study that also aids us in building a strategy for
avoiding such paradoxes when constructing theories
of interest.
To illustrate an example of the above,
lets take Russell paradox, it seems that we
can build up a set of those paradoxes where all
can be described by a single simple rule that is:
The set of all sets that are not e_i members
of themselves cannot exist, because otherwise
the i_singleton set of it would be paradoxical
i.e. both e_i of itself and not e_i of itself.
e_i is defined recursively as
x e_0 y iff x e y
x e_i y iff Exist z. z e_i-1 y and x e z
for i=1,2,3,......
0_singleton (x) = y iff y=x
i_singleton (x) = y iff Exist! z. z e y and z = (i-1)_singleton (x)
for i=1,2,3,....
Let's call the above Russell paradox category.
Now this shows this paradox in more depth than the usual
presentation which is only the top tier of the above.
Now we need to study further how can we produce other paradoxes
by working within the above category. For example, lets take
the second tier of Russell paradox category, i.e. that concerned
with e_1 membership, now we can have a sub-paradox of this
which is already known as Lesniewski's paradox which is
the set of all singletons that are not in their members
cannot exist, because the singleton of that set is paradoxical
i.e. it is in its sole member and not in its sole member. This is
exactly the same argument behind the second tier of Russell's paradox
category shown above. Seeing this connection one can go further
and define further Lesniewski's paradox category in an exactly
similar manner as to how it is defined above for Russell's.
Another example is Russell's paradox of second order logic,
i.e. on predicates, this is also can be viewed as extending
the same argument but to a higher language, and i think
this can also be extended into a category like the above,
and possibly has sub-categories of it similar to Lesniewski's
paradox category, that is besides many other possible
sub-categories.
I personally think that a comprehensive study of paradoxes
would be fruitful in the sense of increasing the awareness
about them and thus facilitating constructing theories
of general interest that can avoids them.
So what FOM would say about that?
Regards
Zuhair
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