[FOM] About Paradox Theory

[David Auerbach]

James Thomson (Thomson, 'On Some Paradoxes', in Analytical Philosophy (Ist series) ed,. R. J. Butler, pp, 104-119) makes this point, with some nuances and somewhat long ago, in wonderful detail.  (It's a hard article to find, but I have a PDF if anyone wants it.)



David Auerbach                                                      auerbach at ncsu.edu
Department of Philosophy and Religious Studies
NCSU
Raleigh, NC 27695-8103

On Sep 14, 2011, at 4:03 PM, charlie wrote:

> 	I'm sure your project has merit, but I can never overcome "Russell's Paradox" because of the following theorem of  first-order logic.
> 
> 	   ~EyAx[F(xy) <--> ~F(xx)]
> 	
> 	       As a consequence, I tend to dismiss R's Paradox as having nothing to do with sets or anything else in particular.   I'm sure I must be wrong in this, as his paradox regarding classes or sets enjoys such wide popularity, perhaps due to the Frege connection, and was perhaps a guiding light for improved set theories.   So, perhaps someone will please give me a good reason not to trivialize it so.  Perhaps I'm making some significant mistake. I'd be glad to be corrected.
> 
> Charlie
> 
> 
> 
> 
> 
> 
> On Sep 13, 2011, at 3:56 PM, Zuhair Abdul Ghafoor Al-Johar wrote:
> 
>> 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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David Auerbach                                                      auerbach at ncsu.edu
Department of Philosophy and Religious Studies
NCSU
Raleigh, NC 27695-8103