[FOM] About Paradox Theory

Fri Sep 16 13:31:44 EDT 2011

There are first order theorems corresponding to the fact that  
separation implies that there is no universal set (in ZF) and even one  
corresponding to the paradox of grounded classes.  There is an  
interesting list of such theorems in Kalish, Montague, and Mar:  
Techniques of Formal Reasoning, but I'm sorry I don't have a copy  
available so I can't give the page number right now.

Harry

Quoting David Auerbach <auerbach at ncsu.edu>:

> 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
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> _______________________________________________
>>> FOM mailing list
>>> FOM at cs.nyu.edu
>>> http://www.cs.nyu.edu/mailman/listinfo/fom
>>
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>
>
> David Auerbach                                                       
> auerbach at ncsu.edu
> Department of Philosophy and Religious Studies
> NCSU
> Raleigh, NC 27695-8103
>
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