[FOM] About Paradox Theory

hdeutsch at ilstu.edu hdeutsch at ilstu.edu
Sat Sep 17 10:46:45 EDT 2011

The claim that there is a valid first order argument corresponding to  
the "paradox of grounded classes" is made concerning exercise 44,  
p.285 of Logic: Techniques of Formal Reasoning by Kalish, Montague,  
and Mar.  There is a reference to Montague, R., "On the Paradox of  
Grounded Classes," JSL, Vol 20 (1955), p.140.  Does Montague's  
"paradox of grounded classses" differ significantly from Mirimanoff's  

Harry Deutsch

Quoting T.Forster at dpmms.cam.ac.uk:

> Vaughan, Agreed, but how then *would* you characterise the  
> difference between Russell's paradox and other indisputably  
> set-theoretic paradoxes such as Mirimanoff? Charlie has pointed to  
> *something*. what do you want to say about that something?
> On Sep 16 2011, Vaughan Pratt wrote:
>> On 9/14/2011 1:03 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
>> This theorem holds in a Boolean topos, but I don't know how much  
>> further you can take it than that, those better grounded in  
>> category theory should be able to say.  The theorem is  
>> set-theoretic to the extent that the category Set is the canonical  
>> Boolean topos, so I don't think it's fair to say it has nothing to  
>> do with sets.
>> In less categorical language, the semantics with which you give  
>> this sentence meaning is set-theoretic.
>> Vaughan Pratt
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