[FOM] About Paradox Theory

hdeutsch at ilstu.edu hdeutsch at ilstu.edu
Sat Sep 17 11:05:04 EDT 2011

Here is the argument concerning the "paradox of grounded classes" to  
save people from having to look it up:

The following argument is first-order valid:

AyEzAx(F(xz) <--> x=y).  Therefore,

-EwAx(F(xw) <--> Au([F(xu) --> Ey(F(yu) & -Ez{F(zu) & F(zy)])]).

The claim is that this is the paradox of grounded classes "as  
described in [Montague's paper mentioned in my last message]".

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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