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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