[FOM] About Paradox Theory

Vaughan Pratt pratt at cs.stanford.edu
Sat Sep 17 20:36:12 EDT 2011

Very pretty.  Also a convincing demonstration of the power of FOL as a 
Turing universal programming language.

As an answer to Thomas Forster's question, it is more set-theoretic than 
~EyAx[F(xy)<-->  ~F(xx)] only to the extent that it relies on singleton 
formation, for every x there exists (at least one copy of) {x}.  It 
doesn't even depend on extensionality of sets, multiple copies of {x} 
are allowed.

Anyone care to comment on whether it's intuitionistically valid as it 
stands?  (I presume one can hedge by inserting a few "surely"'s, 
formalized as not-not, to make it so, via Goedel's translation.)  Nik 

Vaughan Pratt

On 9/17/2011 8:05 AM, hdeutsch at ilstu.edu wrote:
> 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?

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