# [FOM] About Paradox Theory

Peter Arndt inftycat at gmail.com
Sat Sep 24 08:07:55 EDT 2011

```Hi Taylor!

Indeed this view on contradictions has been taken up by Rene Guitart, he
calls it moving logic. He said something about this in conference talk in
Lisbon 2010, I don't know of a published account; here are an abstract of
some work and some slides which do however not expand on the connection with
contradictions:
http://rene.guitart.pagesperso-orange.fr/textespublications/guitart05ml.pdf
http://saxo.univ-littoral.fr/CT08/slides/Guitart.ppt
Maybe he has some written account that has slipped my attention - you might
ask him.

Also Walter Carnielli has devised a "polynomial semantics" for propositional
logics, which includes semantics in F_4. He gave a course on this and
provided download links on this page (the example F_4 is worked out
somewhere there):
http://www.cle.unicamp.br/ebl2011/logic_school.php
I am not aware of any special attention to the treatment of contradictions
in these notes, but as he has worked on paraconsistent logic, it is well
possible that Carnielli has some thoughts on this...

Cheers,
Peter

On Fri, Sep 23, 2011 at 10:20 PM, Taylor Dupuy <taylor.dupuy at gmail.com>wrote:

> There is an interesting way to view contradictions in Propositional logic
> via arithmetization:
>
> 1. Convert propositions, with propositional variables X1,X2,... into
> polynomials over FF_2=ZZ/ 2 ZZ with indeterminates x1, x2, ...
> X & Y <-> xy
> X or Y <-> xy + x +y
> !X <-> x+1
> True <-> 1
> False <-> 0
>
> 2. Satisfiability of a proposition is equivalent existence of solutions of
> polynomial equations in FF_2. If P(X1,X2,...,Xn) is a proposition and
> p(x1,x2,...,xn) is its associated polynomial the P is satisfiable if and
> only if p=1 admits a solution over FF_2. For example the proposition X & !X
> is clearly not satisfiable which corresponds to the fact that x(x+1)=1 or
> x^2+x +1 =0 has no solutions over FF_2.
>
> One would think then that a canonical way to extend truth values in
> propostional logic would be to allow values in FF_2[x]/(x^2+x+1) (or the
> relevant extension for an unsatisfiable proposition) and carry the truth
> tables back over to logic via (1). I've never heard of a theory but have
> always wondered if its already been developed and what it would look like.
>
> -Does a theory using this idea in propositional logic exist? Can one
> make proofs make sense with these truth values?
> -Does a first order theory using ideas like this make sense?
>
> I could never get past what the methods of proof should be. Thoughts?
>
> Cheers,
> Taylor
>
>
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