[FOM] About Paradox Theory

Taylor Dupuy taylor.dupuy at gmail.com
Fri Sep 23 16:20:46 EDT 2011

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?

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