In what I have read (long ago) on multivalued logics, the classical paradoxes in binary logic can fairly straightforwardly be extended to multi-valued logic.
For instance, S := \{ x | x \in x\; \mbox{is not true} \} sufficies for at least one ternary logic.
i'm going to have to look at this further. since you replaced it with the word false, i'd just like to say that things are not exclusively true or false anymore but possibly the third truth value. just for everyone else, i'll put what i know about 3 valued logic here:
let 0 mean F, .5 mean the third truth value M, and 1 mean T.
V(P) gives the truth value of the property (aka well formed formula) P. to use "max" below, we could just say F < M < T.
V(AvB)=V(BvA)=max{V(A),V(B)}.
V(~A)=1-V(A).
(in this language, your S is S:=\left\{ x\in U:V\left( x\in x\right) \neq T\right\} and i'll think about this. is that something you can state in 1st order logic as the subsets axioms is stated?)
from this, one can derive the truth tables for the other truth values using the rules A^B=~(~A v ~B), A->B = ~A v B, and <-> is what it usually is.
apparently, there are 3072 3 valued logics but i doubt all of those are generalizations of 2 valued logic.
note that V can be any function that generalizes 2valued logic and this one does. let me now write out some truth tables, ending in the truth table for the argument by contradiction, which is \left[ A\rightarrow \left( B\leftrightarrow \symbol{126}B\right) \right] \rightarrow \symbol{126}A. the first two columns will be the truth values of A and ~A. then there will be a double bar ||. the next truth values will be B and ~B, followed by B->~B and ~B->B. the next pair will be B<->~B and A->(B<->~B). the final truth value will be ~A. the main result is that the final values are not always T; ie, this argument is not applicable because it's not a tautology anymore.
1.TF||TF||FT||FF||T
2.TF||MM||MM||MM||M
3.TF||FT||TF||FF||T
4.MM||TF||FT||FM||M
5.MM||MM||MM||MM||M
6.MM||FT||TF||FM||M
7.FT||TF||FT||FT||T
8.FT||MM||MM||MT||T
9.FT||FT||TF||FT||T
the standard russell's paradox is to use the following:
A states the universal set exists (normally, ~A is T)
B states that S ∈ S.
and there's the rub; we need to know what a "safe" set of formulas is. This is a purely metamathematical concern; I can't see any way it could be written formally.
it would require quantifying over wffs as far as i can see. this would be added somewhere:
if there is a wff such that it implies a contradiction, then there is no subset with that wff as a defining property.
yes, this would involve an investigation of "safe" wffs. we would only have to try this if one were to not allow 3-valued logic to get around what russell's paradox is currently.
there's no sign that some other paradox won't rule U out; if there are others, do they also rely on contradiction which is no longer a tautology in 3 valued logic? I'm not suggesting we use 3 valued logic for everything. it is a generalization of 2 valued logic used only when necessary such as to say things like "from one perspective, russell's theorem is a nontautology."
Anyways, it is an interesting exercise to formally write up your ternary logic and see if it really sufficies. I bet that replacing "p \notin q" in Cantor's argument with "p \in q is false or <name of third logical value>", then you can still derive a contradiction.
i'll look at your S. thanks for submitting it.
organic,
i think what I've been doing is what you're after but it only applies to the universal set, not just every infinite set. you said it, in spirit: binary logic cannot handle infinitely many objects. while that's technically incorrect, it is definitely true of the absolute infinity. and i think i can show how 3 valued logic can handle russell's paradox.
for anyone interested, the stanford encyclopedia has nice articles on many-valued logic:
http://plato.stanford.edu/entries/logic-manyvalued/
there it gives the possibility of using it to resolve some paradoxes but it doesn't mention russell's from what i saw.
