A critical note has just appeared in the Review of Symbolic Logic:

"The Inadequcy of a Proposed Paraconsistent Set Theoy," by Frode Bjørdal. This paper claims that the set theory I used in "Transfinite Numbers in Paraconsistent Set Theory" has

for all x, ~(x = x)

as a theorem. That would be pretty lousy news. My reply to Bjørdal's note will appear shortly in the RSL. Here's the short version:

It is not a theorem that ~(x = x) for all x.

Here's a slightly longer version.

In my last post, I indicated that there is now almost overwhelming evidence that the system in TFNPST is not trivial. Still, a system could be logically non-trivial , but still be mathematically useless. Everything being not self identical is an example of the latter.

But here's all that Bjørdal has pointed out. There is a set X such that every x both is and is not in X. Almost anyone who has worked on this kind of crazy set theory knows that. Arief Daynes adds it as an axiom. Just take {x: L}, where L is your favorite true contradiction.

Unfortunately, Bjørdal wants to define identity, x=y, to mean that x and y are in all and only the same sets. Now if x isn't even in all the same sets as x itself, well then, yes, ~(x=x), for every x. Insist on writing identity signs in the wrong places, and you get wrong statements about identity.

This emphasizes -- and it's what I argue in full paper length elsewhere -- that this isn't the right way to define identity in intensional contexts. We already knew that, because Clark Kent is Superman, even if one guy wears glasses and the other does not.

It is true that identical objects are in all and only the same sets. It is, according to the rule of substitution. It's just that Leibniz' law can't be the definition of identity.

I'm reminded of some critiques of dialetheism that go like this. Let's define the zero place sad-face connective :( for the disjunction of all contradictions, and agree to pronounce the connective "so bad!". Now suppose the Russell paradox were a true contradiction; then :( follows. But :( is so bad! Therefore dialetheism is inadequate. Hmm....