Counterexamples to reassurance relative to a relation between models of the logic of paradox are provided. Another relation, designed to fix the problem in logic without equality, is introduced and discussed in connection with the issue of classical recapture.

We prove the cut-elimination theorem, Gentzen's Hauptsatz, for the system for stratified comprehension, i. e. Quine's NF minus extensionality.

We characterize the notion of validity relatively to models, for comprehension axioms, containing gluts.

We characterize the 3-stratifiable theorems of NF as a 3-stratifiable extension of NF 3 ; and show that NF is equiconsistent with TT plus raising type axioms for sentences asserting the existence of some predicate over an atomic Boolean algebra.

It is shown that the 3-stratifiable sentences are equivalent in to truth-functional combinations of sentences about objects, sets of objects, sets of sets of objects, and sentences stating that there are at least urelements. This is then used to characterize the closed 3-stratifiable theorems of with an externally infinite number of urelements, as those that can be nearly proved in with an externally infinite number of urelements. As a byproduct we obtain a rather simple demonstration of the consistency of 3-stratifiable (...) extensions of. (shrink)

It is shown that the 3-stratifiable sentences are equivalent in $\mathit{NFU}$ to truth-functional combinations of sentences about objects, sets of objects, sets of sets of objects, and sentences stating that there are at least $n$ urelements. This is then used to characterize the closed 3-stratifiable theorems of $\mathit{NFU}$ with an externally infinite number of urelements, as those that can be nearly proved in $\mathit{TTU}$ with an externally infinite number of urelements. As a byproduct we obtain a rather simple demonstration of (...) the consistency of 3-stratifiable extensions of $\mathit{NFU}$. (shrink)