NFP is the predicative fragment of NF. In this system we do not allow a set to exist if it cannot be defined without using quantifiers ranging over its type or parameters of a higher type. NFI is a less restrictive fragment located between NFP and NF.We show that NFP is really weaker than NFI; similarly, NFI is weaker than NF. This result will be obtained in the following manner: on the one hand, we will show that NFP can be proved consistent in elementary arithmetic and that second order arithmetic is interpretable in NFI; on the other hand, we will prove the consistency of NFI in third order arithmetic, which is contained in NF.The paper is divided in four sections. In §1, we define the concepts needed and collect a few results together in such a way that they will be ready for later use. In §2, we will present a model-theoretic (quick) proof of the consistency of NFI (and thus of NFP). The proof will be chosen (it is not the quickest!) so as to motivate in a natural manner the details of the proof-theoretical version of it that will be presented in §3. §4 will be devoted to the axiom of infinity in NFP and NFI.

Counterexamples to reassurance relative to a "less inconsistent" 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."it has . . . a certain consonance with common sense which makes it inherently credible. This, however, is not a merit upon which much stress should be laid; for common sense is far more fallible than it likes to believe." (B. Russell) §1. Truth and falsehood. The logic of paradox is the natural paraconsistent logic arising from classical logic by simply dropping the principle of noncontradiction. 1 Thus, an LP-model A, with nonempty universe |A|, is exactly like an ordinary model, except that n-ary relation symbols are interpreted by ordered pairs of their respective extension and antiextension r + A , r − A , such that the exhaustiveness requirement (excluded middle) r + A ∪ r − A = |A| n is met. Constants and function symbols are interpreted, as usual, by objects and functions. Likewise, a valuation is still a function of the set of the variables and the valuation v to |A| extends canonically to an interpretation v A of the terms. The truth and falsehood in a model with respect to a valuation are defined inductively, as follows:

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In this paper, we show the normalization of proofs of NF (Quine's New Foundations; see [15]) minus extensionality. This system, called SF (Stratified Foundations) differs in many respects from the associated system of simple type theory. It is written in a first order language and not in a multi-sorted one, and the formulas need not be stratifiable, except in the instances of the comprehension scheme. There is a universal set, but, for a similar reason as in type theory, the paradoxical sets cannot be formed.It is not immediately apparent, however, that SF is essentially richer than type theory. But it follows from Specker's celebrated result (see [16] and [4]) that the stratifiable formula (extensionality → the universe is not well-orderable) is a theorem of SF.It is known (see [11]) that this set theory is consistent, though the consistency of NF is still an open problem.The connections between consistency and cut-elimination are rather loose. Cut-elimination generally implies consistency. But the converse is not true. In the case of set theory, for example, ZF-like systems, though consistent, cannot be freed of cuts because the separation axioms allow the formation of sets from unstratifiable formulas. There are nevertheless interesting partial results obtained when restrictions are imposed on the removable cuts (see [1] and [9]). The systems with stratifiable comprehension are the only known set-theoretic systems that enjoy full cut-elimination.