The Kuznetsov-Index of a modal logic is the least cardinal $\mu$ such that any consistent formula has a Kripke-model of size $\leq \mu$ if it has a Kripke-model at all. The Kuznetsov-Spectrum is the set of all Kuznetsov-Indices of modal logics with countably many operators. It has been shown by Thomason that there are tense logics with Kuznetsov-Index $\beth_{\omega+\omega}$. Futhermore, Chagrov has constructed an extension of K4 with Kuznetsov-Index $\beth_{\omega}$. We will show here that for each countable ordinal $\lambda$ there are logics with Kuznetsov-Index $\beth_{\lambda}$. Furthermore, we show that the Kuznetsov-Spectrum is identical to the spectrum of indices for $\Pi^1_1$-theories which is likewise defined. A particular consequence is the following. If inaccessible (weakly compact, measurable) cardinals exist, then the least inaccessible (weakly compact, measurable) cardinal is also a Kuznetsov-Index.

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 $\mathit{NFU}$.

This paper presents an exposition of subsystems $\mathit{NFP}$ and $\mathit{NFI}$ of Quine's $\mathit{NF}$, originally defined and shown to be consistent by Crabbé, along with related systems $\mathit{TTP}$ and $\mathit{TTI}$ of type theory. A proof that $\mathit{TTP}$ (and so $\mathit{NFP}$) interpret the ramified theory of types is presented (this is a simplified exposition of a result of Crabbé). The new result that the consistency strength of $\mathit{NFI}$ is the same as that of $\mathit{PA}_2$ is demonstrated. It will also be shown that $\mathit{NFI}$ cannot be finitely axiomatized (as can $\mathit{NF}$ and $\mathit{NFP}$).

The role of foundation with respect to transitive closure in the Zermelo system Z has been investigated by Boffa; our aim is to explore the role of antifoundation. We start by showing the consistency of "Z $+$ antifoundation $+$ transitive closure" relative to Z (by a technique well known for ZF). Further, we introduce a "weak replacement principle" (deductible from antifoundation and transitive closure) and study the relations among these three statements in Z via interpretations. Finally, we give some adaptations for ZF without infinity.

An open formalism for arithmetic is presented based on first-order logic supplemented by a very strictly controlled constructive form of the omega-rule. This formalism (which contains Peano Arithmetic) is proved (nonconstructively, of course) to be complete. Besides this main formalism, two other complete open formalisms are presented, in which the only inference rule is modus ponens. Any closure of any theorem of the main formalism is a theorem of each of these other two. This fact is proved constructively for the stronger of them and nonconstructively for the weaker one. There is, though, an interesting counterpart: the consistency of the weaker formalism can be proved finitarily.

A generalized Veltman semantics developed by de Jongh is used to investigate correspondences between several extensions of intepretability logic $\mathit{IL}$. In this paper we present some new results on independences.

Standard possible world semantics for propositional modal languages ignore truth-value gaps. However, simple considerations suggest that it should not be so. In Section 1, I identify what I take to be a correct truth-clause for necessity under the assumption that some possible worlds are incomplete (i.e., "at" which some propositions lack a truth-value). In Section 2, I build a world semantics, the semantics of TV-models, for standard modal propositional languages, which agrees with the truth-clause for necessity previously identified. Sections 3-5 are devoted to systematic concerns. In particular, in Section 4, Prior's system $Q$ (propositional version) is given a TV-models semantics and proved adequate (i.e., sound and complete) with respect to it.

McTaggart assumed (1) that propositions cannot change in truth value and (2) if (a) there is real change, then (b) events must acquire the absolute property of being present and then lose this property. He held that {1,2b} is an inconsistent set and thus inferred $\sim$2a—that there is no real change. The B theory rejects 2 and the A theory rejects 1. I accept 1, 2, 2a, and consequently, 2b, and argue that this is consistent. There is an absolute property of presentness, but we can never predicate this property without producing an assertion in which that property is "chronologized." If at $t$, we attribute presentness, we thereby attribute presentness-at-$t$. We can grasp the existence of an absolute property which we are unable to attribute absolutely.

This paper is concerned with a natural deduction system for First Degree Entailment (FDE). First, we exhibit a brief history of FDE and of combined systems whose underlying idea is used in developing the natural deduction system. Then, after presenting the language and a semantics of FDE, we develop a natural deduction system for FDE. We then prove soundness and completeness of the system with respect to the semantics. The system neatly represents the four-valued semantics for FDE.

In 1926, Mally proposed the first formal deontic system. As Mally and others soon realized, this system had some rather strange consequences. We show that the strangeness of Mally's system is not so much due to Mally's informal deontic principles as to the fact that he formalized those principles in terms of the propositional calculus. If they are formalized in terms of relevant logic rather than classical logic, one obtains a system which is related to Anderson's relevant deontic logic and not nearly as strange as Mally's own system.