This paper gives a propositional reformulation of the fixed-point problem posed by Gupta and Belnap, using the stipulation logic of Visser. After presenting a solution for clones of three-valued operators that include the constant functions, I determine the maximal three-valued clones with constants that have the fixed-point property, giving different characterizations of them.

It is proved that in a suitable intuitionistic, locally classical, version of the theory ZFC deprived of the axiom of infinity, the requirement that every set be finite is equivalent to the assertion that every ordinal is a natural number. Moreover, the theory obtained with the addition of these finiteness assumptions is equivalent to a theory of hereditarily finite sets, developed by Previale in "Induction and foundation in the theory of hereditarily finite sets." This solves some problems stated there. The analysis is undertaken using for each of these results a limited fragment of the relevant theory.

Some of the basic substructural logics are shown by Ono to have the disjunction property (DP) by using cut elimination of sequent calculi for these logics. On the other hand, this syntactic method works only for a limited number of substructural logics. Here we show that Maksimova's criterion on the DP of superintuitionistic logics can be naturally extended to one on the DP of substructural logics over FL. By using this, we show the DP for some of the substructural logics for which syntactic methods don't work well.

This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. This clarifies the nature of the equivalence of PA and ZF−inf and corrects some errors in the literature. We also survey the restrictions of the Ackermann interpretation and its inverse to subsystems of PA and ZF−inf, where full induction, replacement, or separation is not assumed. The paper concludes with a discussion on the problems one faces when the totality of exponentiation fails, or when the existence of unordered pairs or power sets is not guaranteed.

We deal with the modal logic of cluster-decomposable Kripke interpretations, present an axiomatization, and prove some additional results regarding this logic.

There are simple, purely syntactic axiomatic proof systems for both the logical truths and the logical falsehoods of propositional logic. However, to date no such system has been developed for the logical contingencies, that is, formulas that are both satisfiable and falsifiable. This paper formalizes the purely syntactic axiomatic proof systems for the logical contingencies and proves its soundness as well as completeness.

This paper obtains lower and upper bounds for the number of alternations of bounded quantifiers needed to express all formulas in certain ordered Abelian groups admitting elimination of unbounded quantifiers. The paper also establishes model-theoretic tests for equivalence to a formula with a given number of alternations of bounded quantifiers.