In this paper we present a new modeling for belief revision that is what we term irrevocable. This modeling is of philosophical interest since it captures some features of suppositional reasoning, and of formal interest since it is closely connected with AGM, yet provides for iterated belief revision. The analysis is couched in terms of dynamic doxastic logic.

Using nonstandard methods we construct a model of an induction scheme called $\hat{R}^2_3$ inside a "resource" of the form $\{M(a) : M $ is a Turing machine of code $ \leq r , \mbox{ and } M(a)$ is calculated in less than $2^{\vert\vert a\vert\vert^r}\mbox{ steps}\}$, where $\vert x\vert$ means the length of the binary expansion of $x$ and $a,r$ are nonstandard parameters in a model of $S^1_3$. As a consequence we obtain a model theoretic proof of a witnessing theorem for this theory by functions computable in time $2^{\vert n\vert^{O(1)}}$, a result first obtained by Buss, Krajícek, and Takeuti using proof theory.

In this paper we define the notion of $m$-linearity for $m \in \omega(m>1)$ and discuss interpretability (and noninterpretability) of $m$-linear orders in structures and theories.

Semi-contraction is a withdrawal operation defined by Fermé in "On the logic of theory change: Contraction without recovery." In this paper we propose: (1) an axiomatic characterization of semi-contraction; (2) an alternative construction for semi-contraction based on semi-saturatable sets, inspired by Levi's saturatable sets; (3) a special kind of semi-contraction that satisfies the Lindström and Rabinowicz interpolation thesis.

This paper is an attempt to show that my work to establish numerically flexible quantifiers for the syllogism can be aptly combined with the term logic advanced by Sommers, Englebretsen, and others.

The notion of strictly primitive recursive realizability is further investigated, and the realizable prenex sentences, which coincide with primitive recursive truths of classical arithmetic, are characterized as precisely those provable in transfinite progressions $\{\mathrm{PRA}(b) \vert b \in \underline{\mathrm{O}}\}$ over a fragment $\mbox{PR-}(\Sigma^{0}_{1}\mbox{-IR})$ of intuitionistic arithmetic. The progressions are based on uniform reflection principles of bounded complexity iterated along initial segments of a primitive recursively formulated system $\mathrm{\underline{O}}$ of notations for constructive ordinals. A semiformal system closed under a primitive recursively restricted $\omega$-rule is described and proved equivalent to the transfinite progressions with respect to the prenex sentences.

We consider a language containing partial predicates for subjective knowability and truth. For this language, inductive hierarchy rules are proposed which build up the extension and anti-extension of these partial predicates in stages. The logical interaction between the extension of the truth predicate and the anti-extension of the knowability predicate is investigated.

We modify the semantics of topological modal logic, a language due to Moss and Parikh. This enables us to study the corresponding theory of further classes of subset spaces. In the paper we deal with spaces where every chain of opens fulfils a certain finiteness condition. We consider both a local finiteness condition relevant to points and a global one concerning the whole frame. Completeness of the appearing logical systems, which turn out to be generalizations of the well-known modal system G, can be obtained in the same manner as in the case of the general subset space logic. It is our main purpose to show that the systems differ with regard to the finite model property.

Say that a property is topological if and only if it is invariant under homeomorphism. Homeomorphism would be a successful criterion for the equivalence of logical systems only if every logically significant property of every logical system were topological. Alas, homeomorphisms are sometimes insensitive to distinctions that logicians value: properties such as functional completeness are not topological. So logics are not just devices for exploring closure topologies. One still wonders, though, how much of logic is topological. This essay examines some logically significant properties that are topological (or are topological in some important class). In the process, we learn something about the conditions under which the meaning of a connective can be "given by the connective's role in inference."