This paper deals with the modal logics associated with (possibly nonstandard) provability predicates of Peano Arithmetic. One of our goals is to present some modal systems having the fixed point property and not extending the Gödel-Löb system GL. We prove that, for every $ n\geq 2, K+\square(\square^{n-1}p\rightarrow p)\rightarrow\square p$ has the explicit fixed point property. Our main result states that every complete modal logic L having the Craig's interpolation property and such that $ L\vdash\Delta(\nabla(p)\rightarrow p)\rightarrow\Delta(p)$, where $ \nabla(p)$ and $ \Delta(p)$ are suitable modal formulas, has the explicit fixed point property.

In the context of a distributive lattice we specify the sort of mappings that could be generally called ''negations'' and study their behavior under iteration. We show that there are periodic and nonperiodic ones. Natural periodic negations exist with periods 2, 3, and 4 and pace 2, as well as natural nonperiodic ones, arising from the interaction of interior and quasi interior mappings with the pseudocomplement. For any n and any even $s<n$, negations of period n and pace s can also be constructed, but in a rather ad hoc and trivial way.

A pseudotree $ \langle T, \leq \rangle$ is a partially ordered set for which $ \{ u \in T: u \leq t \}$ is a linear ordering for each $ t \in T$. Define $ \mathcal{B}(T)$, the pseudo treealgebra over T, as the subalgebra of the power set of T generated by $ \{ b_{t} : t \in T \}$ where $ b_{t} = \{ u \in T : t \leq u \}$. It is shown that every pseudo treealgebra is embeddable into an interval algebra; thus it is a retractive Boolean algebra. Moreover, superatomicity of $ \mathcal{B}(T)$ is described using conditions on $ \langle T, \leq \rangle$.

We construct two recursive models of fragments of set theory. We also show that the fragments of Kripke-Platek set theory that prove $ \varepsilon$-induction for $ \Sigma_1$-formulas have no recursive models but the standard model of the hereditarily finite sets.

This note is a continuation of our former paper ''Complexity of the r-query tautologies in the presence of a generic oracle.'' We give a very short direct proof of the nonexistence of t-generic oracles, a result obtained first by Dowd. We also reconstitute a proof of Dowd's result that the class of all r-generic oracles in his sense has Lebesgue measure one.

This note exploits back-and-forth characteristics to construct, using a single method, short proofs for ten classics of first-order and modal logic: interpolation theorems, preservation theorems, and Lindström's theorem.