This paper discusses the relation between the minimal positive relevant logic B$_{+}$ and intersection and union type theories. There is a marvelous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking B$\wedge$T of B$_{+}$, which saves implication $\to$ and conjunction $\wedge$ but drops disjunction $\vee$. The filter models of the $\lambda$-calculus (and its intimate partner Combinatory Logic CL) of the first author and her coauthors then become theory models of these calculi. (The logician's "theory" is the algebraist's "filter".) The coincidence extends to a dual interpretation of key particles—the subtype $\leq$ translates to provable $\to$, type intersection $\cap$ to conjunction $\wedge$, function space $\to$ to implication, and whole domain $\omega$ to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper, and to be expected that type union $\cup$ should correspond to the logical disjunction $\vee$ of B$_{+}$. But the simulation of functional application by a fusion (or modus ponens product) operation $\circ$ on theories leaves the key Bubbling lemma of work on ITD unprovable for the $\vee$-prime theories now appropriate for the modeling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of $\lambda$ and CL.

Paraconsistent logics are, by definition, inconsistency tolerant: In a paraconsistent logic, inconsistencies need not entail everything. However, there is more than one way a body of information can be inconsistent. In this paper I distinguish {contradictions} from {other inconsistencies}, and I show that several different logics are, in an important sense, "paraconsistent" in virtue of being inconsistency tolerant without thereby being contradiction tolerant. For example, even though no inconsistencies are tolerated by intuitionistic propositional logic, some inconsistencies are tolerated by intuitionistic predicate logic. In this way, intuitionistic predicate logic is, in a mild sense, paraconsistent. So too are orthologic and quantum propositional logic and other formal systems. Given this fact, a widespread view—that traditional paraconsistent logics are especially repugnant because they countenance inconsistencies—is undercut. Many well-understood nonclassical logics countenance inconsistencies as well.

We show that if N and M are transitive models of ZFA such that N $\subseteq$ M, N and M have the same kernel and same set of atoms, and M $\models$ AC, then N is a Fraenkel-Mostowski-Specker (FMS) submodel of M if and only if M is a generic extension of N by some almost homogeneous notion of forcing. We also develop a slightly modified notion of FMS submodels to characterize the case where M is a generic extension of N not necessarily by an almost homogeneous notion of forcing.

Shortest possible axiomatizations for the strict implicational fragments of the modal logics S4 and S5 are reported. Among these axiomatizations is included a shortest single axiom for implicational S4—which to our knowledge is the first reported single axiom for that system—and several new shortest single axioms for implicational S5. A variety of automated reasoning strategies were essential to our discoveries.

We show that if R is a nonconstant regular (semi-)local subring of a rational function field over an algebraically closed field of characteristic zero, Hilbert's Tenth Problem for this ring R has a negative answer; that is, there is no algorithm to decide whether an arbitrary Diophantine equation over R has solutions over R or not. This result can be seen as evidence for the fact that the corresponding problem for the full rational field is also unsolvable.