We study pairs of reals that are mutually Martin-Löf random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgen’s theorem holds for noncomputable probability measures, too. We study, for a given real $A$, the independence spectrum of $A$, the set of all $B$ such that there exists a probability measure $\mu$ so that $\mu \{A,B\}=0$ and $(A,B)$ is $(\mu \times \mu )$-random. We prove that if $A$ is computably enumerable (c.e.), then no ${\Delta }_2^0$-set is in the independence spectrum of $A$. We obtain applications of this fact to PA degrees. In particular, we show that if $A$ is c.e. and $P$ is of PA degree so that $P{\ngeqq }_{\mathrm{T}}A$, then $A\oplus P{\ge }_{\mathrm{T}}\varnothing \text{'}$.

A “new” criterion for set existence is presented, namely, that a set $\left\{x\right|\varphi \}$ should exist if the multigraph whose nodes are variables in $\varphi$ and whose edges are occurrences of atomic formulas in $\varphi$ is acyclic. Formulas with acyclic graphs are stratified in the sense of New Foundations, so consistency of the set theory with weak extensionality and acyclic comprehension follows from the consistency of Jensen’s system NFU. It is much less obvious, but turns out to be the case, that this theory is equivalent to NFU: it appears at first blush that it ought to be weaker. This paper verifies that acyclic comprehension and stratified comprehension are equivalent by verifying that each axiom in a finite axiomatization of stratified comprehension follows from acyclic comprehension.

The sequential form of a statement

$$\forall \xi \left(B\right(\xi )\to \exists \zeta A(\xi ,\zeta \left)\right)\left(†\right)$$

is the statement

$$\forall \xi (\forall nB({\xi }_n)\to \exists \zeta \forall nA({\xi }_n,{\zeta }_n\left)\right).$$

There are many classically true statements of the form (†) whose proofs lack uniformity, and therefore the corresponding sequential form is not provable in weak classical systems. The main culprit for this lack of uniformity is of course the law of excluded middle. Continuing along the lines of Hirst and Mummert, we show that if a statement of the form (†) satisfying certain syntactic requirements is provable in some weak intuitionistic system, then the proof is necessarily sufficiently uniform that the corresponding sequential form is provable in a corresponding weak classical system. Our results depend on Kleene’s realizability with functions and the Lifschitz variant thereof.

Relatively recently nested sequent systems for modal logics have come to be seen as an attractive deep-reasoning extension of familiar sequent calculi. In an earlier paper I showed that there was a strong connection between modal nested sequents and modal prefixed tableaux. In this paper I show that the connection continues to intuitionistic logic, both standard and constant domain, relating nested intuitionistic sequent calculi to intuitionistic prefixed tableaux. Modal nested sequent machinery generalizes one-sided sequent calculi—intuitionistic nested sequents similarly generalize two-sided sequents. It is noteworthy that the resulting system for constant domain intuitionistic logic is particularly simple. It amounts to a combination of intuitionistic propositional rules and classical quantifier rules, a combination that is known to be inadequate when conventional intuitionistic sequent systems are used.

We show that being low for difference tests is the same as being computable and therefore lowness for difference tests is not the same as lowness for difference randomness. This is the first known example of a randomness notion where lowness for the randomness notion and lowness for the test notion do not coincide. Additionally, we show that for every incomputable set $A$, there is a difference test ${\mathcal{T}}^A$ relative to $A$ which cannot even be covered by finitely many unrelativized difference tests.

We investigate the reverse-mathematical status of a version of the Baire category theorem known as $\mathrm{BCT}$. In a 1993 paper Brown and Simpson showed that $\mathrm{BCT}$ is provable in ${\mathsf{RCA}}_0$. We now show that $\mathrm{BCT}$ is equivalent to ${\mathsf{RCA}}_0$ over ${\mathsf{RCA}}_0^{\ast }$.

A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds; Watson (2010) suggests that Bohn’s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo–Fraenkel set theory $\mathsf{ZF}$, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: topology (and possibly other “algebraic” mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds.

The present paper continues the investigation initiated in an earlier work. After a short introduction, the notion of relative productivity is defined and a technical apparatus is developed in order to evaluate the classification of the axioms previously obtained. Some results on the semilattice of simple relative degrees are proved at the end of Section 2. Section 3 adds some concluding remarks.

Brady has shown how to define a class of deep relevant logics from Meyer’s crystal lattice CL. The aim of this paper is to generalize Brady’s result by showing how to define a class of deep relevant logics from each weak relevant matrix (weak relevant matrices only verify logics with the variable-sharing property). A class of deep relevant logics not included in R-Mingle is defined.

Béziau developed the paraconsistent logic $\mathbb{Z}$, which is definitionally equivalent to the modal logic $\mathbb{S}5$, and gave an axiomatization of the logic $\mathbb{Z}$: the system HZ. Omori and Waragai proved that some axioms of HZ are not independent and then proposed another axiomatization for $\mathbb{Z}$ that includes two inference rules and helps to understand the relation between $\mathbb{S}5$ and classical propositional logic. In the present paper, we analyze logic $\mathbb{Z}$ in detail; in the process we also construct a family of paraconsistent logics that are characterized by different properties that are relevant in the study of logics.