This paper continues earlier work and extends it to an intuitionistic setting. Kripke frames are used to semantically define a family of intuitionistic-like logics for which the “local” part of the truth definition is supplied by many-valued logics whose semantics are algebraically simple and natural. A uniform tableau system is given and soundness and completeness are proved. The tableau connection entails that the semantic family collectively determines just four logics, intuitionistic logic itself, and intuitionistic-like versions of FDE, K3, and LP. These, apparently, are new logics, and are of natural interest. For instance, all have the disjunction property, and standard double negation embeddings are applicable. In addition, intuitionistic analogues of ST (strict-tolerant logic) and TS (tolerant-strict) logics are defined, and shown to have the same relationships to intuitionistic logic that the usual ST and TS have to classical logic.

We investigate the existence of “generic derivations” in exponential fields. We show that exponential fields without additional compatibility conditions between derivation and exponentiation cannot support a generic derivation.

A computable graph G is computably categorical relative to a degree d if and only if for all d-computable copies B of G, there is a d-computable isomorphism f:G→B. In this paper, we prove that for every computable partially ordered set P and computable partition P=P0⊔P1, there exists a computable, computably categorical graph G and an embedding h of P into the c.e. degrees where G is computably categorical relative to all degrees in h(P0) and not computably categorical relative to any degree in h(P1). This is a generalization of a result by Downey, Harrison-Trainor, and Melnikov.

This study introduces Gentzen-style natural deduction systems for Gurevich logic, Nelson logic, intuitionistic logic, classical logic, and modal logic. Gurevich logic is an extended constructive three-valued logic obtained from intuitionistic logic by adding strong negation, and Nelson logic is the intuitionistic-negation-less fragment of Gurevich logic. The proposed natural deduction systems are constructed in a modular manner based on primitive rules for negation, that is, rules of explosion, of negation introduction, and of excluded middle. Theorems for equivalence between these natural deduction systems and the corresponding previously proposed cut-free Gentzen-style sequent calculi are proved. Moreover, normalization theorems for the proposed natural deduction systems are proved.

Given a rough definably amenable rough approximate subgroup A of a group in some first-order structure there is a type-definable subgroup N normalized by A and contained in A4 of bounded index in the subgroup ⟨A⟩ generated by A.

This paper studies class theory over the logic HYPE recently introduced by Hannes Leitgeb. We formulate suitable abstraction principles and show their consistency by displaying a class of fixed-point (term) models. By adapting a classical result by Brady, we show their inconsistency with standard extensionality principles, as well as the incompatibility of our semantics with weak extensionality principles introduced in the literature. We then formulate our version of weak extensionality (appropriate to the behavior of the conditional in HYPE) and show its consistency with one of the abstraction principles previously introduced. We conclude with observations and examples supporting the claim that, although arithmetical axioms over HYPE are as strong as classical arithmetical axioms, the behavior of classes over HYPE is akin to the one displayed by classes in other nonclassical class theories.

In the literature, the question regarding how to axiomatize the transitive logic of false belief is thought of as hard and left as an open problem. In this paper, among other contributions, we deal with this problem. In more detail, although the standard doxastic operator is undefinable with the operator of false belief, the former is almost definable with the latter. On one hand, the involved almost definability schema guides us to find the desired core axioms for the transitive logic and the Euclidean logic of false belief. On the other hand, inspired by the schema and other considerations, we propose a suitable canonical relation, which can uniformly handle the completeness proof of various logics of false belief, including the transitive logic. We also extend the results to the logic of reliable belief, due to the interdefinability of the operators of false belief and reliable belief.