逻辑

It is well known that if a diophantine equation turns out not to have a solution over the integers modulo p, for some p, then it does not have a solution over the integers per se. This is because the integers modulo p are a homomorphic…

Let \(\vec F(2^{\mathbb Z^2})\) be the directed Schreier graph on the free part of the Bernoulli shift \(\mathbb Z^2\curvearrowright 2^{\mathbb Z^2}\), with arcs in the two coordinate directions. We prove that the continuous oriented…

In nonstandard analysis the halo of a point in a topological space is the intersection of the nonstandard extensions of all its open neighbourhoods. We define a parametric family of modal operators from the halo by varying which elements of…

We develop translations from relevant logics into normal modal logics, and use them to clarify structural connections between relevant and modal logic, obtain a few corollary results, and raise questions for future work.

We define the notion of IK-bisimulation between the relational semantics for the intuitionistic modal logic IK, and prove that IK arises as the IK-bisimulation-invariant fragment of intuitionistic first-order logic. En route, we provide an…

We study interpretations of modal logics in one another where the Boolean connectives are interpreted identically and the modal operator diamond is interpreted by an arbitrary formula A(p). Clearly, such a formula A(p) defines an…

We investigate a modal extension of the infinitary classical logic with countable meets and joins, formulated with an eye toward measure-theoretic work in dynamical systems and in point-free ergodic theory. We define a modal formalism in…

A coequivalence relation over a modal logic L is a formula in two tuples of propositional variables of the same length such that the logic L proves it to be an equivalence relation. They were introduced by Ghilardi and Zawadowski in the…

We investigate logics that generalize both intuitionistic logic and quantum logic. In earlier work, we introduced Ex-logic, an extension of Holliday's fundamental logic that coincides with the intersection of orthologic and the…

This paper investigates recovery operators in quasi-Nelson logic, the algebraizable logical counterpart of quasi-Nelson algebras. These form a variety of three-potent, distributive, but not necessarily involutive residuated lattices that…

Let $\mathbb{H}_{n+1}$ denote a computable copy of the $(n+1)$-clique free universal homogeneous Henson graph, $G$ denote a finite subgraph of $\mathbb{H}_{n+1}$, and $k(G,n)$ denote the big Ramsey degree of $G$ in $\mathbb{H}_{n+1}$. We…

In this manuscript, we provide an independent equational basis for the variety of reflexive Nelson algebras, a generalization of the variety of SNA-algebras. The proof of this result relies on a substantial number of technical arguments and…

Fundamental logic is a non-classical logic based only on the introduction and elimination rules for conjunction, disjunction, negation, and the quantifiers in a Fitch-style natural deduction system. In this paper, we attempt to obtain a…

In this note, we give some generalizations of G\"{o}del's second incompleteness theorem and study their surroundings. We revisit it from two perspectives. One perspective is the relationship between the definable complexity of a theory and…

The solutions of algebraic differential equations in certain valued differential fields, including the differential field of transseries, can be analyzed using a Newton diagram method. In this paper, we show that (eventual) equalizers, a…

Rado's Conjecture (RC) is a compactness principle for a certain class of partial orders, namely trees $T$ of height $\omega_1$ without cofinal branches, postulating that a partial order $P$ from this class can be decomposed into at most…

Casanovas and Potier proved that algebraic quantification preserves stability of formulas. They also gave a nonsimple example, answering a question of Laskowski, showing that the algebraicity hypothesis cannot simply be replaced by NFCP,…

We settle the long-standing open question whether there exists a $3$-ladder of cardinality $\aleph_2$. Given a positive integer $n$, an $n$-ladder is a lower finite lattice whose elements have at most $n$ lower covers. In 1984, Ditor proved…

We give applications of the properties $\mathrm{NSOP}_{r}$ for non-integer values of $r$ to problems on the original hierarchy $\mathrm{NSOP}_{n}$ for integer values of $n$. We first show that the properties $\mathrm{NSOP}_{r}$, previously…

In a previous paper, we introduced the notion of hyper swap structures, a novel class of hyperalgebras that naturally generalizes swap structures semantics. In this paper we introduce the concept of hyper Boolean algebras based on Morgado…