A theory $T$ is shown to have an ICT pattern of depth $k$ in $n$ variables iff it interprets some $k$-maximum VC class in $n$ parameters.

We shed light on an old problem by showing that the logic LP cannot define a binary connective $\odot$ obeying detachment in the sense that every valuation satisfying $\phi$ and $\left(\phi \odot \psi \right)$ also satisfies $\psi$, except trivially. We derive this as a corollary of a more general result concerning variable sharing.

We show that a standard axiomatization of mereology is equivalent to the condition that a topological space is discrete, and consequently, any model of general extensional mereology is indistinguishable from a model of set theory. We generalize these results to the Cartesian closed category of convergence spaces.

Let $\Gamma$ be the random bipartite graph, a countable graph with two infinite sides, edges randomly distributed between the sides, but no edges within a side. In this paper, we investigate the reducts of $\Gamma$ that preserve sides. We classify the closed permutation subgroups containing the group $Aut(\Gamma {)}^{\ast }$, where $Aut(\Gamma {)}^{\ast }$ is the group of all isomorphisms and anti-isomorphisms of $\Gamma$ preserving the two sides. Our results rely on a combinatorial theorem of Nešetřil and Rödl and a strong finite submodel property for $\Gamma$.

We extend first-order logic to include variadic function symbols and prove a substitution lemma. Two applications are given: one to bounded quantifier elimination and one to the definability of certain Borel sets.

This paper shows that the basic logic induced by the parallel recurrence $\Yup$ of computability logic (i.e., the one in the signature $\{ ¬, ∧,∨,⅄,\small{𝖸} \}$) is a proper superset of the basic logic induced by the branching recurrence $⫰$ (i.e., the one in the signature $\{ ¬,∧,∨, ⫰, ⫯ \}$). The latter is known to be precisely captured by the cirquent calculus system CL15, conjectured by Japaridze to remain sound—but not complete—with $⅄$ instead of $⫰$. The present result is obtained by positively verifying that conjecture. A secondary result of the paper is showing that $⅄$ is strictly weaker than $⫰$ in the sense that, while $⫰F$ logically implies $⅄F$, the reverse does not hold.

The requirement of proof-theoretic harmony has played a pivotal role in a number of debates in the philosophy of logic. Different authors have attempted to precisify the notion in different ways. Among these, three proposals have been prominent in the literature: harmony–as–conservative extension, harmony–as–leveling procedure, and Tennant’s harmony–as–deductive equilibrium. In this paper I propose to clarify the logical relationships between these accounts. In particular, I demonstrate that what I call the equivalence conjecture—that these three notions essentially come to the same thing—is erroneous.

In this paper a logic suitable for reasoning disquotationally about truth, ${\mathsf{TJK}}^{+}$, is presented and shown to have a standard model. This work improves on Hartry Field’s recent results establishing consistency and $\omega$-consistency of truth theories with strong conditional logics. A novel method utilizing the Banach fixed point theorem for contracting functions on complete metric spaces is invoked, and the resulting logic is shown to validate a number of principles which existing revision theoretic methods have so far failed to provide.

This paper continues the study of the metric topology on $2^{\mathbb{N}}$ that was introduced by S. Binns. This topology is induced by a directional metric where the distance from $Y\in 2^{\mathbb{N}}$ to $X\in 2^{\mathbb{N}}$ is given by

$${\displaystyle \underset{n}{\mathrm{lim\; sup}}}\frac{C(X\upharpoonright n\mid Y\upharpoonright n)}{n}.$$

This definition is closely related to the notions of effective Hausdorff and packing dimensions. Here we establish that this is a path-connected topology on $2^{\mathbb{N}}$ and that under it the functions $X\mapsto {dim}_{\mathcal{H}}X$ and $X\mapsto {dim}_{p}X$ are continuous.

We also investigate the scalar multiplication operation that was introduced by Binns. The multiplication of a real $X\in 2^{\mathbb{N}}$ by an element $\alpha \in [0,1]$ represents a dilution of the information in $X$ by a factor of $\alpha$.

Our main result is to show that every regular real is the dilution of a real of Hausdorff dimension 1. That is, that the information in every regular real can be maximally compressed.