Assuring and critical labels for relations between maximal consistent sets for interpretability logics

Modal Matters in Interpretability Logics

This paper from 2008 is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the foundations are laid for later results. These foundations consist of a thorough treatment of a construction method to obtain modal models. This construction method is used to reprove some known results in the area of interpretability like the modal completeness of the logic ${\textbf{IL}}$. Next, the method is applied to obtain new results: the modal completeness of the logic ${\textbf{IL}}{\sf M_0}$, and modal completeness of ${\textbf{IL}…

弱可解释性逻辑上的持续性原则

The persistence principle over weak interpretability logic

本文研究在弱可解释性逻辑中加入持续性原则所得逻辑，证明其弱不动点性质、切消与 Craig 插值、有限标架性质及算术语义下的可靠完全性。

Modal completeness of sublogics of the interpretability logic $\mathbf{IL}$

We study modal completeness and incompleteness of several sublogics of the interpretability logic $\mathbf{IL}$. We introduce the sublogic $\mathbf{IL}^-$, and prove that $\mathbf{IL}^-$ is sound and complete with respect to Veltman prestructures which are introduced by Visser. Moreover, we prove the modal completeness of twelve logics between $\mathbf{IL}^-$ and $\mathbf{IL}$ with respect to Veltman prestructures. On the other hand, we prove that eight natural sublogics of $\mathbf{IL}$ are modally incomplete. Finally, we prove that these incomplete logics are complete with respect to general…

Provability and interpretability logics with restricted realizations

The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $Γ$. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set $Γ$, where each sentence in $Γ$ has a well understood (meta)-mathematical content in T, the result is the logic of linear frames. Ho…

可解释性逻辑的统一插值

Uniform interpolation for interpretability logic

本文对可解释性逻辑 IL 进行证明论研究，提出良基与非良基相继式演算并利用切消与循环证明，证明了 IL 及其推论 ILP 的统一插值性质。

Two series of formalized interpretability principles for weak systems of arithmetic

The provability logic of a theory $T$ captures the structural behavior of formalized provability in $T$ as provable in $T$ itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability logics. Where provability logics are the same for all moderately sound theories of some minimal strength, interpretability logics do show variations. The logic IL(All) is defined as the collection of modal principles that are provable in any moderately sound theory of some minimal strength. In this paper we raise the previously known lower bound of IL(All) by…

The Poset of All Logics I: Interpretations and Lattice Structure

A notion of interpretation between arbitrary logics is introduced, and the poset Log of all logics ordered under interpretability is studied. It is shown that in Log infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between Log and the lattice of interpretability types of varieties are investigated.

Feferman 可解释性

Feferman Interpretability

本文引入模态逻辑 FIL 以处理 Feferman 可解释性，通过带标签的模态调整底层理论的公理集，证明了多种可解释性原理在弱算术理论中的可靠性。

A new principle in the interpretability logic of all reasonable arithmetical theories

The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper from 2011 revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by ${\textbf{IL}}({\rm All})$. In this paper we present a new principle $\sf R$ in ${\textbf{IL}}({\rm All})$. We show that $\sf R$ does not follow from the logic ${\textbf{IL}}{\sf P_0W^*}$ that contain…

Interpretability in PRA

In this paper from 2009 we study IL(PRA), the interpretability logic of PRA. As PRA is neither an essentially reflexive theory nor finitely axiomatizable, the two known arithmetical completeness results do not apply to PRA: IL(PRA) is not ILM or ILP. IL(PRA) does of course contain all the principles known to be part of IL(All), the interpretability logic of the principles common to all reasonable arithmetical theories. In this paper, we take two arithmetical properties of PRA and see what their consequences in the modal logic IL(PRA) are. These properties are reflected in the so-called Beklemi…

Logic Blog 2018

Some notions from algorithmic randomness are extended to measures and to quantum states. There is a lot on group theory and its relation to logic. This includes some new results on oligomorphic groups. There's also metric spaces and Scott rank, and interpretability.

Veltman语义的超滤扩展

Ultrafilter Extensions for Veltman Semantics

本文提出可解释性逻辑的一阶框架条件，证明其不可模态定义，但适用于ILM和ILP框架，并为此发展超滤扩展的代数工具。

双范畴、双等价与双可解释性

Bicategories, Biequivalence, and Bi-Interpretability

本文通过构造相干理论双范畴与相干范畴双范畴之间的双等价，给出双可解释性的范畴刻画，并利用正合完备化得到两个理论双可解释的充要条件，推广到一阶逻辑常见片段。

可证性模型

Provability Models

本文提出名为可证性模型的经典模态逻辑新型克里普克式语义，证明了多类模态逻辑、解释性逻辑ILM对该类模型的可靠性与完全性，并将其改进为谓词可判定版本。

The closed fragment of IL is PSPACE hard

In this paper from 2011 we consider $\textbf{IL}_0$, the closed fragment of the basic interpretability logic $\textbf{IL}$. We show that we can translate $\textbf{GL}_1$, the one variable fragment of Gödel-Löb's provabilty logic $\textbf{GL}$, into $\textbf{IL}_0$. Invoking a result on the PSPACE completeness of $\textbf{GL}_1$ we obtain the PSPACE hardness of $\textbf{IL}_0$.

$\boldsymbol{\text{IL}}$子逻辑的Visser标架

Visser frames for sublogics of $\mathbf{IL}$

本文研究可解释性逻辑$\text{IL}$的若干子逻辑在Visser标架下的模态完备性与有穷标架性质，证明保守性逻辑$\text{CL}$具有该标架下的有穷标架性质，正面解决了Ignatiev提出的问题。

The fixed point and the Craig interpolation properties for sublogics of $\mathbf{IL}$

We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic $\mathbf{IL}$. We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.

A dichotomy in classifying quantifiers for finite models

We consider a family U of finite universes. The second order quantifier Q_R, means for each u in U quantifying over a set of n(R)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Q_R, ever Q_R is interpretable by quantifying over subsets of u and one to one functions on u both of bounded order, or the logic L(Q_R) (first order logic plus the quantifier Q_R) is undecidable.