Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2403.07948

A Study on Actions for Atomic Logics

Raül Espejo-Boix
View original source ↗Read source document ↗Export metadata ↓

Abstract

Nowadays there is a large number of non-classical logics, each one best suited for reasoning about some issues in abstract fields, such as linguistics or epistemology, among others. Proving interesting properties for each one of them supposes a big workload for logicians and computer scientists. We want an approach into this problematic that is modular. To adress this issue, the report shows new insights in the construction of Atomic Logics introduced by Guillaume Aucher. Atomic Logics let us represent very general left and right introduction rules and they come along a new kind of rules based on display logics and residuation. A new approach is taken into the definition of Atomic Logics, which is now built on a class of actions for which we prove cut-elimination. We show that some of them are equivalent to Aucher's Atomic Logics and we prove cut-elimination and Craig Interpolation for a class of them. The introduced theory is applied to the non-associative Lambek Calculus throughout the report. It is accompanied by a computer-checked formalisation of the original syntax in the proof assistant Coq.

Explore related subjects

math.LOcs.LO

Keep this discovery

BibTeXRIS

Raül Espejo-Boix. 2024-03-11. A Study on Actions for Atomic Logics. https://arxiv.org/abs/2403.07948

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

There is no maximal $K$-degree

The Kolmogorov complexity of a string characterize how complex it is to describe the string. If every prefix of a real $x$ is more complex to describe than every prefix (of the same length) of real $y$, then it is seen as $x$ is more complex to describe than $y$. It is wondered if there is a real $x$ so that no other reals are strictly more complex (to describe) than $x$. The behavior of Kolmogorov complexity functions generated by reals (namely $n\mapsto$ the minimal description length of the real) is quite chaos. Therefore, it is widely believed that there are many reals that are maximally complex to describe. For instance, it is conjectured that all random enough reals have maximal $K$-degree. In this paper, it is shown that there is no real with maximal $K$-degree. Actually, for almost all real $x$, we can uniformly computably find another real whose $K$-degree is strictly above $x$.

math.LO↗
arXiv2026

Quadruples and cubes

We prove, in $\mathsf{ZFC}$, that the $\lambda$-terraced cube relation fails whenever $\lambda$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $\lambda$. If $\lambda$ is $\aleph_0$ then the pretinent terraced relation has consistency strength of at least one Woodin cardinal. We prove positive polarized relations at a successor and a double successor from wondrous ideals. We show, however, that there are no such ideals over two consecutive cardinals simultaneously.

math.LO↗
arXiv2026

Possibilistic Logic over a Logic of Formal Inconsistency

In this article, we have introduced a new possibilistic logic on a logic of formal inconsistency with the aim of developing a possibility theoretic framework to deal with uncertainty and inconsistency meaningfully without leading to a system collapse. We have discussed the syntax and semantics for this logic and have proved the soundness and completeness theorems. A set of new measures of consistency, contradictoriness, and triviality of a set of formulas have been defined. These have then been put to use in an example to show that this framework can provide better means of machine reasoning.

math.LO↗