Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.19930

Uniform validity of atomic Split rule in monotonic proof-theoretic semantics

Antonio Piccolomini d'Aragona
View original source ↗Read source document ↗Export metadata ↓

Abstract

Proof-theoretic semantics (PTS) is normally understood today as Base-Extension Semantics (B-eS), i.e., as a theory of proof-theoretic consequence over atomic proof systems. Intuitionistic logic (IL) has been proved to be incomplete over a number of variants of B-eS, including a monotonic one where introduction rules play a prior role (miB-eS). In its original formulation by Prawitz, however, PTS consequence is not a primitive, but a derived notion. The main concept is that of argument structure valid relative to atomic systems and assignments of reductions for eliminating generalised detours of inferences in non-introduction form. This is called Proof-Theoretic Validity (P-tV), and it can be given in a monotonic and introduction-based form too (miP-tV). It is unclear whether, and under what conditions, the incompleteness results proved for IL over miB-eS can be transferred to miP-tV. As has been remarked, the main problem seems to be that the notion of argumental validity underlying the miB-eS notion of consequence is one where reductions are either forced to be non-uniform, or non-constructive. Building on some Prawitz-fashion incompleteness proofs for IL based on the notion of (intuitionistic) construction, I provide in what follows a set of reductions which are surely uniform (however uniformity is defined) and constructive, and which make the atomic Split rule logically valid over miP-tV, thus implying the incompleteness of IL over a Prawitzian (monotonic, introduction-based) framework strictly understood.

Explore related subjects

math.LO

Keep this discovery

BibTeXRIS

Antonio Piccolomini d'Aragona. 2025-03-22. Uniform validity of atomic Split rule in monotonic proof-theoretic semantics. https://arxiv.org/abs/2503.19930

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↗