Probability and natural deduction

IF 0.7 4区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Journal of Logic and Computation Pub Date : 2024-03-06 DOI:10.1093/logcom/exae007

Marija Boričić Joksimović, Nebojša Ikodinović, Nenad Stojanović

引用次数: 0

Abstract

We develop a system of basic probability reasoning founded on two great logical concepts, Gentzen’s natural deduction systems and Carnap–Popper probability of sentences. Our system makes it possible to manipulate with probabilized sentences and justify their causal relationships: if probabilities of sentences $A$ and $B$ are in $[r,1]$ and $[s,1]$, respectively, then the probability of sentence $C$ belongs to $[t,1]$, i.e. $A^{r},B^{s}\vdash C^{t}$, for $r,s,t\in [0,1]$. We prove that our system is sound and complete with respect to the traditional Carnap–Popper type probability semantics. This approach opens up a new perspective of proof-theoretic treatment of sentence probability, potentially allowing immediate algorithmic use of the pure syntactic convenience of natural deductions in programming.

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概率与自然演绎

我们基于两个伟大的逻辑概念--根岑的自然演绎系统和卡纳普-波普尔的句子概率--开发了一套基本概率推理系统。我们的系统使我们能够处理概率句子并证明它们之间的因果关系：如果句子 $A$ 和 $B$ 的概率分别在 $[r,1]$ 和 $[s,1]$，那么句子 $C$ 的概率属于 $[t,1]$，即 $A^{r},B^{s}\vdash C^{t}$，对于 $r,s,t\in [0,1]$ 来说。我们证明，相对于传统的卡纳普-波普尔式概率语义，我们的系统是健全和完整的。这种方法为句子概率的证明论处理开辟了一个新视角，有可能在编程中直接使用自然演绎的纯语法便利。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Logic and Computation 工程技术-计算机：理论方法

CiteScore

1.90

自引率

14.30%

发文量

审稿时长

6-12 weeks

期刊介绍： Logic has found application in virtually all aspects of Information Technology, from software engineering and hardware to programming and artificial intelligence. Indeed, logic, artificial intelligence and theoretical computing are influencing each other to the extent that a new interdisciplinary area of Logic and Computation is emerging. The Journal of Logic and Computation aims to promote the growth of logic and computing, including, among others, the following areas of interest: Logical Systems, such as classical and non-classical logic, constructive logic, categorical logic, modal logic, type theory, feasible maths.... Logical issues in logic programming, knowledge-based systems and automated reasoning; logical issues in knowledge representation, such as non-monotonic reasoning and systems of knowledge and belief; logics and semantics of programming; specification and verification of programs and systems; applications of logic in hardware and VLSI, natural language, concurrent computation, planning, and databases. The bulk of the content is technical scientific papers, although letters, reviews, and discussions, as well as relevant conference reviews, are included.