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Falsity and refutations: Understanding the negative side of logic

虚假与反驳：理解逻辑的消极面

基本信息

批准号：
397512418
负责人：
Dr. Luca Tranchini
金额：
--
依托单位：
Institut für Philosophie I
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2018
资助国家：
德国
起止时间：
2017-12-31 至 2021-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/397512418?language=en
关键词：
Falsity refutations Understanding negative side

项目摘要

In spite of the central role played by the notions of falsity and refutation, as of today there is no agreement on which should be their correct understanding in logic. On the most common view, falsity, refutation and denial play a merely ancillary role compared to their positive counterparts, truth, proof and assertion. Such a view has been opposed by some authors , who called for treating positive and negative notions on a par. How this should be done, however, is still the object of controversies. The goal of the project is to propose a novel picture of logic that does justice to its negative side. Its major contribution will be an analysis of the notion of refutation on its own, i.e. independently of the notion of proof. The project will begin by criticising the traditional view, on which negative notions are subordinated to positive ones, and by proposing to identify the source of this view in a realist conception of meaning. On such a conception, whereas portions of reality - i.e. facts - correspond to true sentences, nothing corresponds to false ones. On current alternative views, which rather rely on a `constructive' conception of meaning, the assertion of a proposition is justified by the possession of a proof of it, and the denial of a proposition is justified by the possession of a refutation of it. The common weakness of both the traditional conception and these alternative views is that, although implication expresses the most important kind of relation between propositions in logic, the conditions at which an implication `if A then B' is false (refuted) are formulated by making reference to the notion of truth (proof): `if A then B' is false (refuted) whenever A is true (proved) and B is false (refuted).The main part of the project will consists in delivering a novel operational understanding of refutations, analogous to the operational understanding of proofs underlying intuitionism. The operative conception of proofs is based on the `Curry-Howard correspondence', that establishes a close connection between logic and computer science, in that a formal derivation can be seen as encoding an algorithm, i.e. the abstract representation of a computer program, that taken a proof of the assumptions as input yields a proof of the conclusion as output. In order to make the notion of refutation precise in this context, we will show how to associate `dual programs' to formal derivations, i.e. programs that take a refutation of the conclusions as input and yield a (collective) refutation of the assumptions as output.On the basis of the proposed understanding of refutations, we will investigate the conditions at which a logical system lends itself to be understood as a ``logic of proof'', as a ``logic of refutation'', or as a synthesis of the two. Moreover, we will introduce tools for developing new logical systems in which proofs and refutations are encoded in different ways and connected by different kinds of negation operators.

尽管虚假和反驳的概念发挥了核心作用，但到今天为止，对于哪一个应该是它们在逻辑上的正确理解还没有达成一致。在最常见的观点，虚假，反驳和否认只发挥辅助作用相比，其积极的同行，真理，证据和断言。这种观点遭到一些作者的反对，他们呼吁平等对待积极和消极的概念。然而，如何做到这一点，仍然是有争议的对象。该项目的目标是提出一个新颖的逻辑图片，公正地对待其消极的一面。它的主要贡献将是对反驳概念本身的分析，即独立于证明概念。该项目将开始，批评传统的观点，消极的概念是从属于积极的，并提出确定这种观点的来源，在现实主义的意义概念。根据这样一种概念，现实的一部分--即事实--对应于真的句子，而没有任何东西对应于假的句子。当前的替代观点相当依赖于意义的“建构性”概念，对一个命题的断言是通过拥有它的证明来证明的，而对一个命题的否认是通过拥有它的反驳来证明的。传统观念和这些替代观点的共同弱点是，虽然蕴涵表达了逻辑中命题之间最重要的一种关系，但蕴涵"如果A则B"为假（被反驳）的条件是通过引用真（证明）的概念来表述的：每当A为真（被证明）且B为假（被反驳）时，“如果A，则B”为假（被反驳）。该项目的主要部分将在于提供对反驳的新颖操作理解，类似于直觉主义基础上的证据的操作理解。证明的操作概念是基于“Curry-Howard对应”，它建立了逻辑和计算机科学之间的密切联系，因为形式推导可以被视为编码算法，即计算机程序的抽象表示，将假设的证明作为输入，产生结论的证明作为输出。为了使反驳的概念在这种情况下精确，我们将展示如何将“对偶程序”与形式推导相关联，即程序将结论的反驳作为输入并产生一个结果。（集体）反驳的假设作为输出。在提出的理解反驳的基础上，我们将研究一个逻辑系统适合于被理解为"证明逻辑"、"反驳逻辑"或两者的综合的条件。此外，我们将介绍开发新的逻辑系统的工具，其中证明和反驳以不同的方式编码，并通过不同类型的否定运算符连接。