Properties, Paradox, and Circularity. A New, Type-Free Account

关于世界的理论需要一个命题、性质和关系理论(PPR)。命题是真的、相信的、期望的、已知的属性适用于的实体。自然属性和自然关系是自然科学研究的物理实体的特征。社会属性和关系塑造了人类的互动。一个令人满意的PPR理论的发展是建立在科学和哲学探究的基础上的。然而，命题、性质和关系的理论受到逻辑悖论的制约：诸如“语言中的每个谓词对应一个性质”这样的直观原则实际上不符合标准的逻辑规律。这就是为什么PPR理论需要坚实的正式基础。该项目旨在为命题、性质、关系理论提供一个新的形式和哲学基础。提供这种基础的最流行的方式是以逻辑类型的层次结构来安排PPR。类型化方法的突出之处可以用它的一些理论优点来解释：它与经典逻辑是一致的，而且它可以达到相当大的数学实力。然而，类型化的方法忽略了语言和思维的一些重要特征，如循环性和自我适用性。例如，像“有必要做某事是必要的”这样的说法在打字的环境中不能被适当地组织起来。更广泛地说，循环现象可以在哲学(自我参照)、计算机科学(流、自应用程序)和数学(图形、无充分基础集合)的许多理论背景中找到。循环性也可以在自然语言中找到(自我预测)。这种现象很难用一种典型的PPR理论来建模。该项目将详细分析现有的PPR理论，并发展无类型方法的逻辑和哲学，这是一种有前途但不发达的PPR方法及其悖论。该项目将发展两种PPR的无类型理论。第一种建立在PPR的幼稚原则和经典逻辑的限制之上。第二种理论是在古典逻辑中铸就的。它们将捕捉到比当前PPR理论更广泛的循环现象，同时保留了类型化方法的理论优势，如在各种背景下的适用性和数学实力。该项目将详细分析语义学和形而上学理论的含义，这是PPR应用的两个核心领域。该项目的一个关键组成部分是其内在的跨学科方法；基本上都使用了数学、理论计算机科学、语言学和哲学的技术。该项目将以语料库为基础，对与PPR有关的经验语言学证据进行系统分析，并对PPR理论的Desiderata进行全面的哲学评估。将使用数学和计算机科学中的理论构建和分析技术来构建和测试新的无类型理论的形式方面。该项目的发现将对正规科学和哲学的基础产生影响。该项目的主要成果将是由PI撰写的一本关于圆形内涵实体的专著，以及由COI和PI撰写的一本简短的专著，调查现有的形式内涵实体理论的基本思想。除了两个传统的讲习班外，该项目还将组织一个影响讲习班，向从事数据建模工作的非学术利益攸关方介绍在该项目中构建的理论。

Theorizing about the world requires a theory of propositions, properties, and relations (PPRs). Propositions are the entities to which the properties of being true, believed, desired, known apply. Natural properties and relations are the features of physical entities studied by natural science. Social properties and relations shape human interactions. The development of a satisfactory theory of PPRs lies at the foundation of scientific and philosophical enquiry. However, theories of propositions, properties, and relations are constrained by logical paradoxes: intuitive principles such as "To every predicate in language, there corresponds a property" are in fact inconsistent with the standard laws of logic. This is why a solid formal foundation for theories of PPRs is required. The project aims to provide a new formal and philosophical foundation to theories of propositions, properties, relations. The most popular way of providing such a foundation arranges PPRs in hierarchies of logical types. The prominence of the typed approach is explained by some of its theoretical virtues: it is consistent with classical logic and it may achieve substantial mathematical strength. However, the typed approach misses out some important features of language and thought such as circularity and self-applicability. For instance, claims such as 'it's necessary that something is necessary' cannot be properly regimented in the typed setting. More generally, circular phenomena can be found in many theoretical contexts in philosophy (self-reference), computer science (streams, self-applicative programs), and mathematics (graphs, non-wellfounded sets). Circularity can also be found in natural language (self-predication). Such phenomena cannot be easily modelled in a typed theory of PPRs. The project will analyse in detail extant theories of PPRs and develop the logic and philosophy of the type-free approach, a promising but underdeveloped approach to PPRs and their paradoxes. The project will develop two kinds of type-free theories of PPRs. The first builds on naive principles for PPRs and on a restriction of classical logic. The second kind of theories is cast in classical logic. They will capture a much wider range of circular phenomena than current theories of PPRs, while retaining the theoretical benefits of the typed approach such as applicability in various contexts and mathematical strength. The project will analyze in detail the implications of the theories in semantics and metaphysics, two of the core domains of applications of PPRs. One key component of the project is its intrinsic interdisciplinary methodology; techniques from mathematics, theoretical computer science, linguistics, and philosophy are all essentially employed. The project will be informed by a corpus-based, systematic analysis of empirical linguistic evidence bearing on PPRs and by a comprehensive philosophical evaluation of desiderata for theories of PPRs. Techniques of theory building and analysis from mathematics and computer science will be employed to construct and test formal aspects of the new type-free theories. The project findings will impact on the foundations of formal sciences and philosophy. Major outputs of the project will be a monograph on circular intensional entities by the PI, and a short monograph by the CoI and the PI surveying the fundamental ideas of the available formal theories of intensional entities. Besides two traditional workshops, the project will organize an impact workshop in which the theories constructed in the project will be presented to non-academic stakeholders working on data modelling.