Inexpressibility and reflection in the formal sciences

这个项目将致力于研究形式系统是否足以描述各自形式科学的主题，并表达在抽象科学中接受理论所隐含的内容。为获取抽象科学中的理论而设计的形式系统，如数学、语言学、计算机科学和哲学，仅表达了当我们接受那些系统应该捕获的非正式理论时，我们隐含支持的部分内容。这些形式系统是关于它们的表达资源和它们的结果不完整的。例如，有人认为，在某些一般条件下，绝对的一般量化不能在这样的形式框架中表达。同样，有人声称，某些话语领域的开放性在标准的形式系统中是不可表达或不可描述的。此外，隐含在对理论的接受中的理论的合理性，不能在旨在捕捉我们关于某些抽象主题的形式前理论的形式系统中得到充分的表达或证明。特别是，在形式框架中，人们甚至不能表达理论的所有结果都为真的主张，甚至对这一一般主张的近似在系统中也是不可证明的。我们将调查如何克服形式系统对抽象科学理论的表达和反思能力方面的这些不足。因此，我们将使接受理论的过程中所隐含的东西变得明确起来。为此，我们将用超越通常一阶公理的限制的工具来丰富形式系统。我们将调查在多大程度上可以使用真值谓词、二阶量词、情态运算符和其他手段来克服标准一阶系统的表达和反射弱点。因此，我们的结果也将揭示情态对话或二阶量化的表达能力。我们的工作将对形式系统对抽象对象理论的捕获的充分性以及形式演绎系统的适用性和重要性以及它们的范围和意义产生新的见解。我们将研究的理论包括算术、句法、集合论和部分形式语义。在本项目中，我们不是分开而是从一个共同的角度来研究不同学科的基础问题。我们的工作是跨学科的，因为我们希望将概念和想法应用于跨学科。例如，如果将真理的概念添加到数学理论中，我们将考察它可能起到的作用，也就是说，我们将形式语义学中的概念应用到与形式语义学没有直接关系的理论中。同样，我们将把情态逻辑中熟悉的情态算子与集合论和其他学科的形式理论结合起来；我们将应用数理逻辑和数学的结果来研究它们对本体论和语言哲学的影响。

The project will be devoted to a study of the adequacy of formal systems for describing the subject matters of the respective formal sciences and for expressing what is implicit in the acceptance of theories in the abstract sciences.Formal systems designed to capture theories in the abstract sciences such as Mathematics, Linguistics, Computer Science and Philosophy express only partially what we implicitly endorse when we accept the informal theories that are supposed to be captured by those systems. These formal systems are with respect to their expressive resources and to their consequences incomplete.For instance, it has been argued that under certain general conditions absolute general quantification cannot be expressed in such formal frameworks. Similarly it has been claimed that the open-endedness of certain domains of discourse is not expressible or describable in the standard formal systems. Moreover, the soundness of a theory, which is implicit in the acceptance of a theory, cannot be adequately expressed or proved within the formal systems aimed to capture our pre-formal theories of certain abstract subject matters. In particular, in the formal frameworks one cannot even express the claim that all consequences of the theory are true and even approximations to this general claim are not provable in the system.We will investigate how these deficiencies in the expressive and reflective power of formal systems for theories in the abstract sciences can be overcome. Thereby we will make explicit what is implicit in the acceptance of the theories.To this end we will enrich the formal system with devices going beyond the confines of the usual first-order axiomatisations. We will investigate to what extent the addition of truth predicates, second-order quantifiers, modal operators and other devices can be used to overcome the expressive and reflective weaknesses of the standard first-order systems. Thus our results will also reveal the expressive strength of modal talk or second-order quantification.Our work will yield new insights in the adequacy of formal systems for capturing theories of abstract objects and therefore also in the applicability and significance of formal deductive systems and in their scope and significance.The theories we will study comprise arithmetic, syntax, set theory, and parts of formal semantics.In the project we study foundational issues in various disciplines not in separation but from a common viewpoint. Our work is interdisciplinary in the sense that we want to apply concepts and ideas across disciplines. For instance, we look at the role a notion of truth could play if added to a mathematical theory, that is, we are applying a notion from formal semantics to a theory not directly concerned with formal semantics. Likewise, we will combine modal operators familiar from modal logic with set theory and other formal theories from other disciplines; and we shall apply results from mathematical logic and mathematics to study their implications on ontology and philosophy of language.