Inexpressibility and reflection in the formal sciences

该项目将致力于研究形式系统是否足以描述各个形式科学的主题，并表达抽象科学中接受理论时隐含的内容。形式系统旨在捕捉抽象科学中的理论，如数学，语言学，计算机科学和哲学只是部分地表达了我们在接受那些系统应该捕获的非正式理论时所隐含的支持。这些形式系统在表达资源和结果方面都是不完整的，例如，在一定的一般条件下，绝对一般量化不能在这些形式系统中表达。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描述的。此外，一个理论的可靠性，这是隐含在接受一个理论，不能充分表达或证明的正式系统，旨在捕捉我们的前形式理论的某些抽象的主题。特别是，在形式框架中，人们甚至不能表达理论的所有结论都为真的主张，甚至对这一一般主张的近似在系统中也是不可证明的。我们将研究如何克服形式系统对抽象科学理论的表达和反思能力的这些缺陷。因此，我们将使接受理论所隐含的东西明确化，为此，我们将用超出通常的一阶公理化的范围的方法来丰富形式系统。我们将研究在多大程度上增加真值谓词，二阶量词，模态算子和其他设备可以用来克服标准的一阶系统的表达和反射的弱点。因此，我们的研究结果也将揭示模态话语或二阶量化的表达能力。我们的工作将在形式系统捕获抽象对象理论的充分性方面产生新的见解，因此也在形式演绎系统的适用性和意义以及它们的范围和意义方面产生新的见解。我们将研究的理论包括算术，句法，集合论，和部分形式语义学。在这个项目中，我们研究的基础问题，在不同的学科，而不是在分离，但从一个共同的观点。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各个学科。例如，我们研究了一个真的概念如果被加到一个数学理论中所能发挥的作用，也就是说，我们将一个来自形式语义学的概念应用到一个与形式语义学没有直接关系的理论中。同样，我们将把模态逻辑中熟悉的联合收割机模态算子与集合论和其他学科的其他形式理论结合起来;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.