INEXPRESSIBILITY AND REFLECTION

in the formal sciences

INEXPRESSIBILITY AND REFLECTION

in the formal sciences

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 implicit 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 conceptual strength of modal talk or quantification over second-order objects such as concepts or properties.

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, property 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 cross-disciplinary. 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 Similarly, we shall apply results from mathematical logic and mathematics to study their implications on ontology and philosophy of language.

Welcome to the AHRC Research Project

Inexpressibility and Reflection in the Formal Sciences

Project Leaders: Prof. Volker Halbach and Prof. Philip Welch