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

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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.