The overall goal of PLEXUS is to advance the knowledge of radically substructural logics and deepen our understanding of the broader phenomenon of substructurality, by coordinating the efforts of researchers across the globe, across generations, and across traditions.
Saul Kripke is a towering figure in contemporary philosophy, widely held to be one of the greatest living philosophers and logicians. This project, if funded, will bring to publication a series of previously unpublished seminars on the theory of truth, which develop ideas first sketched in his legendary paper “Outline of a Theory of Truth” (1975).
In this project, we will explore: i) the precise sense in which operational and substructural approaches weaken classical logic. ii) to what extent these proposals are revenge-free and what are the prospects of achieving such a goal. iii) whether the problem of restricted quantification can be satisfactorily dealth with in paracomplete and paraconsistent theories. iv) if it is possible to provide plausible philosophical interpretations of non-contractive consequence relations and in particular whether the is a plausible reading of the quantifiers. v) in what sense the cut-free approaches are classical and in what sense they are not.
The main goal of this project will be to find answers to questions that might lead this topic into some new directions, such us: (i) Does the truth predicate have explanatory power after all, contra the deflationist approach to truth? Could a good theory of truth have no standard models? And if one considers the second-order case with standard semantics, could a good theory of truth have no models while being consistent? (ii) Is there a theory of truth capable of dealing with semantic paradoxes and revenge problems in which every classical principle is valid? Or does the best explanation of these phenomena involve adopting a different notion of logical consequence? (iii) Is there a consistent, systematic, and philosophically illuminating way of expressing intensional notions such as necessity and knowledge in terms of predicates? How can the notorious paradoxes that have been claimed to affect such a treatment of intensional notions be avoided?
The aim of the project is to establish a collaboration between two active research groups (“Reflection and Incompleteness”, which is based in Oxford and funded by the AHRC and “Unrestricted Quantification, Expressiveness and Contextualism”, which is based in Buenos Aires and funded by the Argentinian National Research Council for Science and Technology). There is much overlap between the agendas of our research projects, and we would benefit from an opportunity to exchange our results and facilitate a longstanding cooperation between the groups. Although the senior members of each group have already had some limited opportunity to visit each other, the project would be particularly beneficial for the younger members of each group for whom the opportunities to travel from Oxford to Buenos Aires are much more limited. Three specific areas of overlap in our research projects concern type-free theories of truth, the expressive limitations caused by open endedness of the set-theoretic universe and the means of overcoming them.