Final Report Summary - PPP (Plurals, Predicates, and Paradox: Towards a Type-Free Account) The project set out to transform our understanding of the logical paradoxes, their solution and significance for mathematics, philosophy and semantics. Substantial progress has been made on this overarching aim and on the more specific objectives. This progress is evidenced in part by scientific publications resulting from the project: 20 articles in peer reviewed journals, with a further 8 under review, and 10 articles in anthologies. Furthermore, the project has made a huge contribution to the research communities in London and Oslo, where it has been based. We have organized 16 workshops and a major conference with an associated summer school; run one or two research seminars each semester, open to the academic community in general; and hosted numerous visitors and guest speakers. Four areas of progress are especially noteworthy. (1) We have advocated a reconception of higher-order logic (HOL). We have defended the distinctness of plural and second-order quantification, as well as the legitimacy of very high (even infinite) orders of each of these forms of quantification (Linnebo 2011, Linnebo & Rayo 2012). Moreover, by developing a set-free Henkin-style semantics for plural logic, we have developed new challenges to its alleged ontological innocence and semantic determinacy (Florio & Linnebo forthc.). (2) We have developed a radically new conception of the relation between HOL and first-order logic (FOL). Traditionally, any kind of collapse of the former to the latter has been prohibited in order to block paradoxes. In a series of high-profile publications, we argue that these prohibitions are unfounded and the mentioned forms of collapse are resources that can be harnessed with great explanatory benefit—provided that alternative measures are developed to block the paradoxes. (Two such measures are described in (3) and (4) below.) More specifically, we have (a) defended the intuitive idea that any objects can form a set (Linnebo 2010); (b) that defenders of HOL are forced to adopt logics of ever higher order, and that once the order becomes infinite, the logic changes its qualitative character to become much more like a first-order set theory (Linnebo & Rayo 2012); and (c) that defenders of HOL are under great pressure to compare the values of variables of different order, which we show that they can consistently do, but only by turning their system into one which effectively is first-order (Hale & Linnebo forthc.). Together, these arguments pose a formidable challenge to the current renaissance of HOL in philosophy and demand a radical rethinking of the usual response to the logical paradoxes. (3) We have developed a modal approach to mathematics, capturing the idea that mathematical objects such as sets are generated successively and that this generation cannot be completed. In this modal setting we retain the natural thought that any objects can form a set and thus provide a novel solution to the set theoretic paradoxes enabling us to retain more of our intuitive assumptions than customary. The modal approach has also been used to provide a new and very natural motivation for the axioms of ZFC set theory based on a logic of modals and plurals, some strong extensionality axioms which characterize what sets are, and the intuitive modal idea mentioned above as the single set existence assumption. (Linnebo 2010 and 2013) The limits of this modal approach, especially with regard to set-theoretic reflection principles, are examined in Sam Roberts’ PhD thesis. (4) Drawing inspiration from Kripke’s notion of “groundedness”, we have developed new tools to make the world safe for the kinds of “collapses” of HOL to FOL mentioned above. This has resulted in several new theories of grounded classes (Horsten & Linnebo forthc., Kriener 2014); a general account of groundedness and its philosophical significance (Jönne Kriener’s PhD thesis); and major new works on the logic of ground (Litland forthc.).
