New theories of logic
Logicians typically avoid paradoxes. However, some say that paradoxes can be put to good use in mathematics, philosophy and semantics, thereby becoming a source of insight. The EU-funded PPP (Plurals, predicates, and paradox: Towards a type-free account) project argued that an acceptable solution to logical paradoxes had not been found previously. The argument opposed the prevailing view in the field. The consortium's work addressed four main areas. PPP identified a need for higher-order logics (HOL), which allow quantification into positions of predicates and plural noun phrases. A further development was a radically new idea of the relation between HOL and first-order logic (FOL). The team argued that conventional blockages of logical paradoxes are unnecessary and actually beneficial. The argument added the proviso that the paradoxes must be resolved in other ways. Researchers created a modal approach to mathematics. The development embodies the idea that mathematical objects such as sets can be generated without end. The new solution to the paradox enables retention of more common-sense interpretation concerning sets than generally the case in logic. The results permit a new and natural motivation for the axioms of ZFC set theory. The team also developed new tools, based on Kripke's notion of groundedness, which protect against collapses of HOL to FOL. Results include several new theories of grounded classes, a general description of groundedness and new works on the logic of ground. Project work yielded 20 peer-reviewed journal papers with a further 8 under review, plus 10 anthology articles. The members also organised 16 workshops, 1 conference with an associated summer school, and a seminar series. The PPP project demonstrated that logical paradoxes need not be troubling. The team has provided alternative solutions.
Keywords
Logic, paradoxes, higher-order logic, groundedness, first-order logic, ZFC set theory
