Speaker: Carlo Nicolai (Utrecht University)
Title: Intensional Paradoxes and the Maxim of Maximal Recapture
Time: 16:00 - 17:30
Location: ILLC Seminar Room F1.15
According to Gödel, there never were any set theoretic paradoxes, but intensional paradoxes involving semantic, property theoretic, modalities and propositional attitudes, were still unresolved. Gödel clearly referred to the absence of consistent treatments of these notions that allowed for their self-application. Much progress has been made since Gödel's remarks: we have at our disposal a wide array of theories of self-referential truth, predication, modalities. A standard way to classify them is by looking at the underlying logic: classical theories are based on a restriction of the naïve rules for intensional notions, nonclassical theories keep naïveté by giving up some classically valid patterns of reasoning. These two approaches are hard to compare: pragmatic or metaphysical considerations are often involved, and the dialogue between the advocates of such approaches often resolves in a clash of intuitions. In the talk I focus on a more tractable way of comparing them, based on the comparison of the non-intensional, non-logical patterns of reasoning available in the two approaches. In particular, I will consider theories of satisfaction/predication over a fixed syntax/mathematical theory: classical on the one hand and paracomplete/paraconsistent approaches on the other (I'll leave aside substructural approaches). Advocates of the latter often emphasize that for the satisfaction/predication-free fragment of the language, classical reasoning can be restored. I will question the coherence of these recapture strategies: the fail to provide a faithful recapture of schematic, satisfaction/predication-free reasoning. Moreover, paracomplete/paraconsistent approaches also fall short of non-intensional consequences that are mathematically significant: I will give what I believe to be the first example of a theorem of standard mathematics separating these two clusters of theories. But these considerations may not immediately lead to supporting classical approaches: I conclude the talk by showing that this mathematical gap may be covered by enriching the four-valued core of paracomplete/paraconsistent approaches with a simple conditional.