Following the previous post, and considering the project I sketched in October, maybe indexicalism (see here for a broad idea of what it is) is less a (paradoxico-)metaphysics or a cosmology of incompleteness than a Logic in the sense of Hegel's Logik, formal but thoroughly inhaltliche. It is perhaps another Logic of being and its exterior, of borders and its determinations etc. It is a different Logic of negation and concrescence. Maybe it should be developed as in equally paraconsistent way but where a thorough rejection of the principle of explosion (that from contradiction anything can be derived) comes from the work of the Other, of the Outside - through supplement.
I have been developing a logic of supplement with a group of people; our starting point is to study, in the way of universal (or abstract) logic, thoroughly non-monotonic formal systems, we call them antimonotonic. In this system, every argument is non-monotonic. As a consequence, no addition can be made to any valid argument - and nothing can be removed from it. These systems are in a sense minimal but not to confuse with the minimal (intuitionistic) logic proposed by Johansson, we call them clean logics. It is easy to see that to clean a logic is a form of paraconsistetization, as developed by my colleague and co-author Alexandre Costa-Leite, Edelcio de Souza and Diego Dias (here is their 2016 paper). This is because if we add a contradiction to the premisses of any valid argument, it derives nothing - instead of deriving anything. Clean logics are paraconsistent in the strong sense that nothing makes the system explode.
I have been developing a logic of supplement with a group of people; our starting point is to study, in the way of universal (or abstract) logic, thoroughly non-monotonic formal systems, we call them antimonotonic. In this system, every argument is non-monotonic. As a consequence, no addition can be made to any valid argument - and nothing can be removed from it. These systems are in a sense minimal but not to confuse with the minimal (intuitionistic) logic proposed by Johansson, we call them clean logics. It is easy to see that to clean a logic is a form of paraconsistetization, as developed by my colleague and co-author Alexandre Costa-Leite, Edelcio de Souza and Diego Dias (here is their 2016 paper). This is because if we add a contradiction to the premisses of any valid argument, it derives nothing - instead of deriving anything. Clean logics are paraconsistent in the strong sense that nothing makes the system explode.
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