On degrees of exclusion within and among systems

Abstract

The Color Exclusion Problem imposes inter alia that logic must look to conceptual arrangements in the world. This represents the collapse of what Wittgenstein held in his tractarian account of Logic (cf. 5.473 and 5.551). The kind of logical structure that we find inside elementary propositions in systems cannot be expressed in terms of truthfunctionality, i.e., with logical products and sums. Moreover, since elementary propositions cannot be properly captured by the negation of a repeated item, the kind of exclusion within systems cannot be the contradiction. This means the necessary emergence of semantics in an alleged syntactical domain. We require then new tools to tackle non-neutral operators with restrictive scope and finer semantic exclusions. I introduce here, then, the notion of degrees of exclusion both internal and external (of second order) to a system. These degrees of exclusion may not be homogeneous, but must be infinite depending on the system of proposition that we are working with. Along the way, I advocate that we cannot reduce one system to another without distortions and that doing so is not necessary either.