Boole Algebra in a Contemporary Setting. Boole-Operations, Types as Propositions and Immanent Reasoning
Contributor(s)Savoirs, Textes, Langage (STL)
Université de Lille, Sciences et Technologies - Université de Lille, Sciences Humaines et Sociales - Centre National de la Recherche Scientifique (CNRS)
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The work of Souleymane Bachir Diagne has set a landmark in many senses, but perhaps the most striking one is his inexhaustible thrive to build multifarious conceptual links and bridges between traditions and to motivate others to further develop this wonderful realization of unity in diversity. Three main fields of his remarkable work are: history and philosophy of logic (In the present talk I will focus on philosophy of logic, and more precisely on the algebra of logic of George Boole, that launched Bachir Diagne's (1989) academic carrier. However, the framework has bearings for the other both fields as developed in recent publications in collaboration. I will briefly discuss as an example of application the case of suspensive (muʿallaq) condition (taʿliq) in Islamic law and I might discuss this issue more deeply during the discussion, More precisely, the main objective of my presentation is to discuss a novel approach to both, the distinction between Boolean operators and inferentially defined connectives, and the interplay of the former with the latter. The epistemological framework underlying my discussion is the dialogical approach to Per Martin-Löf's (1984) Constructive Type Theory recently developed in Lille I will illustrate the issues by showing how to generalize Boolean operators for finite sets within the dialogical setting and I will take the chance to put the framework into work by developing a demonstration of what is known as the fourth axiom of Peano's arithmetic ("0 is identical to no successor of a natural number": (∀x : ℕ) ¬Id(ℕ, 0, s(x))). Such a demonstration gives us provides a nice example of the difference between, to put it in a dialogical terminology, asserting that the solution to a problem or enquiry is no, and asserting that there is no solution or answer to such an enquiry. Furthermore, the dialogical setting, where an "empirical quantity" is understood as the outcome of a procedure triggered by a question specific to that quantity, provides a new perspective on Willfried Sellars's notion of Space of Reasons. More precisely, the dialogical framework proposed should show how to integrate world-directed thought (that displays empirical content) into an inferentialist approach. This suggests that the dialogical approach to Constructive Type Theory offers a way to integrate within one epistemological framework the two conflicting readings of the Space of Reasons brought forward by John McDowell on one 1 The paper has been developed in the context of the researches for transversal research axis Argumentation (UMR 8163: STL), the research project ADA at the MESHS-Nord-pas-de-Calais and the research projects: ANR-SÊMAINÔ (UMR 8163: STL).