Rocco Gangle

Computer scientists, logicians and other researchers have recently paid renewed attention to the model of computation based on the logic of combinators. We develop a new formalization of the syntax of combinators employing the ‘generic figures’ approach to the categories of presheaves that provides an intuitive and easily interpreted diagrammatic syntax for combinators, one that eschews the need to label proper combinators with variable names. Furthermore, we show that this formal framework provides the possibility of representing choices of operand from among given alternatives at each level of a combinatory term, allowing for a more general type of combinatory expression, which we call a multi-combinator.

This chapter proposes a reframing of the current debate in French phenomenology over the “theological turn” by contrasting Husserlian reduction with Peirce’s method of Musement in “A Neglected Argument for the Reality of God.” It proceeds by first situating the “theological turn” in the context of the historical reception of Husserlian phenomenology in France and examining the core views of three participants in the controversy: Jean-Luc Marion, Michel Henry and Dominique Janicaud. The place of reduction in Husserl’s phenomenology is then examined more closely and understood from the point of view of its relation to intuition at the heart of the Husserlian “principle of all principles” in Ideas I. Peirce’s methodological approach of Musement is then contrasted with that of reduction in the French Husserlians to show how the correlative role of the category of Firstness in Peirce’s phaneroscopy is in this respect relevantly distinct from that of intuition in Husserl.

The hypothesis of the mathematician is always the conception of a system of relations. In order that they may be reasoned about mathematically, these relations must be conceived as embodied in some kind of objects; but the character of the objects, apart from the relations, is utterly immaterial. They are always made as bare, skeleton-like, or diagrammatic as possible. With mathematicians not born blind, they are always visual objects of the simplest kind, such as dots, or lines, or letters, and the like. The mathematician often passes from one mode of embodiment to another. Such a change is no change in the hypothesis but only in the diagrammatic embodiment of the hypothesis. The hypothesis itself consists in the system of relations alone.

Peirce’s diagrammatic system of Existential Graphs (EGα)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha })$$\end{document} is a logical proof system corresponding to the Propositional Calculus (PL). Most known proofs of soundness and completeness for EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document} depend upon a translation of Peirce’s diagrammatic syntax into that of a suitable Frege-style system. In this paper, drawing upon standard results but using the native diagrammatic notational framework of the graphs, we present a purely syntactic proof of soundness, and hence consistency, for EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document}, along with two separate completeness proofs that are constructive in the sense that we provide an algorithm in each case to construct an EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document} formal proof starting from the empty Sheet of Assertion, given any expression that is in fact a tautology according to the standard semantics of the system.

This article examines types of abductive inference in Hegelian philosophy and machine learning from a formal comparative perspective and argues that Robert Brandom’s recent reconstruction of the logic of recollection in Hegel’s Phenomenology of Spirit may be fruitful for anticipating modes of collaborative abductive inference in human/A.I. interactions. Firstly, the argument consists of showing how Brandom’s reading of Hegelian recollection may be understood as a specific type of abductive inference, one in which the past interpretive failures and errors of a community are explained hypothetically by way of the construction of a narrative that rehabilitates those very errors as means for the ongoing successful development of the community, as in Brandom’s privileged jurisprudential example of Anglo-American case law. Next, this Hegelian abductive dynamic is contrasted with the error-reducing backpropagation algorithms characterizing many current versions of machine learning, which can be understood to perform abductions in a certain sense for various problems but not (yet) in the full self-constituting communitarian mode of creative recollection canvassed by Brandom. Finally, it is shown how the two modes of “error correction” may possibly coordinate successfully on certain types of abductive inference problems that are neither fully recollective in the Hegelian sense nor algorithmically optimizable.

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the formal investigation of relations between syntax and semantics in such diagrammatic systems.

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic (Peirce’s Existential Graphs α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}). The straightforward construction of a presheaf category EGα∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}{{\mathcal {G}}}_{\alpha ^{*}}$$\end{document} of cuts-only Existential Graphs (equivalent to the well-studied category of finite forests) provides a basis for the further construction of the category EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}{{\mathcal {G}}}_\alpha $$\end{document} which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the formal investigation of relations between syntax and semantics in such diagrammatic systems.

A formal model of abductive inference is provided in which abduction is conceived as expansive and contractive movements through a topological space of theoretical and practical commitments. A pair of presheaves over the space of commitments corresponds to communities sharing commitments on the one hand and possible obstructions to commitments on the other. In this framework, abductive inference is modeled by the dynamics of redistributed communities of commitment made in response to obstructive encounters. This semantic-pragmatic model shows how elementary category theory tools can be used to formalize abductive inference while hewing close to ordinary intuitions about collective agency and reasoning.

In this chapter, we look at the problem of abductive inference as refracted through a mathematical context quite different from the topological areas and cuts of Peirce’s graphs as analyzed in the previous chapter. Rather than the relational nestings of areas within graphs and the categorical structures induced in a holistic way through the relations holding among the graphs conceived as a deductive system, we will examine the model of abduction that emerges from Alain Badiou’s provocative conflating of philosophical ontology and Zermelo–Fraenkel set theory in Being and Event.

The semantics of picturing, broadly understood as an isomorphism between relevant relations among parts of a picture and relations constituting a state of affairs in some target domain, are a core feature of Wittgenstein’s Tractarian theory of representation. This theory was subsequently developed by Wilfrid Sellars into a rich theory of language and cognition. In this paper we show that by recasting the positive fragment (without negation) of C.S. Peirce’s beta level of Existential Graphs as a category of presheaves, the iconic coordination of syntax and semantics in the Wittgensteinian-Sellarsian picturing-relation may be represented formally in terms of the natural transformations in this category.

Brandom's method of analyzing pragmatic relations among different practices and vocabularies through meaning-use diagrams is used to specify how Laruelle's nonphilosophical suspension of the Principle of Sufficient Philosophy may be distinguished from the philosophical auto-critiques of such thinkers as Badiou and Derrida. A superposition of diagrams modeling philosophical sufficiency on the one hand and supplementation through the Other on the other provides a schematic representation of the core duality of what Laruelle calls The-Philosophy. In contrast to this self-implicating and self-reproducing structure, Laruelle's axiomatic method is shown to enable a unilateral usage of the practices and vocabularies of philosophy as mere material that is no longer subject to the presuppositions or ineluctable restitution of philosophy as such.

The foundation of Badiou’s ontological schema relies on the notion of a transcendental T. As emerged from the analysis in the last chapter, T is in fact best understood as a two-fold entity: It is used to measure the degree of identity of relations between objects in the world, and, on the other hand, is a structured system of relations by itself, namely a complete distributive lattice (Heyting algebra), with the property of being essentially optimal with respect to the contexts it is meant to model.

The notion of iconicity developed in the previous chapter might appear open to the objection that the use of any such conception of iconic “structure” to model abduction must err either on the side of being too formal and thus insensitive to the concrete, existential details of the situation at hand, or that of being too concretely figural or “intuitive” and thus inappropriate for the rigorous purposes of logical analysis. In fact, we aim to show that the particular power of iconic relations to support robust abductive modeling rests precisely in their capacity to reduce this Scylla-and-Charybdis problem of dry formalism versus muddy intuition to the status of a false dichotomy. Especially when modeled and analyzed via the categorical mathematics of presheaves and sheaves, iconicity as such provides a conceptual framework that is capable of varying smoothly and continuously (and tractably) from the most concrete, material domains to the most formal and abstract.

Everything you need to understand both Laruelle's critique of difference and his project of non-philosophyGilles Deleuze described Laruelle's thought as 'one of the most interesting undertakings of contemporary philosophy'. Now, Rocco Gangle - who translated Laruelle's philosophy into English - takes you through Laruelle's trailblazing book Philosophies of Difference, helping you to understand both Laruelle's critique of Difference and his project of non-philosophy, which has become one of the most intriguing avenues in contemporary thought. He explains the context within which Laruelle's thought developed and takes you through the challenging argument and conceptual scaffolding of 'Philosophies of Difference'.

This book consolidates and extends the authors’ work on the connection between iconicity and abductive inference. It emphasizes a pragmatic, experimental and fallibilist view of knowledge without sacrificing formal rigor. Within this context, the book focuses particularly on scientific knowledge and its prevalent use of mathematics. To find an answer to the question “What kind of experimental activity is the scientific employment of mathematics?” the book addresses the problems involved in formalizing abductive cognition. For this, it implements the concept and method of iconicity, modeling this theoretical framework mathematically through category theory and topoi. Peirce's concept of iconic signs is treated in depth, and it is shown how Peirce's diagrammatic logical notation of Existential Graphs makes use of iconicity and how important features of this iconicity are representable within category theory. Alain Badiou’s set-theoretical model of truth procedures and his relational sheaf-based theory of phenomenology are then integrated within the Peircean logical context. Finally, the book opens the path towards a more naturalist interpretation of the abductive models developed in Peirce and Badiou through an analysis of several recent attempts to reformulate quantum mechanics with categorical methods. Overall, the book offers a comprehensive and rigorous overview of past approaches to iconic semiotics and abduction, and it encompasses new extensions of these methods towards an innovative naturalist interpretation of abductive reasoning.