arXiv subjects

Clarence Protin

Publications and source records attributed to Clarence Protin.

7 recordsLinked to original sources

arXiv2025

Hegel and Modern Topology

In this paper we sketch how some fundamental concepts of modern topology (as well as logic and category theory) can be understood philosophically in the light of Hegel's Science Logic as well how modern topological concepts can provide concrete illustrations of many of the concepts and deductions that Hegel used. Also these modern concepts can in turn be very powerful hermeneutic tools permitting a more rigorous and thorough grasp of Hegelian concepts. This paper can be seen as a continuation of our paper \cite{pro} where we argued that the prototypes of many fundamental notions of modern topology were already found in Aristotle's Physics. More generally it is hoped that this note makes a case for the possibility of a rigorous enriching interaction and mutual support between philosophy on one hand and modern logic and mathematics on the other. This paper is obviously meant only as a preliminary sketch and to offer some motivation for exploring in a more detailed and thorough way the subjects discussed.

math.HO↗

arXiv2024

Natural Term Logic

In this paper we develop a formal system called Natural Term Logic (NTL). NTL aims to represent key aspects of the logical and grammatical mechanisms of natural language as well as grammatical transformations which preserve core logical meaning. NTL can be seen as a refinement of the ideas of Quine's paper `Variables Explained Away' and the technical concepts introduced by Bealer and Zalta. NTL is more fine-grained than Bealer's first-order intensional logic (BL): there is a many-to-one correspondence $\nu$ between NTL terms and closed BL terms as well as a canonical map $\beta$ which assigns to each closed BL term a corresponding NTL term. The map $\nu$ can be seen as assigning a core logical content of the NTL term. We define a series of reductions on NTL terms which intuitivelyy speaking capture meaning-preserving syntactic transformations ( transformations which preserved the basic logical meaning of a term) and our main result is that each NTL term $T$ reduces to a unique normal term $N$. The reductions fall into the structural, predicative and pushing-in categories. Predicative reductions decompose NTL terms so that predication is only applied to a primitive term (such terms are called prenormal). A key ingredient in the proof is the fact that $\beta \nu N = N$ when $N$ is normal. This suggests that within NTL the normal form of a term expresses the core logical content of the term.

math.LO↗

arXiv2023

Aristotle's Second-Order Logic and Natural Deduction

This paper has two goals. The first goal is to show how an extension of second-order logic is a natural framework to formalize portions of Aristotle's \emph{Topics} and to bring to the foreground the logical, linguistic and philosophical interest of this work, showing in particular that we are in the presence of a richly intensional and modal conception of logic. Aristotelian logic and its related traditions in antiquity are often held to have been equivalent to monadic predicate logic and as such inadequate to formalize mathematics as well as scientific and philosophical discourse in general. The second goal of this paper is to argue that on the contrary the logical theories of Aristotle (which we argue correspond to a variant of natural deduction) and ancient authors such as Galen and Boethius were in fact quite sufficient to account for the logically complex expressions and reasoning involving multiple generality fundamental to the aforementioned disciplines.

math.HO↗

arXiv2022

Modern Definition and Ancient Definition

In this essay we examine some aspects of the classical theory of definition as codified in Aristotle's \emph{Topics} and Porphyry's \emph{Eisagog\^e} in the light of the way definition is carried out in modern mathematical practice. Our goal is to contribute to the understanding of the alleged gap existing between ancient and modern logic and science as well as the reasons behind allegations of inadequacy and lack of sophistication in the ancient theory of definition. Also to investigate the possibility of a co-interpretation between modern mathematical definitional practice and ancient definitional practice in particular in the light of topos theory. We find the ancient definitional practice asks relevant and overlooked questions about modern mathematical practice which apparently have escaped current philosophical and mathematical logical literature. We also present some general considerations about the structure and development of theories as these relate to the theory of definition.

math.HO↗

arXiv2021

A Logic for Aristotle's Modal Syllogistic

We propose a new modal logic endowed with a simple deductive system to interpret Aristotle's theory of the modal syllogism. While being inspired by standard propositional modal logic it is also a logic of terms that admits a (sound) extensional semantics involving possible states-of-affairs in a given world. Applied to the analysis of Aristotle's modal syllogistic as found in the \emph{Prior Analytics} A8-22 it sheds light on various fine-grained distinctions which when made allow us to clarify some ambiguities and obtain a completely consistent system and prove all of the modal syllogisms considered valid by Aristotle.This logic allows us also to make a connection with the axioms of modern propositional modal logic and to perceive to what extent these are implicit in Aristotle's reasoning. Further work wil involve addressing the question of the completeness of this logic (or variants thereof) together with the extension of the logic to include a calculus of relations (for instance the relational syllogistic treated in Galen's \emph{Introduction to Logic}) which Slomkowsky has argued is already found in the \emph{Topics}.

math.LO↗

arXiv2021

Modern incarnations of the Aristotelian concepts of Continuum and Topos

The aim of this paper is i) to argue for the feasibility and fruitfulness of a balance between the phenomenological method seeking intuitive evidence and the axiomatic-deductive method and ii) that there should be a mutual understanding between philosophy and mathematics and a cultivation of a historical self-awareness with regards to their common source in Greek philosophy. To this end we show how Aristotle's theory of \emph{sunekh\^es, apeiron} and \emph{topos} and related notions can be given a rigorous interpretation in terms of modern topology and geometry as well as category theory. This is facilitated by the fact that in Aristotle himself we already find a balance between intuition and formal logic. We also show how these powerful Aristotelian intuitions and concepts are found incarnated in diverse domains of modern mathematics.

math.HO↗

arXiv2020

On the Soundness of Bealer's Logics

Bealer's intensional logics T1 and T2 were proposed and expounded most fully in his book \emph{Quality and Concept} (1982) \cite{QC} as well in \cite{C}. These logics are unique in being extensions of classical first-order associated to a non-nominalist or non-inscriptionalist ontology and theory of meaning. Structurally they are similar to the second-order systems proposed about the same time by Zalta \cite{zalta}. In the book and article referenced above Bealer presents a detailed sketch of a proof of soundness and completeness for T1 and T2 (something which seems to be lacking for Zalta's systems). However there are key steps to the soundness proofs which are stated without proof and which seem to be non-trivial. In this paper we both simplify the original presentation of systems T1 and T2 and supply the rather complex and involved proofs of Bealer's missing lemmas.

math.LO↗