The Advent of Formal Diagrammatic Reasoning Systems

Abstract

Abstract. In knowledge representation and reasoning systems, diagrams have many practical applications and are used in numerous settings. Indeed, it is widely accepted that diagrams are a valuable aid to intuition and help to convey ideas and information in a clear way. On the other side, logicians have viewed diagrams as informal tools, but which cannot be used in the manner of formal argumentation. Instead, logicians focused on symbolic representations of logics. Recently, this perception was overturned in the mid 1990s, first with seminal work by Shin on an extended version of Venn diagrams. Since then, certainly a growth in the research field of formal reasoning with diagrams can be witnessed. This paper discusses the evolution of formal diagrammatic logics, focusing on those systems which are based on Euler and Venn-Peirce diagrams, and Peirces existential graphs. Also discussed are some challenges faced in the area, some of which are specifically related to diagrams.

Syllogisms in Rudimentary Linear Logic, Diagrammatically

“…Existential graphs, see [6,7,47,66,50] were introduced by Peirce as a diagrammatic system for representing and reasoning with relations. The system is divided into three parts named Alpha, Beta and Gamma.…”

“…Its meaning is "there is a cat which is cute and young", whereas the meaning of the rightmost graph above is "there exists a man who will not die". The two graphs were taken from [6] and [7], respectively.…”

“…See [19]. is the one associated with proof (7) and captures its essential geometric content. It has to be observed that it is a planar graph, namely one without crossing edges.…”

“…To contextualize this paper in the framework of logical reasoning with diagrams, in this section we recall some of the main features of the diagrammatic formal reasoning systems that extend the expressiveness of Euler diagrams, see [55,7]. We go from Euler diagrams to existential graphs, through Venn and Venn-Peirce diagrams, Venn-I and Venn-II diagrams, Euler/Venn diagrams, spider diagrams and constraint diagrams.…”

“…Alpha and Beta respectively correspond to propositional logic and first order logic with predicates and equality without functions or constants, whereas Gamma includes features of higher order logic and modal logic, as well as the possibility of expressing self-reference. Gamma was never finished by Peirce and, as pointed out in [7], mainly its modal logic fragment has been elaborated to the contemporary mathematical standards. We here focus our interest on Alpha and Beta, which are proved to be sound and complete in [6].…”