Abstract. Spider diagrams are a visual notation for expressing logical statements. In this paper we identify a well known fragment of first order predicate logic, that we call ESD, equivalent in expressive power to the spider diagram language. The language ESD is monadic and includes equality but has no constants or function symbols. To show this equivalence, in one direction, for each diagram we construct a sentence in ESD that expresses the same information. For the more challenging converse we show there exists a finite set of models for a sentence S that can be used to classify all the models for S. Using these classifying models we show that there is a diagram expressing the same information as S. 1

Constraint diagrams are designed for the formal specification of software systems. However, their applications are broader than this since constraint diagrams are a logic that can be used in any formal setting. This document summarizes the main results presented in my PhD thesis, the focus of which is on a fragment of the constraint diagram language, called spider diagrams, and constraint diagrams themselves. In the thesis, sound and complete systems of spider diagrams and constraint diagrams are presented and the expressiveness of the spider diagram language is established. 1

Query graphs with cuts are inspired by Sowa’s conceptual graphs, which are in turn based on Peirce’s existential graphs. In my thesis ‘The Logic System of Concept Graphs with Negations’, conceptual graphs are elaborated mathematically, and the cuts of existential graphs are added to them. This yields the system of concept graphs with cuts. These graphs correspond to the closed formulas of first order predicate logic. Particularly, concept graphs are propositions which are evaluated to truth-values. In this paper, concept graphs are extended to so-called query graphs, which are evaluated to relations instead. As the truth-values TRUE and FALSE can be understood as the two 0-ary relations, query graphs extend the expressiveness of concept graphs. Query graphs can be used to elaborate the logic of relations. In this sense, they bridge the gap between concept graphs and the Peircean Algebraic Logic, as it is described in Burch’s book ’A Peircean Reduction Thesis’. But in this paper, we focus on deduction procedures on query graphs, instead of operations on relations, which is the focus in PAL. Particularly, it is investigated how the adequate calculus of concept graphs can be transferred to query graphs.

Diagrammatic reasoning is a tradition of visual logic that allows sentences that are equivalent to first order logic to be written in a visual or structural form: typically for improved usability. A calculus for the diagram is then defined that allows well-formed formulas to be derived. This calculus is intended to simulate logical inference. Description logics (DLs) have become a popular sub-set of first order logic that have decidable tableau theorem provers and are sound and complete. Our paper explores whether several existing well-known diagrammatic reasoning systems are compatible with DLs. We provide translations between the DL ALCI and a appropriate subset of Peirce’s relation graphs, termed RG ALCI. A precise formal elaboration, where relation graphs are for example defined in terms of mathematical graph theory, goes beyond the scope of this paper. We will provide a semiformal approach instead.

It is well accepted that diagrams play a crucial role in human reasoning. But in mathematics, diagrams are most often only used for visualizations, but it is doubted that diagrams are rigor enough to play an essential role in a proof. This paper takes the opposite point of view: It is argued that rigor formal logic can carried out with diagrams. In order to do that, it is first analyzed which problems can occur in diagrammatic systems, and how a diagrammatic system has to be designed in order to get a rigor logic system. Particularly, it will turn out that a separation between diagrams as representations of structures and these structures themselves is needed, and the structures should be defined mathematically. The argumentation for this point of view will be embedded into a case study, namely the existential graphs of Peirce. In the second part of this paper, the theoretical considerations are practically carried out by providing mathematical definitions for the semantics and the calculus of existential Alpha graphs, and by proving mathematically that the calculus is sound and complete.

In the research field of diagrammatic reasoning, there are some attempts for providing diagrammatic forms of logic. Probably the most important one is Sowa's system of conceptual graphs from which Sowa claims that they have at least the expressiveness of first order predicate logic (FOPL). But a closer observation shows that their definitions lack (mathematical) preciseness, which yields several ambiguities, gaps and flaws. In my dissertation

Abstract. Conceptual Graphs are a common knowledge representation system which are used in conjunction with an explicit type hierarchy of the domain. However, this means the interpretation of information expressed in conceptual graphs requires the combined use of information from different sources, which is not always an easy cognitive task. Though it is possible to explicitly represent the type hierarchy with Conceptual Graphs with Cuts, this less natural expression of the type hierarchy information is not as easy to interpret and soon takes up a lot of space. Now, one of the main advantages of Euler diagram-based notations like Spider diagrams is the natural diagrammatic representation of hierarchies. However, Spider diagrams lack facilities such as the ability to represent general relationships between objects which is necessary for knowledge representation tasks. We bring together the most pertinent features of both of these notations, creating a new hybrid notation called Conceptual Spider Diagrams. We provide formal syntax and semantics of this new notation, together with examples demonstrating its capabilities. 1

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. 1