Reasoning with Spider Diagrams

Spider diagrams combine and extend Venn diagrams and Euler circles to express constraints on sets and their relationships with other sets. These diagrams can usefully be used in conjunction with object-oriented modelling notations such as the Unified Modelling Language. This paper summarises the main syntax and semantics of spider diagrams and introduces four inference rules for reasoning with spider diagrams and a rule governing the equivalence of Venn and Euler forms of spider diagrams. This paper also details rules for combining two spider diagrams to produce a single diagram which retains as much of their combined semantic information as possible and discusses disjunctive diagrams as one possible way of enriching the system in order to combine spider diagrams so that no semantic information is lost

A graph-theoretic account of logics

A graph-theoretic account of logics is explored based on the general notion of m-graph (that is, a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as m-graphs. After defining a category freely generated by a m-graph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness theorems are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with nondeterministic semantics, and subsume all logics endowed with an algebraic semantics

Euler diagram-based notations

Euler diagrams have been used for centuries as a means for conveying logical statements in a simple, intuitive way. They form the basis of many diagrammatic notations used to represent set-theoretic relationships in a wide range of contexts including software modelling, logical reasoning systems, statistical data representation, database search queries and file system management. In this paper we survey notations based on Euler diagrams with particular emphasis on formalization and the development of software tool support

The Semiotics of Spider Diagrams

Spider diagrams are based on Euler and Venn/Peirce diagrams, forming a system which is as expressive as monadic first orderlogic with equality. Rather than being primarily intended for logicians,spider diagrams were developed at the end of the 1990s in the context of visual modelling and software specification. We examine the original goals of the designers, the ways in which the notation has evolved and itsconnection with the philosophical origins of the logical diagrams of Euler, Venn and Peirce on which spider diagrams are based. Using Peirce's concepts and classification of signs, we analyse the ways in which different sign types are exploited in the notation. Our hope is that this analysis may be of interest beyond those readers particularly interested in spider diagrams, and act as a case study in deconstructing a simple visual logic. Along the way, we discuss the need for a deeper semiotic engagement in visual modelling

Euler Diagram Transformations

Euler diagrams are a visual language which are used for purposes such as the presentation of set-based data or as the basis of visual logical languages which can be utilised for software specification and reasoning. Such Euler diagram reasoning systems tend to be defined at an abstract level, and the concrete level is simply a visualisation of an abstract model, thereby capturing some subset of the usual boolean logic. The visualisation process tends to be divorced from the data transformation process thereby affecting the user's mental map and reducing the effectiveness of the diagrammatic notation. Furthermore, geometric and topological constraints, called wellformedness conditions, are often placed on the concrete diagrams to try to reduce human comprehension errors, and the effects of these conditions are not modelled in these systems. We view Euler diagrams as a type of graph, where the faces that are present are the key features that convey information and we provide transformations at the dual graph level that correspond to transformations of Euler diagrams, both in terms of editing moves and logical reasoning moves. This original approach gives a correspondence between manipulations of diagrams at an abstract level (such as logical reasoning steps, or simply an update of information) and the manipulation at a concrete level. Thus we facilitate the presentation of diagram changes in a manner that preserves the mental map. The approach will facilitate the realisation of reasoning systems at the concrete level; this has the potential to provide diagrammatic reasoning systems that are inherently different from symbolic logics due to natural geometric constraints. We provide a particular concrete transformation system which preserves the important criteria of planarity and connectivity, which may form part of a framework encompassing multiple concrete systems each adhering to different sets of wellformedness conditions