Diagram Semantics and Schema Algebra: A Framework for the Analysis and Simulation of Diagrammatic Reasoning [Abstract of an article]

(1) Diagrams constitutes a basic type in the classification of the different representational forms based on relations of similarity (i.e. iconic signs). According to C. S. Peirce, there are three basic types of iconic signs: pictures, diagrams and metaphors. Peirce asserted that "all necessary reasoning is diagrammatic" and that "necessary reasoning makes its conclusions evident" by a constructive process leading to the observation of a mental model of the object of the reasoning process. Asserting facts about some situation, introducing hypothetical state of affairs, and drawing conclusions, are somehow similar to constructing, manipulating and observing a diagram. (2) This similarity is often stated as an isomorphism between the diagram and (significant parts of) the object it is intended to represent. But a diagram is never purely iconic in this sense. "Diagrams are not generally the direct image of a certain reality ... but the figural expression of an already elaborated conceptual structure", and in this respect it is organized "like any other symbolic system" (E. Fischbein). (3) From this point of view logic diagrams should make a good casestudy. In logic diagrams we would expect to find a simple relation between our logical intuitions and the diagrammatic structure. Logic diagrams has been defined as "a two-dimensional geometric figure with spatial relations that are isomorphic with the structure of a logical statement" (M. Gardner). This isomorphism is often assumed to be self-evident, and consequently an analysis of the topological, algebraic and cognitive conditions of representations appears almost unnecessary. In my Ph.d. I have proposed an analysis of logic diagrams in terms of three levels of constraints and two levels of representation. The constraints are related to (a) the geometric-figural elements, (b) the semantic interpretation of the diagram, and (c) the algebraic structure, that makes the interpretation possible. An ex.: the Jordan curve theorem imposes a topological constraint on the geometric-figural elements. There has to be two levels of representation in logic diagrams: a primary level that establishes a correspondence between possible combinations of logical expressions and possible compositions of areas (constructed from the geometric-figural elements), and a secondary level of selection of a particular logical class or proposition corresponding to the specification of a particular area in the logic diagram. Thus there are topological and algebraic constraints that contributes to the conditions of representation for such diagrams (i.e. they make them possible). But at the level of the semantic interpretation we should be able to analyse specific cognitive conditions pertaining to the process of constructing, manipulating and observing schematic cognitive structures and mental models involved in diagrammatic reasoning in general. (4) In cognitive semantics there has been a rediscovery of a kind of highly abstract diagrammatic reasoning that organizes our categorial and intuitive understanding in natural language as well as in perceptual experience. This reasoning utilizes a small inventory of very abstract and domain independent schematic structures, like the image schemas suggested by L. Talmy for containment, connectivity, path, part-whole relations and other basic geometric or topological relations and properties. (5) It has been important for cognitive semantics as a research paradigm to emphasize the embodied nature of image schemas and to oppose "objectivist theories" of meaning in AI etc., but from the point of view of AI, there is no reason why we should not try to encode the relational and inferential properties of schematic structures in a logical syntax in order to simulate parts of this kind of reasoning (i.e. the construction, manipulation and observation of schematic structures). Some recent work on knowledge representation formalism for logical databases suggests, that it should be possible to devise a concept algebra for conceptual structures described as feature structures on which a set of algebraic (lattice) operations are performed. It should investigated whether this framework can be extended to deal with schematic structures and operations, thereby forming a schema algebra.