A uniform model of computational conceptual blending

Abstract

We present a mathematical model for the cognitive operation of conceptual blending that aims at being uniform across di↵erent representation formalisms, while capturing the relevant structure of this operation. The model takes its inspiration from amalgams as applied in case-based reasoning, but lifts them into the theory of categories so as to follow Joseph Goguen's intuition for a mathematically precise characterisation of conceptual blending at a representationindependent level of abstraction. We prove that our amalgam-based categorytheoretical model of conceptual blending is essentially equivalent to the pushout model in the ordered category of partial maps as put forward by Joseph Goguen. But unlike Goguen's approach, our model is more suitable for a computational realisation of conceptual blending, and we exemplify this by concretising our model to computational conceptual blends for various representation formalisms and application domains.

Image Schemas and Conceptual Blending in Diagrammatic Reasoning: The Case of Hasse Diagrams

“…The integration of the image-schematic space with the geometric space follows the principles of conceptual blending, i.e., establishing a cross-space correspondence between these two spaces. Formalising these correspondences allows us to compute the conceptual blend that characterises the diagram as the combination of image schemas with the geometric configuration based on category-theoretic colimits [33].…”

“…Having specified some correspondences between elements of these theories, the computation of their colimit yields a new theory where all counterpart elements (types, predicates or functions) are merged into the same element, and the remaining elements and axioms of both theories are also included (see also Appendix A). This mathematical framework is apt to model conceptual blending [33].…”

“…Finally, regarding a formal view of conceptual blending, the blending process has been described though a general, mathematical theory [3,10,33]. This is done through amalgams, obtained by generalising the input spaces as much as necessary to find commonalities, and blending parts of them towards a consistent and novel output.…”

“…Building on this work, Schorlemmer et al [32] modeled the process of solving a riddle, using blending and typed FOL specifications of image schemas. The interrelations between amalgams, Goguen's framework, and our current model of blending are discussed in [33]. All aforementioned work contributes valuable, useful formalisations of blending as a creative process.…”

A Cognitively-Inspired Model for Making Sense of Hasse Diagrams

“…(c) A formalisation of suitable correspondences between image-schema structures and the geometric configuration. Given such correspondences, each blend can be modeled as a particular kind of category-theoretical colimit [9].…”