When is a bunch of marks on paper a diagram? Diagrams as homomorphic representations

Abstract

That diagrams are analog, i.e., homomorphic, representations of some kind, and sentential representations are not, is a generally held intuition. In this paper, we develop a formal framework in which the claim can be stated and examined, and certain puzzles resolved. We start by asking how physical things can represent information in some target domain. We lay a basis for investigating possible homomorphisms by modeling both the physical medium and the target domain as sets of variables, each with a constraint structure. When a homomorphism exists, the causality of the physical medium can provide a "free ride," automatically determining the information that is needed to solve a problem. The modeling technique enables us to show how in sentential representations the structure of the 2-D space is used non-homomorphically to define tokens, and to provide a minimal homomorphism with respect to the ordering of the arguments for n-ary functions, predicates, and operators, but otherwise there is no homomorphism with the target domain. Our treatment also characterizes diagrams as a type of a more general class of physical representations in which physical causality plays a role in providing the information needed to solve a problem.

Human inference beyond syllogisms: an approach using external graphical representations

“…Before the inference tasks, the participants were given instructions on the (informal) semantics of representations and then their comprehensions were checked by a pretest (for details, see below). Given the experimental settings, it is assumed in our framework that the tasks for participants involve not just syntactic transformations of the pieces of some plane figure puzzle, but a kind of diagrammatic inference, in which the entailment from premise 6 This kind of difficulty in premise integration can also arise in reasoning with Euler diagrams in cases of syllogisms having an existential premise (e.g., All B are A; some C are B. Therefore, some C are A); see Sato and Mineshima (2015).…”

“…Cases in which premise diagrams do not uniquely match when merging them are relatively complex in that the reasoner has to consider more than one possibility; we call them 'non-matching' cases. 6 Solving non-matching cases like Fig.7 could involve processes requiring more cognitive effort than 'matching' cases like Fig.5(c-d) in the non-matching cases is harder than reasoning in the matching cases. 7…”