Heterogeneous Logic

Abstract

A major concern to the founders of modern logic—Frege, Peirce, Russell, and Hilbert—was to give an account of the logical structure of valid reasoning. Taking valid reasoning in mathematics as paradigmatic, these pioneers led the way in developing the accounts of logic which we teach today and that underwrite the work in model theory, proof theory, and definability theory. The resulting notions of proof, model, formal system, soundness, and completeness are things that no one claiming familiarity with logic can fail to understand, and they have also played an enormous role in the revolution known as computer science. The success of this model of inference led to an explosion of results and applications. But it also led most logicians—and those computer scientists most influenced by the logic tradition—to neglect forms of reasoning that did not fit well within this model. We are thinking, of course, of reasoning that uses devices like diagrams, graphs, charts, frames, nets, maps, and pictures. The attitude of the traditional logician to these forms of representation is evident in the quotation of Neil Tennant in Chapter I, which expresses the standard view of the role of diagrams in geometrical proofs. One aim of our work, as explained there, is to demonstrate that this dogma is misguided. We believe that many of the problems people have putting their knowledge of logic to work, whether in machines or in their own lives, stems from the logocentricity that has pervaded its study for the past hundred years. Recently, some researchers outside the logic tradition have explored uses of diagrams in knowledge representation and automated reasoning, finding inspiration in the work of Euler, Venn, and especially C. S. Peirce. This volume is a testament to this resurgence of interest in nonlinguistic representations in reasoning. While we applaud this resurgence, the aim of this chapter is to strike a cautionary note or two. Enchanted by the potential of nonlinguistic representations, it is all too easy to overreact and so to repeat the errors of the past.

Heterogeneous inferences with maps

“…I will argue that maps can stand in logical relation with sentences through heterogeneous inferences (see also Aguilera & Castellano, forthcoming). Contrary to the traditional view, we saw that Camp's (2007) and Rescorla's (2009b) Barwise and Etchemendy (1996), who have mainly focused on the role of diagrams in mathematical proofs. These authors put forward an informational approach to inference.…”

Towards a diagrammatic classification

“…As Barwise and Etchemendy (1996) argue, ‘despite the obvious importance of visual images in human cognitive activities, visual representation remains a second-class citizen in both the theory and practice of mathematics’ (Barwise & Etchemendy, 1996: 3). If we accept this logocentric approach to mathematics, what would be left of its epistemology?…”

Rectifying the Mischaracterization of Logic by Mental Model Theorists

“…We believe the answer is Yes, and in fact believe that since MMT posits models that are at once iconic and (via annotations) symbolic, will turn out to be what is called a heterogenous logic (Arkoudas & Bringsjord, 2009; Barwise & Etchemendy, 1995), 17 but whether or not our beliefs are correct, one thing is clear: If MMT theorists continue to “inform” cognitive scientists in the business of searching for such things as that logic is to be identified with a logic based on the material conditional (or for that matter any particular feature of a particular logic or class thereof), the odds of falsifying or vindicating our belief are lowered by forces that no genuine science should countenance. For the record, we are not in this short commentary advocating the use of formal logic to model the mind (or even, more reasonably, its inferential and declarative dimension).…”