Constructing internal diagrammatic proofs from external logic diagrams

Constructing internal diagrammatic proofs from external logic diagrams Yuri Sato, Koji Mineshima, and Ryo Takemura Department of Philosophy, Keio University {sato, minesima, takemura}@abelard. ﬂet. keio.ac. jp Abstract Internal syntactic operations on diagrams play a key role in accounting for efﬁcacy of diagram use in reasoning. How- ever, it is often held that in the case of complex deductive reasoning, diagrams can serve merely as an auxiliary source of information in interpreting sentences or constructing mod- els. Based on experiments comparing subjects’ performances in syllogism solving where logic diagrams of several different forms are used, we argue that internal manipulations of dia- grams, or what we call internal constructions of diagrammatic proofs, actually exist, and that such constructions are naturally triggered even for users without explicit prior knowledge of their inference rules or strategies. Keywords: External representation; Diagrammatic reasoning; Logic diagram; Deductive reasoning Introduction People have tried to enhance reasoning abilities by the use of artiﬁcial devices since ancient times. In particular, symbol manipulation is a distinctive tool-use of human beings. Cer- tainly, symbolic logic may be considered to be a tool for de- ductive reasoning. However, it should be noted that symbolic logic (e.g., ﬁrst-order logic) is not always a usable system for untrained people. By contrast, visual-spatial representations are considered to be much more intuitive and effective for novices’ actual reasoning. Consequently, over the past few decades, many researchers have shown an interest in the ef- ﬁcacy of diagrammatic reasoning (e.g. Allwein & Barwise, 1996; Glasgow, Narayanan, & Chandrasekaran, 1995). An important assumption in the study of diagrammatic rea- soning is that diagrams are syntactic objects to be manipu- lated in certain ways; we make an inference about a diagram itself, transforming it into another form or combining it with other diagrams. Such syntactic manipulations of diagrams play a crucial role in accounting for their efﬁcacy in deduc- tive problem solving. For example, consider the following process of checking the validity of a syllogism using Euler diagrams. B A D 1 : All A are B D 2 : No C are B Therefore, No C are A D 1 C j C B D 2 B A D 3 Figure 1: A diagrammatic proof of syllogism All A are B, No C are B; therefore No C are A with Euler diagrams. The premise All A are B is represented by D 1 , and the premise No C are B by D 2 . By unifying D 1 with D 2 , we can obtain di- agram D 3 . Here the exclusion relation holds between circles A and C, from which we can extract the correct conclusion “No C are A”. In what follows, we call such a syntactic ma- nipulation of diagrams to derive a conclusion of deductive reasoning a construction of a diagrammatic proof. The point here is that by unifying two diagrams in premises and ob- serving the topological relationship between the circles, one can automatically read off the correct conclusion. Shimojima (1996) calls this a “free ride” property, and shows that it can be seen to exist in other kinds of diagram use in reasoning and problem solving. In general, a deductive reasoning task would be easy if it could be replaced with a task of constructing a concrete diagrammatic proof. Typically, such a construction is sup- posed to be triggered by external diagrams and carried out internally, without actual drawing or movement of physical objects. However, the existence of such internal manipula- tions of diagrams has been the subject of controversy (see, e.g. Schwartz, 1995). Indeed, it is widely held that diagrams can serve merely as a memory-aid or an auxiliary source of information in deductive problem solving. Thus, Larkin and Simon (1987) argue that reasoning is largely independent of ways of representing information, hence diagrams are less beneﬁcial in reasoning. Bauer and Johnson-Laird (1993) dis- cuss efﬁcacy of diagrams in deductive reasoning with double disjunction and argue that diagrams are used to keep track of alternative models, as postulated in mental model the- ory. Also in many logic textbooks, diagrams are used to de- pict models and aid understanding of logical representations, rather than as objects of syntactic manipulations. In view of this situation, it is of central importance to in- vestigate whether internal manipulations of diagrams really exist in actual reasoning with diagrams. Trafton and Trick- ett (2001) argue that there are mental processes of “spatial transformations” to extract information from graphs or visu- alization, based on an analysis of how expert scientists col- lect data in their researches. Shimojima and Fukaya (2003) and Shimojima and Katagiri (2008) argue for the existence of “inference by hypothetical drawing”, internal transformations of external diagrams, based on eye-tracking data of subjects working with position diagrams in transitive inferential tasks. In this paper, we focus on more complex deductive reasoning tasks, namely, syllogistic reasoning tasks, and on the effects of logic diagrams externally given therein. We present evi- dence for the existence of internal constructions of diagram- matic proofs, on the basis of experiments comparing subjects’ performances in syllogism solving where logic diagrams of several different forms are given. Our claim is consistent with the inﬂuential view in the study of external representations in general, namely, that (a) external representations can be used without being interpreted, and that (b) they can change