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 fsato, minesima, takemurag@abelard.flet.keio.ac.jp Abstract “No C are A”. In what follows, we call such a syntactic manipulation of diagrams to derive a conclusion of deductive Internal syntactic operations on diagrams play a key role in accounting for efficacy of diagram use in reasoning. How- reasoning a construction of a diagrammatic proof. The point ever, it is often held that in the case of complex deductive here is that by unifying two diagrams in premises and ob- reasoning, diagrams can serve merely as an auxiliary source serving the topological relationship between the circles, one of information in interpreting sentences or constructing models. Based on experiments comparing subjects’ performances can automatically read off the correct conclusion. Shimojima in syllogism solving where logic diagrams of several different (1996) calls this a “free ride” property, and shows that it can forms are used, we argue that internal manipulations of dia- be seen to exist in other kinds of diagram use in reasoning grams, or what we call internal constructions of diagrammatic proofs, actually exist, and that such constructions are naturally and problem solving. triggered even for users without explicit prior knowledge of In general, a deductive reasoning task would be easy if their inference rules or strategies. it could be replaced with a task of constructing a concrete Keywords: External representation; Diagrammatic reasoning; Logic diagram; Deductive reasoning diagrammatic proof. Typically, such a construction is sup- posed to be triggered by external diagrams and carried out Introduction internally, without actual drawing or movement of physical objects. However, the existence of such internal manipula- People have tried to enhance reasoning abilities by the use of tions of diagrams has been the subject of controversy (see, artificial devices since ancient times. In particular, symbol e.g. Schwartz, 1995). Indeed, it is widely held that diagrams manipulation is a distinctive tool-use of human beings. Cer- can serve merely as a memory-aid or an auxiliary source of tainly, symbolic logic may be considered to be a tool for de- information in deductive problem solving. Thus, Larkin and ductive reasoning. However, it should be noted that symbolic Simon (1987) argue that reasoning is largely independent of logic (e.g., first-order logic) is not always a usable system for ways of representing information, hence diagrams are less untrained people. By contrast, visual-spatial representations beneficial in reasoning. Bauer and Johnson-Laird (1993) dis- are considered to be much more intuitive and effective for cuss efficacy of diagrams in deductive reasoning with double novices’ actual reasoning. Consequently, over the past few disjunction and argue that diagrams are used to keep track decades, many researchers have shown an interest in the ef- of alternative models, as postulated in mental model the- ficacy of diagrammatic reasoning (e.g. Allwein & Barwise, ory. Also in many logic textbooks, diagrams are used to de- 1996; Glasgow, Narayanan, & Chandrasekaran, 1995). pict models and aid understanding of logical representations, An important assumption in the study of diagrammatic rea- rather than as objects of syntactic manipulations. soning is that diagrams are syntactic objects to be manipu- In view of this situation, it is of central importance to in- lated in certain ways; we make an inference about a diagram vestigate whether internal manipulations of diagrams really itself, transforming it into another form or combining it with exist in actual reasoning with diagrams. Trafton and Trick- other diagrams. Such syntactic manipulations of diagrams ett (2001) argue that there are mental processes of “spatial play a crucial role in accounting for their efficacy in deduc- transformations” to extract information from graphs or visu- tive problem solving. For example, consider the following alization, based on an analysis of how expert scientists col- process of checking the validity of a syllogism using Euler lect data in their researches. Shimojima and Fukaya (2003) diagrams. and Shimojima and Katagiri (2008) argue for the existence of “inference by hypothetical drawing”, internal transformations B C B D1 : All A are B A of external diagrams, based on eye-tracking data of subjects D2 : No C are B j ¼ working with position diagrams in transitive inferential tasks. D1 D2 In this paper, we focus on more complex deductive reasoning Therefore, No C are A C B A tasks, namely, syllogistic reasoning tasks, and on the effects of logic diagrams externally given therein. We present evi- D3 dence for the existence of internal constructions of diagram- Figure 1: A diagrammatic proof of syllogism All A are B, No C are B; therefore No C are A with Euler diagrams. matic proofs, on the basis of experiments comparing subjects’ performances in syllogism solving where logic diagrams of The premise All A are B is represented by D1, and the premise several different forms are given. Our claim is consistent with No C are B by D2. By unifying D1 with D2, we can obtain di- the influential view in the study of external representations agram D3. Here the exclusion relation holds between circles in general, namely, that (a) external representations can be A and C, from which we can extract the correct conclusion used without being interpreted, and that (b) they can change 2668 the nature of tasks, namely, tasks with and without external representations are completely different from users’ point of view (see Zhang & Norman, 1994; Scaife & Rogers, 1996). The efficacy of logic diagrams has been investigated in the context of the studies of logic teaching method (Stenning, 1999; 2002; Dobson, 1999). In these studies, subjects are provided with substantial training in ways of manipulating diagrams. In contrast to this, our interest is in the question whether diagrams can be useful for those who are not trained in rules or strategies of diagrammatic deductive reasoning. This question is important because, in contrast to logical for- mulas in symbolic logic, logic diagrams in general have been Figure 3: Cognitive model for diagrammatic reasoning expected to be much more intuitive and effective for novices’ reasoning, not for experts’ nor for machine reasoning. In view Interpretational efficacy Firstly, diagrams can help fix the of the complexities of solving processes of deductive reason- correct interpretations of sentences and thereby avoid deduc- ing (e.g. Levesque, 1988), it is interesting to ask whether tive reasoning errors due to misinterpretation. For example, a logic diagrams can have this surprising property. sentence “All A are B” is sometimes misinterpreted as equiva- Logic diagrams have also been studied in the field of for- lent to “All B are A”. This is known as illicit conversion error mal diagrammatic logic since the 1990s (e.g. Shin, 1994; in the literature (e.g. Newstead & Griggs, 1983). Subjects Howse, Stapleton & Taylor, 2005), and inference systems for presented with diagrams such as the ones in Figure 2 could various diagrams such as Euler and Venn diagrams have been immediately see that the diagrams corresponding to these two developed. Currently, however, there are few empirical re- sentences are topologically different, and hence deliver dif- searches to investigate their cognitive foundations. Our study ferent semantic information. In our model, such processes is also intended to provide a bridge between logical and cog- are formulated as processes of matching the semantic infor- nitive studies of diagrammatic reasoning. mation obtained from diagrams with the one obtained from sentences. In this case, the validity of an argument is checked Cognitive model for reasoning with diagrams based on the same kind of process as the one in linguistic rea- Typical examples of deductive reasoning problems with ex- soning. Here diagrams are used in a static way, merely as a ternal logic diagrams are shown in Figure 2. record of information (Barwise & Etchemendy, 1991). A Inferential efficacy Secondly, and more importantly, dia- All B are A. B All B are A. grams can play a crucial role in reasoning processes them- B selves. More specifically, the solving processes of deductive All C are B. C All C are B. reasoning tasks can be replaced with internal manipulations of diagrams. In other words, one can check the validity of (Therefore, All C are A) (Therefore, All C are A) a deductive argument by means of constructions of diagrammatic proofs. The above model assumes that such construc- Figure 2: Examples of syllogistic reasoning tasks with diagrams tions are conducted through a proof-theoretical component of diagrammatic reasoning. If a task of constructing diagram- Here a syllogism is presented with logic diagrams (Euler and matic proofs consists of simple and intuitive steps, it is ex- Venn diagrams). How can such diagrams contribute to check- pected to be more tractable than usual linguistic inferences. ing the validity of a deductive argument? Let us first hypoth- esize a cognitive model of deductive problem solving with It seems to be generally agreed that logic diagrams have in- diagrams. The model is shown in Figure 3. This model high- terpretational efficacy. For example, Stenning (2002) argues lights two roles of diagrams in deductive reasoning. that (a)symmetricity of diagrams can aid processing of the Regarding sentential reasoning, we assume a standard two- meaning of quantified sentences in syllogisms. Mineshima, staged framework in natural language semantics (see, e.g. Okada, Sato, and Takemura (2008) presented experimental Blackburn & Bos, 2005), according to which sentences are evidence for such interpretational effects, based on a com- first associated with semantic information, and then the valid- parison of the performances of syllogistic solving tasks with ity of the argument is checked using some inferential mech- and without Euler diagrams. In what follows, we assume that anisms (such as model-theoretical or proof-theoretical ones).

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