Abstract
This paper explores the question of what makes diagrammatic representations effective for human logical reasoning, focusing on how Euler diagrams support syllogistic reasoning. It is widely held that diagrammatic representations aid intuitive understanding of logical reasoning. In the psychological literature, however, it is still controversial whether and how Euler diagrams can aid untrained people to successfully conduct logical reasoning such as set-theoretic and syllogistic reasoning. To challenge the negative view, we build on the findings of modern diagrammatic logic and introduce an Euler-style diagrammatic representation system that is designed to avoid problems inherent to a traditional version of Euler diagrams. It is hypothesized that Euler diagrams are effective not only in interpreting sentential premises but also in reasoning about semantic structures implicit in given sentences. To test the hypothesis, we compared Euler diagrams with other types of diagrams having different syntactic or semantic properties. Experiment compared the difference in performance between syllogistic reasoning with Euler diagrams and Venn diagrams. Additional analysis examined the case of a linear variant of Euler diagrams, in which set-relationships are represented by one-dimensional lines. The experimental results provide evidence supporting our hypothesis. It is argued that the efficacy of diagrams in supporting syllogistic reasoning crucially depends on the way they represent the relational information contained in categorical sentences.
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Notes
Bauer and Johnson-Laird (personal communication) in their unpublished study also reported that while Euler diagrams—or, more specifically, what we will classify as a traditional form of Euler diagrams (Euler circles)—improved reasoning with a class of difficult syllogisms but retarded reasoning with a class of easy syllogisms.
It is standard in the literature of diagrammatic logic to distinguish between two layers of diagrammatic syntax: concrete (token) syntax and abstract (type) syntax (Howse et al. 2002). The former is concerned with the surface structure of a diagram that is visible to users, while the latter gives a formal definition that concrete diagrams must obey. Since we are concerned with cognitive properties of concrete diagrams presented to users, by (diagrammatic) syntax we mean concrete syntax throughout this paper. This is consistent with Stenning and Lemon’s (2001) cognitive view. They claimed that a concatenation is self-evident in a diagrammatic representation, and thus semantic information can be directly interpreted from a diagrammatic representation without the mediation of abstract syntax.
Euler invented his diagrams to teach Aristotelian syllogistic logic to a German princess. Indeed, the origin of such diagrams may go back still further. According to Baron’s (1969) historical review, we can find the original idea at least in the thirteenth century scholar Ramon Lull (1617). Furthermore, in Leibniz’s (1903/1988) work in the seventeenth century, there is the description of the use of diagrams to represent syllogisms although it was only much later that his work was published.
Here, “Euler” diagrams refers to diagrams based on topological relations, such as inclusion and exclusion relations, between circles. Thus, both diagrams in Gergonne’s system and those in our EUL system are instances of Euler diagrams, whereas Venn diagrams are not. In fact, the Euler diagrams currently studied in diagrammatic logic are typically based on the Existence-Free Assumption for minimal regions (EFA) rather than the type used in Gergonne’s system (cf. Stapleton 2005).
Linear diagrams for categorical syllogisms were introduced by Leibniz (1903/1988) in the seventeenth century (cf. Politzer et al. 2006). In the later eras, the linear diagrams were developed by the work of Lambert (1764). More recently, Englebretsen (1992) provided a logical system of deductive inference with linear diagrams with two-dimensional spaces. Although it is known that Englebretsen’s diagrams are unable to express complex logical inferences and hence have limited expressive power (cf. Lemon and Pratt 1998), linear diagrams are generally expressive enough to represent basic categorical syllogisms.
The EUL system can derive not only simple syllogisms with two premises but also a chain of syllogisms with more than two premises (the so-called sorites).
It is noted that this process of checking the invalidity of inferences is similar to the one known as “negation as failure” in the AI literature. One can see that the role of diagrams is to limit the possible search space of drawable conclusions.
The relevant constraint on unification, called the constraint (3), is the following: for any circles X, Y and Z, if neither inclusion or exclusion relation holds between X and Y in the combined diagram, put X and Y in such a way that X and Y are partially overlapped each other.
This classification of one/multiple model syllogisms is different from one in the mental model theory. In contrast to this study, for example, Bucciarelli and Johnson-Laird (1999) classified the AI1 syllogism having the premises All B are A and Some C are B to a single model syllogism.
We used the following translation: “Subete no A wa B de aru” for All A are B, “Dono A mo B de nai” for No A are B, “Aru A wa B de aru” for Some A are B, “Aru A wa B de nai” for Some A are not B. Here we use the quantifiers “subete” and “dono” for all, and “aru” for some. One remarkable difference between English and Japanese is in the translation of No A are B. Since in Japanese there is no negative quantifier corresponding to No, we use the translation “Dono A mo B de nai”, which literally means All A is not B. Except this point, we see no essential differences between English and Japanese. So we will refer to English translation in this paper.
They were regarded as the participants who gave up halfway and dropped out.
Here our instruction emphasized that the meaning of categorical sentences used in our experiment does not contain the existential import. Concretely, the following is given: All A are B does not imply that there are some objects which are A; thus, All A are B does not imply Some A are B. Similarly, No A are B does not imply that there are some objects which are A. Thus, No A are B does not imply Some A are not B.
The accuracy rate for those tasks which have more than one correct answer is described as “X/Y % (Z %)”, where X % represents the accuracy rate for the first correct answer, Y % the second, and Z % the rate for those who selected both correct answers.
Our result is consistent with those presented in Chapman et al. (2014), where the ease of comprehension of Linear, Euler and Venn diagrams was examined. Sato et al. (2011) also reported a similar result on the information extraction from Euler and Venn diagrams. Their empirical results suggested that the interpretation of Venn diagrams requires more substantial efforts than that of Euler diagrams.
In our Experiment, the participants in the Venn group were provided with diagrams consisting of two circles that corresponded to the premises of a given syllogism. However, we also tested a situation where participants are initially provided with Venn diagrams consisting of three circles, or “3-Venn diagrams”, namely \(D_3\) and \(D_4\) of Fig. 12. With 3-Venn diagrams, participants could skip the first steps of adding new circles; the only step needed is to superpose the two premise diagrams. Thus, it may be predicted that 3-Venn diagrams are relatively easy to manipulate in syllogism solving, even for novices. In fact, in the experiments of Sato et al. (2010), we obtained results confirming this prediction.
Further evidence that diagrammatic reasoning has features inherent to System 2 comes from the fact that it can be faster than linguistic or sentential reasoning. Several experiments examined the solution time for reasoning tasks with diagrams and reported that it was short than that for reasoning tasks without diagrams; see Bauer and Johnson-Laird (1993), Cheng (2004), Sato et al. (2015).
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Acknowledgments
Parts of this study were presented at the 6th Diagrams Conference (August, 2010) in Portland and the 7th Diagrams Conference (July, 2012) in Canterbury. This study was partially supported by Grant-in-Aid for JSPS Fellows (21\(\cdot \)7357; 25\(\cdot \)2291) to the first author. The authors would like to thank Mitsuhiro Okada, Atsushi Shimojima, Ryo Takemura, and Keith Stenning for the valuable advice and Philip Johnson-Laird and Malcolm Bauer for permission to use unpublished results.
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Appendices
Appendix 1: The Results of Each Syllogistic Type
The results of each syllogistic type are shown in Tables 5 and 6. Numbers indicate the percentage of total responses to each syllogism. Bold type refers to valid conclusion by the standard of predicate logic. For simplicity, we exclude the conclusions of the so-called “weak” syllogisms (syllogisms whose validity depends on the existential import of subject term) from valid answers.
Appendix 2: Solving Processes Using Euler Diagrams
1.1 Single Type
1.2 Multiple Type (Valid)
1.3 Multiple Type (Invalid)
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Sato, Y., Mineshima, K. How Diagrams Can Support Syllogistic Reasoning: An Experimental Study. J of Log Lang and Inf 24, 409–455 (2015). https://doi.org/10.1007/s10849-015-9225-4
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DOI: https://doi.org/10.1007/s10849-015-9225-4