Abstract
How can Euler diagrams support non-consequence inferences? Although an inference to non-consequence, in which people are asked to judge whether no valid conclusion can be drawn from the given premises (e.g., All B are A; No C are B), is one of the two sides of logical inference, it has received remarkably little attention in research on human diagrammatic reasoning; how diagrams are really manipulated for such inferences remains unclear. We hypothesized that people naturally make these inferences by enumerating possible diagrams, based on the logical notion of self-consistency, in which every (simple) Euler diagram is true (satisfiable) in a set-theoretical interpretation. The work is divided into three parts, each exploring a particular condition or scenario. In condition 1, we asked participants to directly manipulate diagrams with size-fixed circles as they solved syllogistic tasks, with the result that more reasoners used the enumeration strategy. In condition 2, another type of size-fixed diagram was used. The diagram layout change interfered with accurate task performances and with the use of the enumeration strategy; however, the enumeration strategy was still dominant for those who could correctly perform the tasks. In condition 3, we used size-scalable diagrams (with the default size as in condition 2), which reduced the interfering effect of diagram layout and enhanced participants’ selection of the enumeration strategy. These results provide evidence that non-consequence inferences can be achieved by diagram enumeration, exploiting the self-consistency of Euler diagrams. An alternate strategy based on counter-example construction with Euler diagrams, as well as effects of diagram layout in inferential processes, are also discussed.
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Notes
Shimojima and Katagiri (2013) is a seminal study of the application of eye-tracking technique to diagrammatic reasoning research. By analyzing participants’ eye-tracking data during transitive inference using position diagrams, they showed that the updating (inference) process substantially relies on spatial constraints.
The insufficiency of results obtained from particular diagrams is related to the well-known philosophical question “Is there a true triangle?” (Kulpa 2009; Shin 2012). Leibniz Leibniz (1677/1956) clearly wrote: we must recognize that these figures [the figures of geometry] must also be regarded as characters, for the circle described on paper is not a true circle and need not be; it is enough that we take it for a circle (p. 281). The diagram written on paper (i.e., external diagram) is just a particular object. In principle, a claim constructed from the particular diagram can hold only in the particular case. Thus, there is no guarantee of the correctness of the claim in other cases. In other words, the particular diagram lacks generality. Therefore, we cannot make a general claim based on the particular diagram.
This system of Euler diagrams does not contain the conventional device of a line connecting distinct diagrams, which represents a disjunctive state of \(D_3\), \(D_4\), and \(D_5\) (cf. “Venn II” system in Shin 1994).
Here only circle C is scalable, since what the matter here is a relative size of circle C to a given circle A. Of course, we can provide the setting such that all circles are scalable. However, this setting would probably confuse reasoners in solving tasks.
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Acknowledgements
Parts of this study were presented at the 8th Diagrams Conference (July, 2014) in Melbourne and the 36th CogSci Conference (July, 2014) in Quebec. This study was supported by Grant-in-Aid for JSPS Research Fellow Grant Number JP25\(\cdot \)2291 to the first author and Grant-in-Aid for JSPS KAKENHI Grant Number JP25280049 as well as JP16H01725 to the third author. The authors would like to thank Prof. Atsushi Shimojima, Dr. Gem Stapleton, and Dr. Hidehito Honda for the valuable comments.
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Sato, Y., Wajima, Y. & Ueda, K. Strategy Analysis of Non-consequence Inference with Euler Diagrams. J of Log Lang and Inf 27, 61–77 (2018). https://doi.org/10.1007/s10849-017-9259-x
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DOI: https://doi.org/10.1007/s10849-017-9259-x