# Strategy Analysis of Non-consequence Inference with Euler Diagrams

## 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.

## Keywords

Diagrammatic reasoning Non-consequence Self-consistency Enumeration Diagram layout Cognitive science## Notes

### 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.

## References

- Aitken, S., & Melham, T. (2000). An analysis of errors in interactive proof attempts.
*Interacting with Computers*,*12*, 565–586.CrossRefGoogle Scholar - Bacon, A., Handley, S., & Newstead, S. (2003). Individual differences in strategies for syllogistic reasoning.
*Thinking and Reasoning*,*9*, 133–168.CrossRefGoogle Scholar - Barwise, J., & Etchemendy, J. (1992). Hyperproof: Logical reasoning with diagrams. In B. Chandrasekaran & H. Simon (Eds.),
*Reasoning with diagrammatic representations: Paper from the 1992 AAAI Spring Symposium, Technical Report SS-92-02*(pp. 77–81). Menlo Park, CA: AAAI Press.Google Scholar - Barwise, J., & Etchemendy, J. (1994).
*Hyperproof, CSLI lecture notes, No. 42*. Stanford, CA: CSLI Publications.Google Scholar - Barwise, J., & Shimojima, A. (1995). Surrogate reasoning.
*Cognitive Studies: Bulletin of Japanese Cognitive Science Society*,*4*, 7–27.Google Scholar - Benoy, F., & Rodgers, P. (2007). Evaluating the comprehension of Euler diagrams. In
*Proceedings of Information Visualization 2007*(pp. 771–778). Los Alamitos, CA: IEEE Computer Society.Google Scholar - Biederman, I., & Ju, G. (1988). Surface versus edge-based determinants of visual recognition.
*Cognitive Psychology*,*20*, 38–64.CrossRefGoogle Scholar - Blake, A., Stapleton, G., Rodgers, P., & Howse, J. (2016). The impact of topological and graphical choices on the perception of Euler diagrams.
*Information Sciences*,*330*, 455–482.CrossRefGoogle Scholar - Blake, A., Stapleton, G., Rodgers, P., Cheek, L., & Howse, J. (2012). Does the orientation of an Euler diagram affect user comprehension? In
*Proceedings of DMS Visual Languages and Computing 2012*(pp. 185–190). Skokie, IL: Knowledge Systems Institute.Google Scholar - Blake, A., Stapleton, G., Rodgers, P., Cheek, L., & Howse, J. (2014). The impact of shape on the perception of Euler diagrams. In
*Proceedings of Diagrams 2014, LNAI 8578*(pp. 123–137). Berlin: Springer.Google Scholar - Blanchette, J., Bulwahn, L., & Nipkow, T. (2011). Automatic proof and disproof in Isabelle/HOL. In
*Proceedings of 8th International Symposium on Frontiers of Combining Systems, LNCS 6989*(pp. 12–27). Springer.Google Scholar - Bucciarelli, M., & Johnson-Laird, P. N. (1999). Strategies in syllogistic reasoning.
*Cognitive Science*,*23*, 247–303.CrossRefGoogle Scholar - Ford, M. (1994). Two modes of mental representation and problem solution in syllogistic reasoning.
*Cognition*,*54*, 1–71.CrossRefGoogle Scholar - Gurr, C. A. (1999). Effective diagrammatic communication: Syntactic, semantic and pragmatic issues.
*Journal of Visual Languages and Computing*,*10*, 317–342.CrossRefGoogle Scholar - Gurr, C. A. (2006). Computational diagrammatics: Diagrams and structure. In D. Besnard, C. Gacek, & C. B. Jones (Eds.),
*Structure for dependability: Computer-based systems from an interdisciplinary perspective*(pp. 143–168). London: Springer.CrossRefGoogle Scholar - Gurr, C. A., Lee, J., & Stenning, K. (1998). Theories of diagrammatic reasoning: Distinguishing component problems.
*Minds and Machines*,*8*, 533–557.CrossRefGoogle Scholar - Hentschel, M., Hähnle, R., & Bubel, R. (2016). An empirical evaluation of two user interfaces of an interactive program verifier. In
*Proceedings of 31st IEEE/ACM International Conference on Automated Software Engineering*(pp. 403–413). ACM.Google Scholar - Howse, J., Molina, F., Shin, S.-J., & Taylor, J. (2002). On diagram tokens and types. In
*Proceedings of Diagrams 2002, LNAI 2317*(pp. 146–160). Berlin: Springer.Google Scholar - Kulpa, Z. (2009). Main problems of diagrammatic reasoning. part I: The generalization problem.
*Foundations of Science*,*14*, 75–96.CrossRefGoogle Scholar - Leibniz, G. (1677/1956).
*Philosophical Papers and Letters; Dialogue*. L.E. Loemker (Trans. & Ed.). Chicago, IL: University of Chicago Press.Google Scholar - Lemon, O. (2002). Comparing the efficacy of visual languages. In D. Baker-Plummer, D. I. Beaver, J. van Benthem, & P. S. di Luzio (Eds.),
*Words, proofs and diagrams*(pp. 47–69). Stanford, CA: CSLI Publications.Google Scholar - Lemon, O., & Pratt, I. (1997). Spatial logic and the complexity of diagrammatic reasoning.
*Machine Graphics and Vision*,*6*, 89–108.Google Scholar - Mineshima, K., Sato, Y., Takemura, R., & Okada, M. (2014). Towards explaining the cognitive efficacy of Euler diagrams in syllogistic reasoning: A relational perspective.
*Journal of Visual Languages and Computing*,*25*, 156–169.CrossRefGoogle Scholar - Price, C. J., & Humphreys, G. W. (1989). The effects of surface detail on object categorization and naming.
*The Quarterly Journal of Experimental Psychology*,*41*, 797–828.CrossRefGoogle Scholar - Purchase, H. C. (1997). Which aesthetic has the greatest effect on human understanding? In
*Proceedings of Graph Drawing 1997, LNCS 1353*(pp. 248–261). Berlin: Springer.Google Scholar - Rodgers, P. (2014). A survey of Euler diagrams.
*Journal of Visual Languages and Computing*,*25*, 134–155.CrossRefGoogle Scholar - Sato, Y., & Mineshima, K. (2015). How diagrams can support syllogistic reasoning: An experimental study.
*Journal of Logic, Language and Information*,*24*, 409–455.CrossRefGoogle Scholar - Sato, Y., & Mineshima, K. (2016). Human reasoning with proportional quantifiers and its support by diagrams. In
*Proceedings of Diagrams 2016, LNCS 9781*(pp. 123-138). Switzerland: Springer.Google Scholar - Sato, Y., Masuda, S., Someya, Y., Tsujii, T., & Watanabe, S. (2015). An fMRI analysis of the efficacy of Euler diagrams in logical reasoning. In
*Proceedings of 2015 IEEE Symposium on Visual Languages and Human-Centric Computing*(pp. 143–151). Los Alamitos, CA: IEEE Computer Society Press.Google Scholar - Shimojima, A. (2015).
*Semantic properties of diagrams and their cognitive potentials*. Stanford, CA: CSLI Publications.Google Scholar - Shimojima, A., & Katagiri, Y. (2013). An eye-tracking study of exploitations of spatial constraints in diagrammatic reasoning.
*Cognitive Science*,*37*, 211–254.CrossRefGoogle Scholar - Shin, S.-J. (1994).
*The logical status of diagrams*. New York: Cambridge University Press.Google Scholar - Shin, S.-J. (2012). The forgotten individual: Diagrammatic reasoning in mathematics.
*Synthese*,*186*, 149–168.CrossRefGoogle Scholar - Stenning, K., & Lemon, O. (2001). Aligning logical and psychological perspectives on diagrammatic reasoning.
*Artificial Intelligence Review*,*13*, 1–34.Google Scholar - Stenning, K., & van Lambalgen, M. (2004). A little logic goes a long way: Basing experiment on semantic theory in the cognitive science of conditional reasoning.
*Cognitive Science*,*28*, 481–529.CrossRefGoogle Scholar - Takemura, R. (2015). Counter-example construction with Euler diagrams.
*Studia Logica*,*103*, 669–696.CrossRefGoogle Scholar - Treisman, A. (1988). Features and objects: The fourteenth Bartlett memorial lecture.
*The Quarterly Journal of Experimental Psychology*,*40*, 201–237.CrossRefGoogle Scholar - Zhang, J., & Norman, D. A. (1994). Representations in distributed cognitive tasks.
*Cognitive Science*,*18*, 87–122.CrossRefGoogle Scholar