Abstract
In this paper we compare two types of diagrams for the representation of logical relations such as contradiction and contrariety, namely Logical Space diagrams (LSD) and Aristotelian diagrams (AD). The cognitive potential of Free Ride – defined in terms of tracking by consequence (Shimojima 2015) – is shown to hold for LSDs but not for ADs. The latter, however, do exhibit a greater inspection potential – defined in terms of tracking by correlation. The translational or informational equivalence between LSDs and ADs is contrasted to their lack of computational equivalence and their different degrees of iconicity.
The first author acknowledges the financial support from the Research Foundation – Flanders (FWO) of his research stay at Doshisha University with the second author. The third author holds a Postdoctoral Fellowship of the Research Foundation – Flanders (FWO) and a Research Professorship (BOFZAP) from KU Leuven.
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Notes
- 1.
A semantic convention is essentially arbitrary in its origin, but once people start conforming to it [...] it becomes a “self-perpetuating” constraint over the representational acts of a group of people [6, p. 26].
- 2.
In model-theoretic semantics, these relations receive a modal definition in terms of the (non-)existence of models/possible worlds in which both formulas are true/false.
- 3.
Although the full 2D potential of LSD diagrams is not exploited in the present analysis, subdivisions of Logical Space can be both vertical and horizontal.
- 4.
The crucial ingredients are basically the same, namely a universe set U and two subsets A and B, which yield four areas to be considered, namely \(A \cap B\), \(A \setminus B\), \(B \setminus A\) and \(U \setminus (A \cup B)\) (see also [9]). Given the definition of a proposition as a class of possible worlds, relations between classes in Euler diagrams straightforwardly correspond to relations between propositions in Aristotelian diagrams.
- 5.
In particular in terms of the tracking of the two disjunctive constraints {C, SA} \(\vdash \) {CD, C, SC, SA, Un} and {SC, SA} \(\vdash \) {CD, C, SC, SA, Un}, where Un stands for unconnectedness, the absence of any Aristotelian relation [7].
- 6.
We ignore the fact that both triangles have a mirror image along the vertical axis.
- 7.
The right and left triangle in Fig. 12(a–b) are not closed under negation, since the negations of \(\gamma \) and \(\delta \) are absent from the respective triangles.
- 8.
- 9.
An analysis in terms of over-specificity or indeterminacy [6, p. 60ff] remains a topic for further research.
- 10.
Both with the diagrams in Fig. 20 and with natural languages, we want to emphasize the bidirectionality of the transformation and translation relations, as opposed to the (basic) unidirectionality of the indication and representation relations.
- 11.
The possible connection with the so-called ‘operational’ conception of similarity and iconicity in Peirce, as elaborated by Stjernfelt [10, chapter 4] constitutes an intriguing topic of further research.
- 12.
For example, on this account, Euler diagrams are more iconic than Venn diagrams because they exhibit more constraint trackings. Similarly, the Euler system 2 is stronger and thus more iconic than system 1 since it generates more Free Rides [9].
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Smessaert, H., Shimojima, A., Demey, L. (2020). Free Rides in Logical Space Diagrams Versus Aristotelian Diagrams. In: Pietarinen, AV., Chapman, P., Bosveld-de Smet, L., Giardino, V., Corter, J., Linker, S. (eds) Diagrammatic Representation and Inference. Diagrams 2020. Lecture Notes in Computer Science(), vol 12169. Springer, Cham. https://doi.org/10.1007/978-3-030-54249-8_33
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