Aspect Shifting in Aristotelian Diagrams

Aristotelian diagrams represent logical relations of opposition and implication between formulas or concepts. In this paper we investigate the cognitive mechanism of Aspect Shifting in order to describe various families of Aristotelian diagrams. Aspect shifting occurs when an ‘ambiguous’ visual representation triggers a perceptual change from one perspective or interpretation to another. In a first part, we consider aspect shifting which takes place on the level of Aristotelian subdiagrams and which switches focus precisely between the oppositional and the implicational perspective. In a second part, aspect shifting is involved in focussing on the ways in which smaller (but complete) Aristotelian diagrams—in particular, Aristotelian squares—are embedded inside bigger diagrams—in particular Aristotelian hexagons. In both parts, special attention is paid to the iterative nature of the aspect shifting.

1. 1.

   Thus \(T(2) = 1+2=3\), \(T(3) = 1+2+3=6\), \(T(4) = 1+2+3+4=10\), and so on.

2. 2.

   In the theory of Shimojima [12, p. 154], aspect shifting involves a layered consequence tracking relation with constraints between two decomposition types of one figure.

3. 3.

   In this case \(\varPi \) = {\(\alpha _1, \alpha _2, \alpha _3, \alpha _4\)} = {\(\Box p\), \(\lnot \Box p \wedge p\), \(\Diamond p \wedge \lnot p\), \(\lnot \Diamond p\)} and for every formula \(\varphi \in F_1\), its bitstring \(\beta (\varphi )\) = \(\beta _1\beta _2\beta _3\beta _4\) is such that \(\beta _n = 1\) iff \(\mathsf {S5}\models \alpha _n \rightarrow \varphi \).

4. 4.

   The hour-glass and bow-tie AsDs have been studied in great detail in our Diagrams 2020 paper [15] in terms of Shimojima’s cognitive potential of Free Rides [12].

5. 5.

   In our Diagrams 2021 paper [16] identifying such triangular shapes is analysed in terms of Shimojima’s cognitive potential of Derivative Meaning [12].

6. 6.

   See [16] for an analysis of these triangular shapes in terms of Derivative Meaning [12]. The ‘undirected’ triangles for the symmetric relations CR/SCR in Fig. 5(b) crucially differ from ‘directed’ triangles for the asymmetric SA relations in Fig. 3(c).

7. 7.

   Notice that it is perfectly possible in theory to highlight other subparts for aspect shifting, as long as the crucial property of closure under negation is observed.

8. 8.

   They are problematic for the Apprehension Principle by which the structure/content of the visualisation should be readily/accurately perceived or comprehended [17].

9. 9.

   Differences between ‘minimal’ SC and ‘radical’ JSB rotations directly carry over.

Authors and Affiliations

1. Department of Linguistics, KU Leuven, Blijde-Inkomststraat 21, 3000, Leuven, Belgium

   Hans Smessaert

2. Center for Logic and Philosophy of Science, KU Leuven, Kardinaal Mercierplein 2, 3000, Leuven, Belgium

   Lorenz Demey

Corresponding author

Correspondence to Hans Smessaert .

Editors and Affiliations

1. CNRS, Ecole Normale Supérieure, Paris, France

   Valeria Giardino

2. Lancaster University in Leipzig, Leipzig, Germany

   Sven Linker

3. West Chester University, West Chester, PA, USA

   Richard Burns

4. Universitá di Bologna, Bologna, Italy

   Francesco Bellucci

5. Université Bourgogne Franche-Comté, Dijon, France

   Jean-Michel Boucheix

6. Universidade Federal Fluminense, Niterói, Brazil

   Petrucio Viana

Reprints and permissions

© 2022 Springer Nature Switzerland AG

Policies and ethics