General Patterns of Opposition Squares and 2n-gons

The Interaction Between Logic and Geometry in Aristotelian Diagrams

“…The idea of working up to symmetry can already be found in[4, p. 315], where it is stated that Aristotelian squares that are symmetrical variants of each other should be "counted as being of the same type". The assumed irrelevance of symmetry considerations for diagram design is also in line with work on other types of diagrams, such as Euler diagrams[27]: several of their visual characteristics have been investigated[1,3], but it has been found that rotation has no significant influence on user comprehension of Euler diagrams[2] 7. Note that both fractions have the same numerator (since the two Aristotelian diagrams have the same logical properties, viz.…”

“…One counterexample is Chow[7], who studies Aristotelian diagrams that satisfy the logical condition, but not the geometrical condition 4. In[20, p. 77] this formula is applied to a fragment of 4 formulas (so n = 2) 5.…”

Aristotelian and Duality Relations Beyond the Square of Opposition

“…The square of opposition, and almost all other Aristotelian diagrams found in the literature as well, are closed under contradiction: if the diagram contains ϕ, then it also contains ¬ϕ. 3 The propositions occurring in an Aristotelian diagram can thus naturally be grouped into pairs of contradictory propositions (PCDs). Consequently, a square of opposition should not simply be seen as consisting of 4 'individual' propositions, but rather of 2 PCDs.…”

“…The correspondence was already noted in medieval logic: influential authors such as Peter of Spain [4], William of Sherwood [29] and John Wyclif [19] discussed the mnemonic rhyme pre contradic, post contra, pre postque subalter, in which external negation (pre) is associated with contradiction, internal negation (post) with contrariety, and duality (pre postque) with subalternation. 7 Despite this close correspondence, there are still some crucial differences between Aristotelian and duality diagrams [3,47]. Regarding the individual relations, it should be pointed out that (i) the duality relations are all symmetric, whereas the Aristotelian relation SA is asymmetric, and that (ii) the duality relations are all functional, whereas the Aristotelian relations C, SC and SA are not (i.e.…”

“…Note that the definitions of contrariety and subcontrariety can both be seen as weakened versions of that of contradiction. It can also be shown that contradiction is the most informative of the Aristotelian relations[43] 3. Furthermore, the contradiction relation is usually visualized by means of central symmetry, so that all pairs of contradictory propositions are represented by diagonals that intersect each other in the Aristotelian diagram's center of symmetry[10,12,14] 4.…”