Aristotelian diagrams have a long and rich history in philosophical logic. Today, they are widely used in nearly all disciplines dealing with logical reasoning. Logical geometry is concerned with the theoretical study of these diagrams, from both a logical and a visual/geometrical perspective. In this paper, we argue that the concrete shape of Aristotelian diagrams can be of great heuristic value in logical geometry. A diagram's shape can be used to visually represent certain logical properties and relations, and hence, through its shape, the diagram can help us to better understand these properties and relations and reason about them. These claims are explained and illustrated by means of a series of examples, consisting mainly of (i) Aristotelian diagrams that represent an entire Boolean algebra of formulas, and (ii) complementarities between Aristotelian diagrams.
Aristotelian diagrams are compact visual representations of the elements of some
logical or conceptual field, and the logical relations holding between them. These diagrams
have a long and rich history in philosophical logic [
Logical geometry3 systematically investigates Aristotelian diagrams as objects of
independent interest (regardless of their role as lingua franca), from both a logical and a
visual/geometric perspective. Typical logical topics include the information contents and
logic-dependence of Aristotelian diagrams [
The main aim of this paper is to argue that shape plays an important heuristic role
in logical geometry. Building on earlier work [
The paper is organized as follows. In Section 2, we first introduce Aristotelian diagrams and their key properties, and then argue that a diagram’s shape can be of great heuristic value in logical geometry. In Section 3 we discuss the heuristic role of shape in Aristotelian diagrams that represent an entire Boolean algebra, and in Section 4 we discuss various shape heuristics for complementarities between Aristotelian diagrams. Finally, Section 5 wraps things up, and mentions some questions for further research.
An Aristotelian diagram visualizes a set of logical formulas and the Aristotelian relations between them (see Fig. 1 for a basic example). These relations are defined as follows (relative to some given logical system S, which is taken to have the usual Boolean connectives, and a model-theoretic semantics |=): the formulas j and y are said to be S-contradictory S-contrary S-subcontrary in S-subalternation iff iff iff iff
S |= ¬(j ^ y ) S |= ¬(j ^ y ) S 6|= ¬(j ^ y ) S |= j ! y and and and and
S |= ¬(¬j ^ ¬y ), S 6|= ¬(¬j ^ ¬y ), S |= ¬(¬j ^ ¬y ), S 6|= y ! j .
This is a straightforward formalization of the traditional perspective on the Aristotelian relations, according to which two formulas are, for example, contrary iff they cannot be true together (cf. S |= ¬(j ^ y )), but can be false together (cf. S 6|= ¬(¬j ^ ¬y )).
Furthermore, almost all Aristotelian diagrams that have appeared in the literature
impose the following additional constraints on the formulas that are visualized: these
formulas are supposed to be (i) contingent and (ii) pairwise non-equivalent, and (iii)
they come in contradictory pairs (i.e. for a given formula j , the diagram contains both
j and ¬j ). The historical and technical motivations for imposing these constraints are
discussed in more detail in [17, Subsection 2.1]. For our current purposes, it suffices to
note that although the contingency constraint means that an Aristotelian diagram should
not contain the non-contingent formulas ? and > at all, some authors [
An Aristotelian diagram trivially depends on the formulas that it visualizes:
Aristotelian diagrams that visualize distinct (sets of) formulas are themselves distinct.
Furthermore, it has recently been realized that the logical system also plays a crucial role
[
Consider, for example, the propositional formulas p ^ q, p _ q, ¬p ^ ¬q and ¬p _ ¬q,
and suppose that we are working in the system of classical propositional logic (CPL).
The Aristotelian relations holding between these formulas are fully determined (e.g. p ^ q
and ¬p ^ ¬q are CPL-contrary, etc.), and we are only left with the task of visualizing
these formulas and relations in an appropriate manner. This, however, can be done in
various ways: the most common approach is to make use of a square, as in Fig. 1(b), but
we could also make use of rectangle, a rhombus, or even move from two-dimensional to
three-dimensional shapes, and make use of a tetrahedron, as in Fig. 1(c). These various
diagrams contain the same (logical) information, and hence they can be said to be
informationally equivalent to each other [
In the next two sections, we will show that in many cases, the concrete shape of an
Aristotelian diagram can have a powerful heuristic function. Roughly speaking, the idea
is that the (sets of) formulas represented by Aristotelian diagrams have various
interesting properties and relations amongst each other, and a diagram’s shape can be used to
visualize these properties and relations, thereby enabling the user to better understand
them and reason about them. Following the work of Gurr [
6The cognitive effectiveness of diagrams has also been explained using notions that are closely related to
isomorphism, such as congruity [
These considerations also imply a partial answer to the question concerning cognitive differences between informationally equivalent diagrams (such as those in Fig. 1(b– c)). If we have Aristotelian diagrams D1 and D2 (for a given set of formulas and logical system), and the shape of D1 triggers more heuristics than that of D2, then ceteris paribus7 D1 can be said to be a better or more effective visualization (of the given formulas in the given logical system) than D2.
Our first series of case studies concerns Aristotelian diagrams that are Boolean closed, i.e. that contain all contingent Boolean combinations of their formulas. Equivalently, these diagrams can be characterized as visualizing an entire finite Boolean algebra, except for its top and bottom elements.8 It is well-known that finite Boolean algebras have 2n elements (with n 2 N), and can be represented as the powerset of an n-element set, or equivalently, as the set {0, 1}n of bitstrings of length n.9
The first interesting case arises when n = 3, which yields 23 = 8 formulas/bitstrings.
After excluding > and ? , we are thus left with 8 2 = 6 contingent formulas/bitstrings,
which stand in various Aristotelian relations to each other, and constitute a diagram
commonly known as a (strong) Jacoby-Sesmat-Blanche´ (JSB) diagram [
It is well-known that the Boolean algebra of bitstrings of length n can be
represented as an n-dimensional hypercube: every bit position corresponds to a dimension,
and the two values that it can take (0 and 1) define line segments in that dimension,
which are edges of the hypercube. The bitstrings thus not only serve as Boolean/logical
entities, but also as coordinates of the hypercube’s vertices. In the case n = 3, we thus
find a 3-dimensional hypercube, i.e. an ‘ordinary’ cube, which is shown in Fig. 3(a). We
now consider the vertex-first projection of this cube along the axis defined by the
noncontingent bitstrings 111 and 000, as shown in Fig. 3(b). The result of this projection
is a hexagon, with the bitstrings occupying the relative positions as in the hexagon in
Fig. 2(b) [
Because of this geometric fact, the hexagon is a more effective diagram than the octahedron. First of all, the hexagonal shape serves to remind the user that she is working with (an isometric projection of) a Boolean cube, and thus with the full Boolean algebra {0, 1}3, i.e. with a Boolean closed set of formulas/bitstrings. Hence, there is an immediate connection between a visual property (being a hexagon) and a logical property (being Boolean closed). By contrast, the octahedral visualization of {0, 1}3 lacks this connection, since the octahedron is not the vertex-first projection of the cube.
One might object at this point that the octahedron is related to the cube in other geometrical ways; most importantly, the octahedron is dual to the cube. However, unlike taking a vertex-first projection, the process of dualizing the cube does not have the right geometric properties. For example, by dualizing the cube we go from a polyhedron with 8 vertices (the cube) to a polyhedron with 6 vertices (the octahedron), but the geometrical reason for this reduction from 8 to 6 vertices is the (convenient but ad hoc) fact that the cube has 6 faces (recall that dualization involves turning faces into vertices and vice versa). The octahedron’s 6 vertices thus have very little to do with the cube’s 8 original vertices. All this stands in sharp contrast to the hexagon. By taking the vertex-first projection of the cube, we also go from a polyhedron with 8 vertices (the cube) to a polygon with 6 vertices (the hexagon), but in this case, the geometrical reason for this reduction from 8 to 6 vertices is the highly relevant fact that two of the cube’s vertices lie exactly on the projection axis, and hence will coincide with each other and lie in the center of the (two-dimensional) result of the projection (i.e. the hexagon). The hexagon’s 6 vertices are thus closely related to the cube’s 8 original vertices, viz. they correspond exactly to the cube’s vertices that do not lie on the projection axis.
Furthermore, recall that the projection axis is defined by the non-contingent
bitstrings 111 and 000. This justifies the intuitive idea (ascribed in Section 2 to [
Finally, note that the Boolean algebra {0, 1}3 can not only be visualized by means of
an Aristotelian diagram, but also by means of a Hasse diagram. In [
To summarize: the hexagonal JSB diagram in Fig. 2(b) embodies a strong isomorphism between visual-geometric properties (e.g. being the vertex-first projection of a cube) and abstract-logical properties (e.g. being Boolean closed). This isomorphism has significant cognitive advantages, since it enables heuristic free rides: several logical properties, which might otherwise easily be forgotten or overlooked, are visually manifested in the diagram’s shape, and thus can no longer escape the user’s attention. By contrast, the octahedral JSB diagram in Fig. 2(c) does not embody an isomorphism, and is thus less powerful from a cognitive-heuristic perspective. Hence, even though these two diagrams are informationally equivalent, they are certainly not cognitively equivalent.
Similar remarks can be made when we move to bitstrings of length 4. There are
24 2 = 14 contingent bitstrings of length 4, and there are various ways to visualize
these bitstrings and the Aristotelian relations between them. The most commonly used
diagram is the rhombic dodecahedron [
11See [
We now turn to our second series of case studies, which concerns Aristotelian diagrams that are not Boolean closed, but can be represented by bitstrings of length 4. Logically speaking, this means that the Boolean closure of these diagrams is (isomorphic to) {0, 1}4, and hence, diagrammatically speaking, these diagrams can be seen as subdiagrams of the rhombic dodecahedron for {0, 1}4 (recall from Section 3 that the rhombic dodecahedron is the best visualization of {0, 1}4).
We begin by considering the Aristotelian diagram shown in Fig. 4(a), which shows
8 propositional formulas and the Aristotelian relations holding between them. This
diagram is sometimes called a Buridan diagram, because the first Aristotelian diagram of
this type can be found in the logical works of the 14th-century philosopher John
Buridan (who used it in the context of syllogistics, rather than propositional logic) [
First of all, note that in the rhombicube, the visual property of verticality perfectly
represents the logical property of level (i.e. the number of 1-bits in the bitstring): the
level-1 bitstrings (1000 and 0001) are at the top of the diagram, the level-3 bitstrings
(1110 and 0111) are at the bottom of the diagram, and the level-2 bitstrings (1100, 1010,
0101 and 0011) all lie on a horizontal plane exactly in the middle. By contrast, in the
octagon, this visual-logical isomorphism cannot be maintained: for example, in Fig. 4(b),
1010 is situated lower than 1100, even though these two bitstrings are of the same logical
level. An obvious solution to this problem seems to involve putting all level-2 bitstrings
on a horizontal line, exactly in between the horizontal line constituted by the level-1
bitstrings and the horizontal line constituted by the level-3 bitstrings. This is problematic,
however, because the collinearity of the level-2 bitstrings entails that the Aristotelian
relations between these bitstrings can no longer be clearly visualized; for example, the line
representing the contradiction between 1010 and 0101 would overlap with the line
representing the contradiction between 1100 and 0011 (see [
Secondly, as is clear from Fig. 5(a), the rhombicube is exactly the shape that a Buridan diagram takes when it is seen as a subdiagram inside the rhombic dodecahedron for {0, 1}4. In particular, the rhombic faces of the rhombicube coincide with the rhombic faces of the rhombic dodecahedron. Through its shape, the rhombicube visualization of the Buridan diagram thus immediately suggests that this diagram can be embedded inside the rhombic dodecahedron for {0, 1}4, and hence, that its formulas can be represented by bitstrings of length 4. By contrast, the octagon in Fig. 4(b) does not in any way suggest that it can be embedded inside the rhombic dodecahedron, and thus does not remind the user that its formulas can be represented by bitstrings of length 4.
Finally, it should be noted that the 8 bitstrings constituting the Buridan diagram all have different values in their first and fourth bit positions, which we will formalize by saying that they satisfy the constraint pos1 6= pos4. Out of the 16 bitstrings in the Boolean algebra {0, 1}4, exactly 8 satisfy the constraint pos1 6= pos4; these are the ones constituting the Buridan diagram. The remaining 6 bitstrings that do not satisfy this constraint— i.e., that satisfy the complementary constraint pos1 = pos4—constitute a JSB diagram.12 There thus exists a logical complementarity between Buridan diagrams and JSB diagrams: the bitstrings constituting these two types of Aristotelian diagrams satisfy two complementary constraints on bit positions.
When a JSB diagram is embedded inside the rhombic dodecahedron, it has the shape of a hexagon, which cuts across that rhombic dodecahedron, as shown in Fig. 5(c).13 Furthermore, this hexagon perfectly cuts across the Buridan rhombicube, as shown in Fig. 5(b); together, the Buridan rhombicube and the JSB hexagon form a perfect partition of the rhombic dodecahedron, i.e. there exists a geometrical complementarity between the Buridan rhombicube and the JSB hexagon. Hence, if the Buridan diagram is visualized by means of a rhombicube, then it is geometrically complementary to (the hexago12There are actually 8, rather than 6, bitstrings that do not satisfy the constraint pos1 6= pos4, but two of them are the non-contingent bitstrings 1111 (which has a 1 in its first and fourth bit positions) and 0000 (which has a 0 in both its first and fourth bit positions), which do not occur in Aristotelian diagrams (cf. Section 2).
13Note that this is yet another reason for visualizing a JSB diagram by means of a hexagon, rather than an octahedron, besides the reasons already given in Section 3. nal visualization of) the JSB diagram, which explicitly reminds the user of the underlying logical complementarity between the two diagrams. By contrast, if the Buridan diagram is visualized by means of an octagon, then there is no geometrical complementarity with the JSB diagram, and the underlying logical complementarity runs the risk of being overlooked.
To summarize: the rhombicube-shaped Buridan diagram in Fig. 5(a) embodies a strong isomorphism between geometric properties (e.g. verticality, complementarity to a hexagon) and logical properties (e.g. level, constraints on bitstrings). This isomorphism has significant cognitive advantages, since it enables heuristic free rides: several logical properties, which might otherwise easily be forgotten or overlooked, are visually manifested in the diagram’s shape, and can thus no longer escape the user’s attention. By contrast, the octagonal Buridan diagram in Fig. 4(b) does not embody an isomorphism, and is thus less powerful from a cognitive-heuristic perspective. Hence, even though these two diagrams are informationally equivalent, they are certainly not cognitively equivalent.
In recent years, logical geometry has studied a wide variety of Aristotelian diagrams. Most of these diagrams belong to fundamentally distinct families, but there are also pairs of diagrams that contain exactly the same logical formulas and relations, and are thus informationally equivalent to each other. We have argued that in such cases, the diagrams’ shape often plays a crucial role in their cognitive-heuristic effectiveness, and can thus help us to determine whether the diagrams are also computationally equivalent to each other.
This claim has been substantiated and illustrated by two series of case studies. First of all, focusing on Boolean closed sets of formulas/bitstrings, we have argued extensively that the hexagon is a more effective visualization of {0, 1}3 than the octahedron, since it embodies a strong isomorphism between the bitstrings’ abstract-logical properties and the diagram’s visual-geometrical properties. Secondly, focusing on logical and geometrical complementarities between Aristotelian diagrams, we have argued that the Buridan diagram can more effectively be drawn as a rhombicube, rather than as an octagon.
In ongoing work [40,41], we are investigating informationally equivalent visualizations of other types of Aristotelian diagrams, such as the so-called Sherwood-Czez˙owski diagram. The principles discussed in this paper (being Boolean closed, complementarities with other diagrams) do not suffice to make cognitively relevant distinctions between these visualizations, so other, more fine-grained criteria will have to be looked for.
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