Alpha-Structures and Ladders in Logical Geometry

Abstract

Aristotelian diagrams, such as the square of opposition and other, more complex diagrams, have a long history in philosophical logic. Alpha-structures and ladders are two specific kinds of Aristotelian diagrams, which are often studied together because of their close interactions. The present paper builds upon this research line, by reformulating and investigating alpha-structures and ladders in the contemporary setting of logical geometry, a mathematically sophisticated framework for studying Aristotelian diagrams. In particular, this framework allows us to formulate well-defined functions that construct alpha-structures and ladders out of each other. In order to achieve this, we point out the crucial importance of imposing an ordering on the elements in the diagrams involved, and thus formulate all our results in terms of ordered versions of alpha-structures and ladders. These results shed interesting new light on the prospects of developing a systematic classification of Aristotelian diagrams, which is one of the main ongoing research efforts within logical geometry today.

Full text

Alexander De Klerck
Lorenz Demey
Alpha-Structures and
Ladders in Logical Geometry
Abstract. Aristotelian diagrams, such as the square of opposition and other, more com-
plex diagrams, have a long history in philosophical logic. Alpha-structures and ladders are
two specific kinds of Aristotelian diagrams, which are often studied together because of
their close interactions. The present paper builds upon this research line, by reformulat-
ing and investigating alpha-structures and ladders in the contemporary setting of logical
geometry, a mathematically sophisticated framework for studying Aristotelian diagrams.
In particular, this framework allows us to formulate well-defined functions that construct
alpha-structures and ladders out of each other. In order to achieve this, we point out the
crucial importance of imposing an ordering on the elements in the diagrams involved, and
thus formulate all our results in terms of ordered versions of alpha-structures and ladders.
These results shed interesting new light on the prospects of developing a systematic classi-
fication of Aristotelian diagrams, which is one of the main ongoing research efforts within
logical geometry today.
Keywords: Alpha-structure, Ladder, Aristotelian diagram, Square of opposition, Logical
geometry.
1. Introduction
Ever since the great mind of Aristotle walked the face of the Earth, people
have been concerned with the logical relations holding among various sets of
statements. Such constellations can be visualized using so-called Aristotelian
diagrams, which have statements as vertices and the relations holding be-
tween them as edges.1By far the most well-known example is the so-called
square of opposition for the categorical statements from syllogistics [44]. Not
only do Aristotelian diagrams have a rich history in philosophy and logic,
today they are also ubiquitous in a wide array of other application contexts,
e.g., in disciplines such as linguistics, psychology and knowledge represen-
tation (see the introduction of [14] for bibliographic references to various
historical and contemporary applications of Aristotelian diagrams).
1Note, however, that Aristotle himself never drew such a diagram [23,39].
Presented by Heinrich Wansing;Received September 21, 2023
Studia Logica
https://doi.org/10.1007/s11225-024-10142-0 c
The Author(s) 2024

A. De Klerck, L. Demey
A special class of Aristotelian diagrams that has received some inter-
est of its own is the class of α-structures, which are also called ‘logical
bi-simplexes’ or ‘n-oppositions’. They were extensively studied by Moretti,
who also coined the term ‘α-structure’ [42,43]. In other research, it has
been shown that α-structures are closely related to Euler and partition di-
agrams [16,18,56], and they occur frequently in scholarship on the logic
of Arthur Schopenhauer [12,35,36] and in ethical research on supereroga-
tion [25,27,28]. In Moretti’s work, we find not only α-structures, but also
so-called ‘β-andγ-structures’. Moretti called the latter also ‘modal n(m)-
graphs’, and used them to search for certain β-structures and to exhibit all α-
structures [43]. Subsequently, Pellissier [45] has demonstrated that Moretti’s
framework can be improved in several ways. For example, he proved that
there are two kinds of α-structures, namely strong and weak ones (while
Moretti only focused on strong α-structures). Pellissier also showed that
there is no need for general modal n(m)-graphs (i.e., γ-structures) to ex-
hibit all α-structures, but that we can reach the same goal using only the
modal 3(m)-graphs. He called the latter ‘simplicial ladder graphs’, and we
will simply call them ladders. Such ladders occur frequently in historical
scholarship on William of Sherwood [31,34] and in research on the logic of
singular statements [5,38,40] and proportional quantifiers [46–48]. Finally,
using a set-theoretic approach, Pellissier [45] figured out a way to construct
α-structures out of ladders (called ‘decorating’), which also allowed him to
find the β-structures Moretti was looking for.
Next to having various applications (e.g., Schopenhauer, supererogation,
Sherwood and proportional quantifiers), α-structures and ladders are also
important for more theoretical reasons: they are naturally associated to two
distinct perspectives on the square of opposition. Historically speaking, the
categorical square from syllogistics was viewed as exhibiting a theory of
negation as well as a theory of logical consequence. These two perspectives
are emphasized in distinct commentary traditions: the former is primarily
found in commentaries on Aristotle’s De interpretatione, while the latter
mainly occurs in commentaries on his Analytica Priora [4,54]. If we focus
on negation, the natural generalization of a square of opposition turns out to
be the notion of α-structure; by contrast, if we focus on logical consequence,
the natural generalization is the notion of ladder. Since α-structures and
ladders constitute two distinct classes of Aristotelian diagrams, it is clear
that the negation and consequence perspectives are conceptually quite dis-
tinct from each other. However, since the classical square of opposition is
simultaneously an α-diagram and a ladder, it is equally clear that this square
simultaneously exhibits the negation and consequence perspectives.

Alpha-Structures and Ladders in Logical Geometry
Over the past 15 years, Aristotelian diagrams in general have been stud-
ied in a thorough and systematic way under the heading of logical geometry
[9,14,15,53,54]. One of the main aims of this research program is to develop
a comprehensive theoretical framework for Aristotelian diagrams, which is
capable of explaining their behaviour in a mathematically satisfying way. A
major breakthrough of logical geometry in this respect is the insight that
classical Aristotelian diagrams always reside within a certain Boolean alge-
bra, and that we can therefore use bitstring semantics to investigate them
in a simple yet powerful fashion [14,53,55]. In this paper, we build upon the
work of Moretti and Pellissier, by revisiting the α-structures and ladders
from the perspective of logical geometry. In particular, we define these two
classes of Aristotelian diagrams in the general setting of Boolean algebra,
and we show how the work of Pellissier can be formalized using the tools
of logical geometry. Moreover, we describe the elegant interaction between
these classes of diagrams, by defining several functions that allow us to eas-
ily construct weak α-structures, strong α-structures and ladders in terms of
each other. Overall, the paper thus illustrates the fruitful interplay and con-
tinuity between Moretti and Pellissier’s pioneering insights (for example, the
latter’s set-theoretical approach uses a prototypical example of a Boolean al-
gebra [45]) and the systematicity and mathematical sophistication of logical
geometry.
The paper is organized as follows. In Section 2, we provide the necessary
background from logical geometry that is needed for the rest of the paper. In
Sections 3and 4, we define α-structures and ladders in the setting of logical
geometry, and we explain how weak α-structures and strong α-structures
can be constructed in terms of each other. Finally, in Section 5,werelate
the α-structures to the ladders, by defining functions that construct ladders
out of (weak/strong) α-structures, and vice versa.
2. Background from Logical Geometry
Logical geometry systematically studies Aristotelian diagrams, such as the
square of opposition and many others. An informal example of a square of
opposition, involving the categorical statements from syllogistics, is given in
Figure 1.
However, recent developments in logical geometry suggest that the most
natural (and general) setting in which to define Aristotelian diagrams is that
of Boolean algebra [7,11,22]. Following this line of research, we formally
define these diagrams as follows.

A. De Klerck, L. Demey
Figure 1. An informal square of opposition (the edges are drawn accord-
ing to the legend given in Figure 2)
Definition 1. An Aristotelian diagram Dis a pair (F,B), where Bis
a Boolean algebra (B, ∧B,∨B,¬B,1B,0B)andFis a fragment of B, i.e.,
F⊆B.2Furthermore, a σ-diagram is an Aristotelian diagram (F,B)such
that Fis closed under ¬B, i.e., for all x∈F,wehave¬Bx∈F as well.
When the Boolean algebra Bis clear from context, it is usually omitted as
a subscript to ∧,∨,etc.
To interpret the arrows and edges in the square diagram above, we need
the following four relations.
Definition 2. Given a Boolean algebra B,wesaythatx, y ∈Bare:
•B-contradictory (CDB)iffx∧By=0
Band x∨By=1
B,
•B-contrary (CB)iffx∧By=0
Band x∨By=1
B,
•B-subcontrary (SCB)iffx∧By=0
Band x∨By=1
B,
•in B-subalternation (SAB)iff¬Bx∨By=1
Band x∨B¬By=1
B.
These four relations are called the Aristotelian relations for B; we also write
ARB:= {CDB,CB,SCB,SAB}. When no confusion is possible, Bis usually
omitted as a prefix and subscript.
Even though these relations are not explicitly mentioned in Definition 1,
they will always appear in visualizations of Aristotelian diagrams. A common
way of visualizing these four relations is indicated in Figure 2.
Example 1. Let us look again at the aforementioned informal example of
an Aristotelian diagram (which is also a σ-diagram) from Figure 1.How
is this informal example captured by Definition 1? First of all, we need
a Boolean algebra to work in. For any logical system Sthat has Boolean
connectives ∧,∨and ¬, there exists a Boolean algebra B(S) whose underlying
set is B(S):={[ϕ]|ϕis a well-formed formula in S}. Here, the notation [ϕ]
stands for the equivalence class of ϕwith respect to the relation ≡Sof logical
equivalence in S. The meet ∧B(S)in this algebra is given by [ϕ]∧B(S)[ψ]:=
2Note that we tacitly identify a Boolean algebra with its underlying set, which is
common practice in the literature on Boolean algebra [22].

Alpha-Structures and Ladders in Logical Geometry
Figure 2. The code for visualizing the Aristotelian relations
[ϕ∧ψ]. The operations of ∨Band ¬Bare defined in a similar way. Finally,
we define 0B(S):= [⊥]and1
B(S):= []. It is not hard to check that all of
this gives rise to a well-defined Boolean algebra, which is usually called the
Lindenbaum-Tarski algebra of the logic S[19].
Now, consider the system of syllogistics, SYL, and its Lindenbaum-Tarski
algebra B(SYL).3We define the fragment Fcat ⊂B(SYL)as
Fcat := {[∀x(Sx →Px)],[∃x(Sx ∧Px)],[∀x(Sx →¬Px)],[∃x(Sx ∧¬Px)]}.
This fragment consists of (the SYL-equivalence classes of ) the categorical
statements from syllogistics, hence the name Fcat. It is now clear from
Definition 1that we have an Aristotelian diagram, and even a σ-diagram,
(Fcat,B(SYL)), which is visualized in Figure 3(we drop the equivalence
class brackets for notational simplicity). Note that these formal definitions
capture precisely the more well-known characterizations from the ancient
history of logic [11]. For example, we say that p:= [∀x(Sx →Px)] and
q:= [∀x(Sx →¬Px)] are B(SYL)-contrary to each other, because they
‘cannot be true together’ (i.e., p∧B(SYL)q=0
B(SYL)), but they ‘can be false
together’ (i.e., p∨B(SYL)q=1
B(SYL)).
In most Aristotelian diagrams (F,B) that are found in the extant liter-
ature, the fragment Fis indeed closed under B-complementation, so these
diagrams are σ-diagrams [7]. (The term ‘σ-diagram’ derives from the fact
3The system SYL has the same language as ordinary first-order logic (FOL), but is
axiomatized by adding ∃xSx, for all unary predicate symbols S, as additional axioms
to FOL. This logical system is naturally interpreted on first-order models D, I (with
domain Dand interpretation function I) such that I(S)=∅[14]. It is not closed under
uniform substitution (for example, ∃xSx is a tautology but ∃x(Sx ∧¬Sx) is not), just
like many of the recently developed systems of dynamic epistemic logic (for example, in
public announcement logic, [!p]pis a tautology, but [!(p∧¬Kp)](p∧¬Kp) is not [24,57]).
The system of SYL has also been called FOL ∃, and shown to be intertranslatable with
Ben-Yami’s Quantified Argument Calculus (QUARC) [1,49].

A. De Klerck, L. Demey
Figure 3. A classical square of opposition (Fcat ,B(SYL))
that ¬Bis usually visualized by means of central symmetry.) For example,
every classical square of opposition is a σ-diagram. Other interesting ex-
amples of σ-diagrams include the so-called Jacoby-Sesmat-Blanch´e(JSB)
hexagons [2,3,26,51], unconnectedness-4 (U4) hexagons [13,21,33], Buridan
octagons [10,32,50] and Keynes-Johnson octagons [17,29,30,41]; see [20]for
precise definitions and further examples.4
The Aristotelian relations are naturally part of two other collections of
logical relations [54], namely the opposition and implication relations:
Definition 3. Given a Boolean algebra B,wesaythatx, y ∈Bare:
•B-contradictory (CDB)iffx∧By=0
Band x∨By=1
B(i.e., x=¬By),
•B-contrary (CB)iffx∧By=0
Band x∨By=1
B(i.e., x<
B¬By),
•B-subcontrary (SCB)iffx∧By=0
Band x∨By=1
B(i.e., x>
B¬By),
•B-non-contradictory (NCDB)iffx∧By=0
Band x∨By=1
B,
•in B-bi-implication (BIB)iffx≤Byand x≥By(i.e., x=y),
•in B-left-implication (LIB)iffx≤Byand x≥By(i.e., x<
By),
•in B-right-implication (RIB)iffx≤Byand x≥By(i.e., x>
By),
•in B-non-implication (NIB)iffx≤Byand y≥Bx.
The first four relations are called the opposition relations for Band the
last four are called the implication relations for B. We also write ORB:=
{CDB,CB,SCB,NCDB}and IRB:= {BIB,LIB,RIB,NIB}.WedefineRB
as the set of relations on Bgiven by RB:= {CDB,CB∪SCB,BIB,LI
B∪
RIB}.5When no confusion is possible, Bis usually omitted as a prefix and
subscript.
4Next to this long list of σ-diagrams, do note that we also find quite a few (naturally
occurring) examples of Aristotelian diagrams that are not closed under complementation,
i.e., that are not σ-diagrams; for example, see [52].
5Unlike ARB,ORBand IRB, the set RBhas not been defined as such in previous
work. Its relevance for our current purposes will become clear later in this paper.

Alpha-Structures and Ladders in Logical Geometry
Note that SA from Definition 2coincides with LI from Definition 3and
therefore, the opposition and implication relations together properly extend
the Aristotelian relations. More concretely, we have that AR ⊂ OR ∪ IR.
There is a way of saying that two Aristotelian diagrams have the same
Aristotelian structure, which is made precise in Definition 4below. This
definition makes use of the relabel function ıB
B, which simply identifies every
relation in Bwith the same relation in B. For example, ıB
B(CDB)=CDB
and ıB
B(SAB)=SAB(see Definition 7 of [7] for a complete characterization
of ıB
B).
Definition 4. Let D=(F,B)andD=(F,B) be Aristotelian diagrams.
We say t h a t f:F→F
is an Aristotelian isomorphism from Dto Diff f
is bijective and for all Aristotelian relations RB∈AR
Band x, y ∈F,we
have
RB(x, y)⇐⇒ ıB
B(RB)(f(x),f(y)).
If such an fexists, we also say that Dand Dare Aristotelian isomorphic.
There is also a way of saying that two Aristotelian diagrams have the
same Boolean structure. In order to make this formally precise, we first
need to introduce the notion of a ‘Boolean closure’.
Definition 5. Let Bbe a Boolean algebra and F⊆Bbe a fragment of B.
The Boolean closure of Fin Bis the smallest subalgebra of Bthat contains
F. It is denoted by ClB(F). Furthermore, if Fis finite, then ClB(F)is
isomorphic to {0,1}nfor some natural number n[22]6; this number nis
called the Boolean complexity of Fin B.
Definition 6. Let D=(F,B)andD=(F,B) be Aristotelian diagrams.
We say t h a t f:F→F
is a Boolean isomorphism from Dto Diff f
is bijective and fextends to a Boolean algebra isomorphism ClB(F)→
ClB(F) from the Boolean closure of Fin Bto the Boolean closure of Fin
B.Ifsuchanfexists, we also say that Dand Dare Boolean isomorphic.
The notions of Aristotelian and Boolean isomorphism were first used in
a less general setting in logical geometry [9,14], and have recently also been
shown to arise naturally from a category-theoretical perspective [6,7]. Using
more classification-oriented terminology, when two Aristotelian diagrams
6This observation lies at the foundation of bitstring semantics [9,14]. Furthermore, note
that diagrams of the form ClB(F)−{0B,1B}correspond to what Moretti and Pellissier
call ‘β-structures’ [43,45]. More specifically, if ClB(F) is isomorphic to {0,1}n+1 ,then
ClB(F)−{0B,1B}is called a ‘βn-structure’ (note the offset by 1 in the notation).

A. De Klerck, L. Demey
are Aristotelian isomorphic, we say they belong to the same Aristotelian
family; when they are Boolean isomorphic, we say they belong to the same
Boolean family. In logical geometry, there has been a lot of interest in the
subtle interplay between Aristotelian and Boolean structure. This interest is
based on the observation that when two Aristotelian diagrams are Boolean
isomorphic to each other, they are also Aristotelian isomorphic to each other,
but the converse need not be true [14]. For example, the Aristotelian family
of JSB hexagons contains precisely two Boolean subfamilies, namely the
so-called strong and weak JSB hexagons [9,45]. A strong and a weak JSB
hexagon are Aristotelian isomorphic to each other (i.e., they belong to the
same Aristotelian family, viz., the family of JSB hexagons), but they are
not Boolean isomorphic to each other (i.e., they belong to different Boolean
subfamilies of the family of JSB hexagons). In recent years, several well-
known Aristotelian families have received a detailed treatment in which they
are dissected into their different Boolean subfamilies. For example, the JSB
hexagons, the U4 hexagons, the Buridan octagons and the Keynes-Johnson
octagons have been studied in this way [10,13,14,17].
Aristotelian diagrams can thus be classified according to their Aristotelian
families (and Boolean subfamilies). However, there are also exist patterns
across certain Aristotelian families. Informally, we can have a countably in-
finite series of Aristotelian families (that have increasingly large |F|) which
all satisfy the same properties. The α-structures and ladders investigated by
Moretti and Pellissier [42,43,45] are two major examples of such series.
3. Alpha-Structures
Wearenowinapositiontodefineα-structures and ladders in a fully general
and mathematically sophisticated way. We first focus our attention on α-
structures, and will turn to ladders in the next section.
Definition 7. Let n∈N0be a natural number. An αn-structure is a σ-
diagram (F,B) such that
•|F|=2n,
•0B,1B∈F and
•∃X⊆F such that |X|=nand CB(a, b) for all distinct a, b ∈X.
An ordered αn-structure is a pair (D, x) such that Dis an αn-structure and
x=(x1,...,x
n)isann-tuple consisting of pairwise contrary elements of the

Alpha-Structures and Ladders in Logical Geometry
fragment of D. Dropping reference to the specific number of elements, any
(ordered) αn-structure is more generally called an (ordered) α-structure.
Given an αn-structure, it can easily be turned into an ordered αn-structure
by fixing an order on its pairwise contrary elements, i.e., turning the un-
ordered n-element set Xinto an ordered n-tuple x. It should be clear
that such tuples xare never unique: there are n! permutations on X, i.e.,
n! different ways to order the set Xinto a tuple x. Now, suppose D=
(F,B)isanαn-structure and X={x1,...,x
n}⊆Fis its set of npair-
wise contrary elements. Since (i) Dis a σ-diagram, (ii) CDBis a func-
tion7and (iii) CDBand CBare mutually exclusive, it now follows that
F={x1,...,x
n,¬x1,...,¬xn}.8From these considerations, together with
Example 1, it can be seen that Definition 7corresponds to the notions of
logical bi-simplex [42]andn-opposition [45].
Proposition 1below shows that the α-structures really constitute an infi-
nite series of Aristotelian families, and determines their Boolean subfamilies.
(This result first occurred as Theorem 2.6 in [12], where it was stated with-
out proof.)
Proposition 1. Let n∈N0be a natural number. The family of all αn-
structures is an Aristotelian family with
•a single Boolean subfamily (with Boolean complexity n+1)ifn∈{1,2},
•two Boolean subfamilies (with Boolean complexities nand n+1)ifn≥3.
Proof. Let us first show that the αn-structures constitute an Aristotelian
family. We do this by proving that all Aristotelian relations in a given αn-
structure (F,B) with a set of pairwise contrary elements X={x1,...,x
n}
are fully determined. It is a well-known fact in logical geometry that when
0Band 1Bare not in play, two elements in a Boolean algebra Bcan be
in at most one Aristotelian relation [8]. It therefore suffices to consider the
following five cases:
•For two distinct elements xi,x
j∈X,wehavebydefinitionthatC(xi,x
j).
7A function f:X→YcanbeviewedasaspecialkindofrelationRfon X×Y
for which the following holds: for every x∈Xthere exists exactly one y∈Ysuch that
Rf(x, y). In this case, X=Y=B,Rf=CDBand y=¬Bx.
8Using terminology from [7], Fis thus the negation closure of Xin B. Note that it
is easy to show that x1,...,x
nare pairwise contrary to each other iff ¬x1,...,¬xnare
pairwise subcontrary to each other [54]. Consequently, we can equivalently characterize
αn-structures by replacing CBwith SCBin the third bulletpoint of Definition 7.

A. De Klerck, L. Demey
•For two distinct elements ¬xiand ¬xj(where xi,x
j∈X), we have that
¬xi∧¬xj=¬(xi∨xj)=¬1=0and¬xi∨¬xj=¬(xi∧xj)=¬0=1,
which means that SC(¬xi,¬xj). (Also cf. Footnote 8).
•For two elements xiand ¬xjwith i=j(where xi,x
j∈X), we have that
¬xi∨¬xj=1andxi∨¬¬xj=xi∨xj= 1, which means that LI(xi,¬xj).
•For all xi∈X, we have by definition that CD(xi,¬xi).
•An element that is not 0 or 1 does not stand in any Aristotelian relation
to itself.
These cases determine all the Aristotelian relations in all αn-structures.
Now, note that Definition 4essentially says that two Aristotelian diagrams
are Aristotelian isomorphic to each other whenever they have the same con-
figuration of Aristotelian relations. Therefore, we have now proven that, for
any fixed n,allαn-structures are Aristotelian isomorphic to each other, and
thus constitute a single Aristotelian family.
Next, we analyze the Boolean subfamilies of the Aristotelian family of
αn-structures. Firstly, if n=1,wecanwriteF={x1,¬x1}.Itisthennot
hard to check that ClB(F)={0,x
1,¬x1,1}∼
={0,1}2. Secondly, if n=2,we
can write F={x1,x
2,¬x1,¬x2}(with C(x1,x
2)). Then it is again not hard
to check that ClB(F)={0,x
1,x
2,¬x1,¬x2,x
1∨x2,¬(x1∨x2),1}∼
={0,1}3.
These observations, together with Definition 6, show that for n∈{1,2},the
αn-structures constitute a single Boolean family, with Boolean complexity
n+1.
Finally, suppose that n≥3. We first need to make an observation. Given
I,J ⊂{1,...,n}such that I=J, it follows that i∈Ixi=j∈Jxj.
Suppose, toward a contradiction, that i∈Ixi=j∈Jxjafter all. Since
I=J, we assume without loss of generality that I\J=∅, and consider
some k∈I\J. Since the elements of {x1,...,x
n}are pairwise contrary, we
have
xk=
i∈I
(xk∧xi)=xk∧
i∈I
xi=xk∧
j∈J
xj=
j∈J
(xk∧xj)=0,
which is the contradiction we were looking for. (The first equality in the chain
above holds because k∈I, while the last one holds because k/∈J.) With
this observation under our belt, we now make the following case distinction:

Alpha-Structures and Ladders in Logical Geometry
•Case 1: n
i=1 xi=1.
9Note that ClB(F)=ClB(X), since Fis itself
just the negation closure of Xin B(recall Footnote 8). We now show
that ClB(X)=i∈Ixi|I⊆{1,...,n}. Since ClB(X) is a Boolean
algebra, it must contain the latter set. To prove the other inclusion,
we show that this set is closed under the Boolean operations. Consider
J, J⊆{1,...,n}. Since the elements of {x1,...,x
n}are pairwise con-
trary, it is clear that
¬
j∈J
xj=
j∈J
¬xj=
j∈J
(¬xj∧1) = 
j∈J
(¬xj∧
n

i=1
xi)
=
j∈J
n

i=1
(¬xj∧xi)= 
j∈J
i=j
xi=
j∈JC
xj,
where JC={1,...,n}\J⊆{1,...,n}. It is also clear that

j∈J
xj∨
j∈J
xj=
j∈J∪J
xj,
with J∪J⊆{1,...,n}and that10

j∈J
xj∧
j∈J
xj=
(j,j)∈J×J
xj∧xj=
j∈J∩J
xj,
again with J∩J⊆{1,...,n}. Also recall that 1 = i∈Ixifor I=
{1,...,n},andthat0=¬1=i∈Ixifor I=∅, which proves that
0,1∈i∈Ixi|I⊆{1,...,n}. This set is therefore closed under
all Boolean operations, and thus coincides with ClB(X).Becauseofthe
observation we made above (viz., that i∈Ixi=j∈Jxjfor distinct
I,J ⊆{1,...,n}}), we thus have ClB(F)=ClB(X)=i∈Ixi|I⊆
{1,...,n}∼
=℘({1,...,n})∼
={0,1}n. Combining this with Definition 6,
we find that for n≥3, the αn-structures such that n
i=1 xi= 1 constitute
one Boolean family, which has Boolean complexity n.
9Note that for n<3, this case simply cannot obtain. In particular, for n=1,the
condition that n
i=1 xi= 1 would mean that x1= 1, which violates the second bulletpoint
of Definition 7;forn= 2, it would mean that x1∨x2= 1, which violates the third
bulletpoint of that same definition.
10Given the definability of ∧in terms of ¬and ∨, this last chain of equalities is actually
redundant. However, we have decided to include it for clarity of exposition.

A. De Klerck, L. Demey
•Case 2: n
i=1 xi= 1. Then we can define a new element xn+1 := ¬n
i=1 xi
/∈{0,1}. It clearly holds that n+1
i=1 xi= 1, so an analogous argument
as in the previous case now shows that ClB(F)=i∈Ixi|I⊆
{1,...,n+1}∼
={0,1}n+1, which again proves that we have one Boolean
subfamily of the αn-structures for which n
i=1 xi=1.Inthiscase,the
Boolean closure of Fin Bhas exactly 2n+1 elements. Combining this
with Definition 6,wefindthatforn≥3, the αn-structures such that
n
i=1 xi= 1 constitute one Boolean family, which has Boolean complex-
ity n+1.
From the proof above it is clear that the two Boolean subfamilies of
αn-structures can be distinguished by whether or not n
i=1 xi=1.Ifthis
equality holds, we say that the diagram is a strong αn-structure (which
has Boolean complexity n), otherwise we say that it is a weak αn-structure
(which has Boolean complexity n+1). We introduce the following notational
conventions.
Notation 1. We denote by Sand Wthe classes of all strong α-structures
and of all weak α-structures, respectively. A subscript nmeans we restrict
ourselves to αn-structures and a superscript omeans we restrict ourselves to
ordered α-structures. For example, we denote by Wo
nthe class of all ordered
weak αn-structures (D, x).
It is easy to see that for all n∈N0, there exist concrete examples of
αn-structures. For n≥3, both weak and strong αn-structures exist, while
for n∈{1,2},weonlyhaveweakαn-structures (recall Footnote 9).11 Fo r
n=1,2,3, the αn-structures are the pairs of contradictories (PCDs), the
classical squares of opposition and the JSB hexagons, respectively. The α4-
structures are the Moretti octagons (which Moretti himself drew as cubes
[43]). In Figure 4, we show an example of a strong Moretti octagon in the
Boolean algebra {0,1}4. In order not to overcomplicate the diagram, we
leave out all the subalternations, but it is not hard to see where they should
go. Adding a 0 after each of the lower four elements and a 1 after each of
11This observation about α-structures is a special case of a more general point: an Aris-
totelian family can have distinct Boolean subfamilies only if its diagrams contain at least
3distinct elements (and their negations, in case of σ-diagrams). Only from that cutoff
point onwards, a diagram’s Boolean properties are no longer fully captured by its Aris-
totelian relations (which are all binary in nature) [9]. Similarly, in classical propositional
logicwehavethat{p∨q, ¬p, ¬q}is inconsistent, even though all of its 2-element subsets
are consistent.

Alpha-Structures and Ladders in Logical Geometry
Figure 4. A strong α4-structure in {0,1}4
the upper four elements would yield an example of a weak Moretti octagon
in the Boolean algebra {0,1}5.
The proof of Proposition 1showsusthatforn≥2, weak αn-structures
and strong αn+1-structures have isomorphic Boolean closures: if (F,B)∈
Wnand (F,B)∈Sn+1,thenClB(F)∼
={0,1}n+1 ∼
=ClB(F). This sug-
gests that we can create strong αn+1-structures out of weak αn-structures,
and vice versa. In the following theorems, we define functions that formal-
ize this insight. Before we turn to these results, we introduce some further
handy notational conventions.
Notation 2. When we need to denote an arbitrary n-tuple, we will use xor
y, which are implicitly given by x=(x1,...,x
n)and y=(y1,...,y
n).Ifxis
an (ordered) n-tuple, then we denote by |x|the (unordered) set {x1,...,x
n}.
Notation 3. Let F⊆Bbe a fragment of a Boolean algebra. Then we
denote by ¬BFthe fragment {¬Bb∈B|b∈F} that contains all negations
of elements in F. When no confusion is possible, we omit Bas a subscript.
Theorem 1. Let n≥2, then we have a well-defined function Addn:Wo
n→
So
n+1, which for any D=(F,B)is given by:
Addn(D, x):=(Addn(D),Add
n(x)),
with (n+ 1)-tuple Addn(x):=(x1,...,x
n,¬n
i=1 xi)and strong αn+1-
structure Addn(D):=(|Addn(x)|∪¬|Addn(x)|,B).12
12Note that this involves a slight abuse of notation, as we write Addnfor three distinct
functions, each with their own domain (viz., ordered weak αn-structures, unordered weak
αn-structures and n-tuples) and their own codomain. We trust that it is clear from the
context which function is being used where. Completely analogous remarks apply to many
of the theorems that follow.

A. De Klerck, L. Demey
Proof. It is trivial that ¬n
i=1 xiand ¬¬ n
i=1 xi=n
i=1 xiare distinct
from each other and from all elements of the fragment of D, so that the
fragment of Addn(D)has2n+2 = 2(n+ 1) elements. Since Dis a weak
α-structure, these two new elements are not equal to 0 or 1. We now prove
that C(xk,¬n
i=1 xi) for any 1 ≤k≤n:
xk∧¬
n

i=1
xi=xk∧
n

i=1
¬xi
=xk∧¬x1∧···∧¬xk∧···∧¬xn
=0,
xk∨¬
n

i=1
xi=xk∨
n

i=1
¬xi
=
n

i=1
(xk∨¬xi)
≤xk∨¬x
=¬x
=1,
where is any number in {1,...,n}\{k}—note that since n≥2, the set
{1,...,n}\{k}is non-empty, so such an certainly exists. (The penultimate
identity in the chain above holds because C(xk,x
).) This already shows
that Addn(D, x)isanαn+1-structure. Moreover, we trivially have that
x1∨···∨xn∨¬
n

i=1
xi=1,
so Addn(D, x)isastrong αn+1-structure, i.e., Addn(D, x)∈So
n+1.
Theorem 2. Let n≥2. Then we have a well-defined function Dropn:
So
n+1 →Wo
n, which for any D=(F,B)is given by:
Dropn(D, x):=(Dropn(D),Drop
n(x)),
with n-tuple Dropn(x):=(x1,...,x
n)and weak αn-structure Dropn(D):=
(|Dropn(x)|∪¬|Dropn(x)|,B).13
13For the sake of clarity, we emphasize that since (D, x)∈So
n+1, the tuple xis an
(n+ 1)-tuple, so that Dropn(x)=Dropn(x1,...,x
n,x
n+1)=(x1,...,x
n)isitselfan
n-tuple.

Alpha-Structures and Ladders in Logical Geometry
Proof. Note that since n≥2, there exist strong αn+1-structures, so the
domain of Dropn, i.e., So
n+1, is non-empty. It is immediate by construction
that Dropn(D)isanαn-structure and that Dropn(x) is an appropriate n-
tuple of pairwise contrary elements. Now suppose toward a contradiction
that n
i=1 xi= 1. Since Dis an αn+1 -structure, we have C(xn+1,x
i) for all
1≤i≤n, and hence
xn+1 =xn+1 ∧1=xn+1 ∧
n

i=1
xi
=
n

i=1
(xn+1 ∧xi)
=
n

i=1
0=0,
which is the contradiction we were looking for. This shows that n
i=1 xi=1,
so Dropn(D, x)isaweak αn-structure, i.e., Dropn(D, x)∈Wo
n.
We now have well-defined functions Addnand Dropnat our disposal,
which allow us to construct strong αn+1-structures out of weak αn-structures,
and vice versa. Furthermore, it is easy to show that each of these functions
undoes the effect of the other one, i.e., they are each other’s inverses.
Theorem 3. Let n≥2.ThenDropn◦Addnis the identity function on Wo
n
and Addn◦Dropnis the identity function on So
n+1.
Proof. The first statement follows immediately because for any n-tuple
x,wehave(Dropn◦Addn)(x)=Dropn(Addn(x)) = Dropn((x1,...,x
n,
¬n
i=1 xi)) = x. For the second statement, note that for any (n+ 1)-tuple
x,wehavethat
Addn(Dropn(x)) = Addn((x1,...,x
n)) = (x1,...,x
n,¬
n

i=1
xi).
To prove that (Addn◦Dropn)(x)=Addn(Dropn(x)) = x=(x1,...,x
n,x
n+1),
it thus suffices to prove that ¬n
i=1 xi=xn+1. Since the tuple xcomes from
a strong αn+1-structure, we have that xn+1 ∨n
i=1 xi=n+1
i=1 xi= 1. Also,
following the same reasoning as in the proof of Theorem 2,wehavethat
xn+1 ∧n
i=1 xi= 0. This proves the desired statement.
It might seem somewhat unsatisfactory that the functions Addnand
Dropnonly concern ordered α-structures, since in practice, we primarily
want to construct weak/strong α-structures out of each other, without hav-
ing to take into consideration the specific ordering on their pairwise contrary

A. De Klerck, L. Demey
elements. However, this is simply not possible to do in a canonical way, by
the following argument.
As Proposition 1shows, weak αn-structures and strong αn+1-structures
both have Boolean complexity n+ 1. Thus, their Boolean closures are iso-
morphic to the Boolean algebra {0,1}n+1. It is not hard to show that within
this Boolean algebra, there is only one strong αn+1-structure, whose set of
pairwise contrary elements we will call X. However, there are exactly n+1
weak αn-structures within {0,1}n+1 , whose sets of pairwise contrary ele-
ments we will call X1,...,X
n+1, respectively. Since X=Xi∪{¬Xi}for
any 1 ≤i≤n+ 1, we can go canonically from Wnto Sn+1 . On the other
hand, to go from Xto any Xi, we need to remove one element of X. Because
there is no canonical way of choosing which element to remove, we cannot go
canonically from Sn+1 to Wn. The most reasonable solution to this problem
is to order X.
The best we can hope for is thus canonicity on the level of ordered di-
agrams, which is provided by Addnand Dropn.Tobeabletoconstruct
(unordered) weak/strong αn-structures out of each other, we compose these
canonical functions with the functions Choose (which ‘essentially captures
all non-canonicity’) and Forget (which is, again, canonical). The situation
is summarized in Figure 5. Note that for any possible function Choose :
Wn→Wo
n, the composition Forget ◦Addn◦C hoose :Wn→Sn+1 is one
and the same function, as expected, which we can denote by Add∗
n.Onthe
other hand, if we start from any possible function Choose :Sn+1 →So
n+1,
given any D∈Sn+1, there are n+ 1 possible outcomes (Forget ◦Dropn◦
Choose)(D), namely one for each element of the set of pairwise contrary
elements of Dthat gets put in the last place of its corresponding tuple by
Choose (and thus subsequently gets deleted by Dropn). We refer to these
n+ 1 options as Drop∗1
n(D),Drop
∗2
n(D),...,Drop
∗n
n(D),Drop
∗n+1
n(D).14
We then have that Add∗
n(Drop∗i
n(D)) = Dfor all i, but it is not true that
Drop∗i
n(Add∗
n(D)) = Dfor all i, as is shown by the following examples.
Example 2. Consider the (unordered) strong Moretti octagon Ds:=
(Fs,{0,1}4)∈S4,withFs:= {1000,0100,0010,0001,0111,1011,1101,1110}.
Let us first choose as an ordered 4-tuple of pairwise contrary elements the
14This notation suggests that we have n+ 1 different functions of the form Drop∗i
n:
Sn+1 →Wn. However, the expression ‘Drop∗i
n(D)’ denotes a specific weak αn-structure
related to D, but cannot be decomposed into a function Drop∗i
nthat gets applied to an
argument D. More specifically, it is not possible to define such functions Drop∗i
non all of
Sn+1 at once. The only thing that comes close is to combine them into a single function
from Sn+1 to ℘(Wn), but this is not the approach we take here.

Alpha-Structures and Ladders in Logical Geometry
Figure 5. Going back and forth between weak and strong (ordered) α-
structures. (Dashed lines indicate non-unique processes.)
tuple xs:= (1000,0100,0010,0001), thus obtaining (Ds,x
s)∈So
4.Both
Dsand (Ds,x
s) are shown on the right-hand side of Figure6(the spe-
cific ordering on xsis not visualized as such). Since (Ds,x
s)∈So
4,we
can apply the function Drop3to it, which yields the ordered weak JSB
hexagon Drop3(Ds,x
s)=(Dw,x
w)∈Wo
3,withxw:= (1000,0100,0010),
Dw:= (Fw,{0,1}4)andFw:= {1000,0100,0010,0111,1011,1101}. By for-
getting about xw, we then arrive at the (unordered) weak JSB hexagon
Dw∈W3.BothDwand (Dw,x
w) are shown on the left-hand side of Fig-
ure 6(the specific ordering on xwis not visualized as such). Likewise, we can
go from Dwback to Ds: start from Dw∈W3, choose the tuple xwto obtain
(Dw,x
w)∈Wo
3, apply Add3to obtain Add3(Dw,x
w)=(Ds,x
s)∈So
4,and
finally, forget about xsto obtain Ds∈S4. This example is an instantiation
of the fact that Add∗
3(Drop∗i
3(Ds)) = Dsfor all 1 ≤i≤4.
Example 3. Consider the classical square of opposition D∈W2with set of
pairwise contraries X:= {100,010}. Then Add∗
2(D)∈S3is uniquely defined
to be the strong JSB hexagon with set of pairwise contraries {100,010,001}.
Next, we consider the three distinct classical squares of opposition
Drop∗1
2(Add∗
2(D)), Drop∗2
2(Add∗
2(D)) and Drop∗3
2(Add∗
2(D)), with sets of
pairwise contraries X1:= {100,010},X2:= {100,001}and X3:= {010,001},
respectively. Only the first one of these squares is identical to the original
classical square D. This example shows that Add∗
2(Drop∗i
2(D)) = Ddoes
not hold true for all 1 ≤i≤3.
4. Ladders
Now, we turn our attention to ladders.
Definition 8. Let n∈N0be a natural number. An n-ladder is a σ-diagram
(F,B) such that
•|F|=2n,
•0B,1B∈F and

A. De Klerck, L. Demey
Figure 6. An instantiation of the fact that Addnand Dropnare each
other’s inverses for all n≥2
•∃y=(y1,...,y
n)∈F
nsuch that LIB(yi,y
i+1) for all 1 ≤i≤n−1.
An ordered n-ladder is a pair (D, y) such that Dis an n-ladder and yis
an n-tuple as described above. Dropping reference to the specific number
of elements, any (ordered) n-ladder is more generally called an (ordered)
ladder.
Given a ladder, it can easily be turned into an ordered ladder by fixing
a specific tuple y. It should be clear that such tuples are never unique: if
(y1,...,y
n) meets the requirements stipulated in Definition 8,then
(¬yn,...,¬y1) does so as well. Now, suppose D=(F,B)isann-ladder
and (y1,...,y
n)∈F
nis a tuple from Definition 8with subalternations
holding between its elements. Since (i) Dis a σ-diagram, (ii) CDBis a func-
tion and (iii) CDBand LIBare mutually exclusive when 0Band 1Bare
not in play, it now follows that F={y1,...,y
n,¬y1,...,¬yn}, i.e., Fis
the negation closure of |y|={y1,...,y
n}in B. From these considerations,
together with Example 1, it can be seen that Definition 8corresponds to the
notions of modal 3(m)-graph [43] and simplicial ladder graph [45]. The fol-
lowing proposition shows that the ladders really constitute an infinite series
of Aristotelian families.
Proposition 2. Let n∈N0be a natural number. The family of all n-
ladders is an Aristotelian family with a single Boolean subfamily (with Boolean
complexity n+1).
Proof. Letusfirstshowthatthen-ladders constitute an Aristotelian fam-
ily. Again, we do this by proving that all Aristotelian relations in a given n-
ladder (F,B), with a tuple of elements y=(y1,...,y
n) such that LI(yi,y
i+1)
for all 1 ≤i≤n−1, are fixed. It is well-known in logical geometry that

Alpha-Structures and Ladders in Logical Geometry
when 0 and 1 are not in play, two elements in a Boolean algebra can be in
at most one Aristotelian relation [8]. We thus distinguish six cases:
•For two elements yi,y
j∈|y|such that i<j,wehavethatLI(yi,y
j).
•For two elements ¬yiand ¬yj(where yi,y
j∈|y|) such that i<j,we
have that LI(¬yj,¬yi).
•For two elements yiand ¬yj(where yi,y
j∈|y|) such that i<j,wehave
that yi∧¬yj=¬(¬yi∨yj)=¬1=0andyi∨¬yj= 1, which means that
C(yi,¬yj).
•For two elements yiand ¬yj(where yi,y
j∈|y|) such that i>j,wehave
that yi∧¬yj=¬(¬yi∨yj)=¬1=0andyi∨¬yj= 1, which means that
SC(yi,¬yj).
•For two elements yiand ¬yj(where yi,y
j∈|y|) such that i=j,wehave
that CD(yi,¬yj).
•An element that is not 0 or 1 does not stand in any Aristotelian relation
to itself.
These cases determine all the Aristotelian relations in all n-ladders. It now
follows from Definition 4that all n-ladders are Aristotelian isomorphic to
each other, and thus constitute a single Aristotelian family.
Next, we analyze the Boolean subfamilies of the Aristotelian family of
n-ladders. First of all, note that ClB(F)=ClB(|y|), since Fis itself just
the negation closure of |y|in B. Next, define the (n+1)-tuple x:= (y1,¬y1∧
y2,...,¬yn−1∧yn,¬yn)15; we will show that ClB(|y|)=ClB(|x|). The ⊇-
direction clearly holds by construction of x.Forthe⊆-direction, it suffices to
see that yi=i
j=1 xjfor all 1 ≤i≤n. Finally, we claim that (|x|∪¬|x|,B)
is a strong αn+1-structure (this claim is proved below). It then follows from
the proof of Proposition 1that ClB(F)=ClB(|y|)=ClB(|x|)∼
={0,1}n+1.
Combining this with Definition 6, we find that the n-ladders constitute one
Boolean family, which has Boolean complexity n+1.
We now prove our claim that (|x|∪¬|x|,B) is a strong αn+1-structure.
We first check that any two elements a, b ∈|x|are contrary to each other.
We distinguish between different cases depending on the form of aand b:
•a=y1and b=¬yn:Inthiscase,LI(y1,y
n) implies that C(a, b).
•a=y1and bis of the form ¬yi∧yi+1:Wehavethata∧b=(y1∧
¬yi)∧yi+1 = 0, where the last equality follows because either yi=y1
15See Example 4for a concrete illustration of this construction.

A. De Klerck, L. Demey
or LI(y1,y
i). We also have that a∨b=(y1∨¬yi)∧(y1∨yi+1)≤(y1∨
¬y1)∧(y1∨yi+1)=y1∨yi+1 =yi+1 = 1. Here, the second step holds
because either yi=y1or LI(¬yi,¬y1), while the final equality holds since
LI(y1,y
i+1). We have now proven that C(a, b).
•a=¬ynand bis of the form ¬yi∧yi+1: Similar to the previous case.
•ais of the form ¬yi∧yi+1 and bis of the form ¬yj∧yj+1: Without loss of
generality, we assume that i<j.Nowa∧b=¬yi∧(yi+1 ∧¬yj)∧yj+1 =0
since either yi+1 =yjor LI(yi+1,y
j). Also, a∨b=(¬yi∧yi+1)∨(¬yj∧
yj+1)≤yi+1 ∨yj+1 =yj+1 = 1. Here, the penultimate step holds because
i<jand thus LI(yi+1,y
j+1). We have now proven that C(a, b).
Since the n+ 1 elements of |x|are clearly distinct from each other, from
their negations, and from 0 and 1, we have shown that (|x|∪¬|x|,B)isan
αn+1-structure. To see that it is a strong one, recall that yn=n
j=1 xjand
note that
n+1

j=1
xj=
n

j=1
xj∨xn+1 =yn∨¬yn=1.
Again, we introduce a notational convention.
Notation 4. We denote by Lthe class of all ladders. A subscript nmeans
we restrict ourselves to n-ladders and a superscript omeans we restrict
ourselves to ordered ladders. For example, we denote by Lo
nthe class of all
ordered n-ladders.
Just like in the case of α-structures, it might seem artificial to distin-
guish between Lnand Lo
n, since we are primarily concerned with ladders as
such, without taking into consideration any specific tuple of consecutively
LI elements. However, in the next section, we will construct weak/strong
α-structures and ladders out of each other, and this can only be done (in
a functional way) at the level of ordered diagrams. We therefore take the
same approach here as in the previous section, and go back and forth be-
tween ladders and ordered ladders by simply choosing or forgetting a tuple.
Once again, forgetting a tuple is canonical (i.e., there is a unique function
(D, y)→ D), but choosing a tuple is not: there are two distinct ways to
choose a tuple,16 i.e., to define a function D→ (D, y).
Much more importantly, it can be shown that for all n∈N0, there exist
concrete examples of n-ladders. For n∈{1,2},then-ladders are simply
16As stated before, if one tuple is (y1,...,y
n), the other one is (¬yn,...,¬y1).

Alpha-Structures and Ladders in Logical Geometry
Figure 7. A 4-ladder in {0,1}5
the PCDs and the classical squares of opposition, respectively, as is sum-
marized in Theorem 4below. It follows from this theorem, together with
Propositions 1and 2, that the classical square of opposition is the largest
Aristotelian diagram that is simultaneously an α-structure and a ladder.17
Theorem 4. For n∈{1,2},theαn-structures coincide with the n-ladders.
Proof. It follows trivially from the definitions that the Aristotelian fam-
ilies of α1-structures and of 1-ladders both coincide exactly with the fam-
ily of PCDs, and thus with each other. It is also not hard to check that
the Aristotelian families of α2-structures and 2-ladders both coincide ex-
actly with the family of classical squares of opposition, and thus with each
other.
Moving on, 3-ladders and 4-ladders are sometimes referred to in the
literature as resp. ‘Sherwood-Cze˙zowski hexagons’ and ‘Lenzen octagons’
[5,20,31,34,37]. For example, Figure 7shows a 4-ladder in the Boolean alge-
bra {0,1}5. In order not to overcomplicate the diagram, we leave out some
of the subalternations on the far left and far right, but it is not hard to see
where they should go.
Figure 7clearly illustrates the intuition behind the terminology ‘ladder’,
since we can use the subalternations to climb up in the diagram, rung by
rung.18 However, there is another way in which we could draw such a dia-
gram, which will make it clearer why ladder diagrams are fruitfully studied
17In [54] it is suggested that considerations like these (albeit without using the specific
terminology of ‘α-structure’ and ‘ladder’) might help to explain the widespread popularity
of the classical square (especially in contrast with other, larger diagrams).
18If n=2kis even (like in Figure 7), then an n-ladder has kcontrariety rungs and k
subcontrariety rungs. If n=2k+ 1 is odd, then an n-ladder has kcontrariety rungs, 1
contradiction rung and ksubcontrariety rungs.

A. De Klerck, L. Demey
Figure 8. The same 4-ladder as in Figure 7, drawn in a different way
side by side with α-structures, like we do in this paper. Sticking with our
example, the same ladder diagram could also be drawn as in Figure 8.This
time, we leave out all of the (sub)contarieties in order not to overcomplicate
the diagram. Comparing the 4-ladder from Figure 8to the α4-structure from
Figure 4, we observe a lot of similarities. Informally, the roles of contrariety
and subcontrariety in the α4-structure correspond in some way to the role
of subalternation in the 4-ladder. Put differently, the tuple (x1,...x
n) from
Definition 7and the tuple (y1,...,y
n) from Definition 8correspond in some
way to each other. This insight is made more precise in Propositions 3and 4
below.
Proposition 3. Let n∈N0be a natural number, and consider a σ-diagram
(F,B)such that |F| =2nand 0B,1B/∈F. Then the following are equiva-
lent:
1. ∃y=(y1,...,y
n)∈F
nsuch that LIB(yi,y
i+1)for all 1≤i≤n−1,
2. ∃Y⊆F such that |Y|=nand such that LIB(a, b)or LIB(b, a)for all
distinct a, b ∈Y,
3. ∃Y⊆Fsuch that |Y|=nand (a, b)∈LIB∪RIBfor all distinct a, b ∈Y.
Proof. The third item is merely a slight reformulation of the second one,
so we focus on proving that the first two items are equivalent. We first prove
1⇒2. Suppose there exists an n-tuple y=(y1,...,y
n) as described in item
1. Then it is easy to check that the set Y:= |y|satisfies the requirements of
item 2. Finally, 2 ⇒1 follows from the well-known result that every finite
total order is well-ordered (in particular, it is easy to check that (Y, ≤B∩Y2)
is a finite total order).
Proposition 4. Let (F,B)be an αn-structure with set Xof pairwise B-
contrary elements of F.Let(F,B)be an n-ladder with a tuple yof elements
of Fsuch that LIB(yi,y
i+1)for all 1≤i≤n−1.Letf:F→F
be

Alpha-Structures and Ladders in Logical Geometry
any negation-preserving bijection such that f[X]=|y|. Then there exists a
bijection F:RB→R
B(recall Definition 3) such that for all z, z∈F
and for all RB∈R
B, we have that RB(z,z)⇐⇒ F(RB)(f(z),f(z)).
Moreover, this bijection Fdoes not depend on the concrete bijection f.
Proof. It is easy to check that Fcan be defined by F(CDB):=CDB,
F(BIB):=BIB,F(CB∪SCB):=LIB∪RIB,andF(LIB∪RIB):=
CB∪SCB.
Proposition 3provides an alternative characterization of ladder diagrams,
which coincides exactly with Definition 7of α-structures, except for the fact
that CBis replaced with LIB∪RIB.19 Continuing along these lines, Propo-
sition 4formalizes the comparison between αn-structures and n-ladders
we sketched above.20 The correspondence between the relations (given by
F) and the correspondence between the sets |x|and |y|(given by f)to-
gether form the requirement that RB(z,z)⇐⇒ F(RB)(f(z),f(z)) for all
z,z∈F and all RB∈R
B. Although these two propositions nicely high-
light the similarities between both classes of Aristotelian families, it does
not seem to have much practical use. It would be more interesting to have
a way to construct ladders out of α-structures and vice versa. This is what
we turn to in the next section.
5. Constructing α-Structures and Ladders
Propositions 1and 2show that αn-structures and n-ladders diverge from
each other as soon as n≥3. These propositions also tell us that n-ladders,
weak αn-structures and strong αn+1 -structures all have Boolean complexity
n+ 1, i.e., they all have Boolean closures that are isomorphic to {0,1}n+1,
and thus to each other. Therefore, when given one of these three kinds of
Aristotelian diagrams, it should be possible to construct both other kinds of
diagrams out of it, using only the Boolean operators. In Theorems 1and 2
we already showed that it is indeed possible to create strong αn+1-structures
19Also recall Footnote 8on the alternative characterization of α-structures that is
obtained upon replacing CBwith SCBin Definition 7.
20This comparison can also be expressed as an analogical proportion: α-structures are
to the opposition relations (esp. C,SC ∈OR) like ladders are to the implications relations
(esp. LI,RI ∈IR). (This is a slight oversimplification, as the opposition relations CD,
NCD and the implication relations BI,NI occur across α-structures and ladders alike.)
Finally, recall from Section 1that α-structures and ladders are naturally associated with
resp. the negation (∼opposition) and consequence (∼implication) perspectives on the
categorical square of opposition.

A. De Klerck, L. Demey
Figure 9. The blueprint for Boolean complexity n+1
out of weak αn-structures, and vice versa. In the remainder of this section,
we will extend this story by incorporating the n-ladders. In other words, we
are going to define the dashed arrows in the central triangle of Figure9,for
every possible value of n.
5.1. From Ladders to α-Structures
In this subsection we investigate how we can create α-structures out of lad-
ders, using nothing but the latter’s elements and the Boolean operators. We
first work out a simple example, and then move on to the general construc-
tions. The proof of Proposition 2already gives us some ideas.
Example 4. Suppose we have a 2-ladder ({y1,y
2,¬y1,¬y2},B)withtuple
(y1,y
2). Since LI(y1,y
2), we have that C(y1,¬y2). Now, the identities
y1∧(¬y1∧y2)=(y1∧¬y1)∧y2=0
and
y1∨(¬y1∧y2)=(y1∨¬y1)∧(y1∨y2)=y1∨y2=y2=1
show that C(y1,¬y1∧y2). Similar identities show that also C(¬y2,¬y1∧y2).
Since we also have that ¬y1∧y2=¬(y1∨¬y2)=¬1=0and¬y1∧y2≤y2<
1, we find that ({y1,¬y1∧y2,¬y2,¬y1,y
1∨¬y2,y
2},B)isanα3-structure,
with set of pairwise contraries {y1,¬y1∧y2,¬y2}. This α-structure is a
strong one, since y1∨(¬y1∧y2)∨¬y2= 1. The situation is summarized by
Figure 10.

Alpha-Structures and Ladders in Logical Geometry
Figure 10. Going from a 2-ladder to a strong α3-structure
It bears emphasizing that, modulo the extreme elements 0 and 1, this
strong α3-structure is closed under the Boolean operators, i.e., for all z1,z
2∈
H:= {y1,¬y1∧y2,¬y2,¬y1,y
1∨¬y2,y
2},wehavethatz1∨z2,z
1∧z2,¬z1∈
H∪{0,1}.21 Furthermore, this α3-structure is the smallest structure that
has this property, while containing {y1,y
2,¬y1,¬y2}. Using standard termi-
nology from logical geometry, we say that the strong JSB hexagon on the
right of Figure 10 is the Boolean closure of the classical square of opposition
on its left [2,14].
The previous example and the proof of Proposition 2suggest that, given
a ladder with tuple (y1,...,y
n), we should consider elements of the form
¬yi∧yi+1 to create a set of pairwise contrary elements. In general, we have
the following theorem.
Theorem 5. Let n≥2. Then we have a well-defined function αs
n:Lo
n→
So
n+1, which for any D=(F,B)is given by:
αs
n(D, y):=(αs
n(D),α
s
n(y)),
with (n+1)-tuple αs
n(y):=(y1,¬y1∧y2,...,¬yn−1∧yn,¬yn)and strong
αn+1-structure αs
n(D):=(|αs
n(y)|∪¬|αs
n(y)|,B).
Proof. Entirely analogous to part of the proof of Proposition 2.
21It is easy to check that (weak) α1-structures, i.e., PCDs of the form ({x, ¬x},B),
are also Boolean closed in this sense. Apart from (weak) α1- and strong α3-structures, no
other α-structures are Boolean closed. In particular, for n≥2, a weak αn-structure is not
Boolean closed, since it omits, for example, the join of its npairwise contrary elements
(0 =n
i=1 xi=1).Forn≥4, a strong αn-structure is not Boolean closed either, since it
omits, for example, each binary join of its npairwise contrary elements (0 =xi∨xj=1,
for 1 ≤i=j≤n).

A. De Klerck, L. Demey
So far, we only have a way of constructing strong α-structures out of
ladders. However, Theorem 2gives us a way of going from strong to weak
α-structures. This suggests the following theorem.
Theorem 6. Let n≥1. Then we have a well-defined function αw
n:Lo
n→
Wo
n, which for any D=(F,B)is given by:
αw
n(D, y):=(αw
n(D),α
w
n(y)),
with n-tuple αw
n(y):=(y1,¬y1∧y2,...,¬yn−1∧yn)and weak αn-structure
αw
n(D):=(|αw
n(y)|∪¬|αw
n(y)|,B).
Proof. The proof is analogous to part of the proof of Proposition 2, but
now the final disjunct, ¬yn, is missing in the chain of equalities at the very
end of that proof. This shows that
y1∨(¬y1∧y2)∨(¬y2∧y3)∨···∨(¬yn−1∧yn)=yn=1,
which implies that αw
n(D)isaweakαn-structure.22
5.2. From α-Structures to Ladders
The final thing left to do is to find ways to create ladders out of α-structures.
Of course, it would be nice if these arrows were in some sense the inverses of
αs
nand αw
n, just like Addnand Dropnare each other’s inverses (recall The-
orem 3). Let us first look at an example which, again, draws its inspiration
from Proposition 2.
Example 5. Suppose we have a 3-ladder D=({y1,y
2,y
3,¬y1,¬y2,¬y3},B)
with tuple y=(y1,y
2,y
3). Applying αs
3to (D, y) yields an ordered strong
α4-structure that has x:= (y1,¬y1∧y2,¬y2∧y3,¬y3)asitstupleofpair-
wise contrary elements. So how can we retrieve the original ladder from this
constructed α-structure? We can retrieve y1by simply taking the first ele-
ment from x, i.e., y1. We can retrieve y2by taking the join of the first two
elements from x, i.e., y1and ¬y1∧y2. Finally, we can retrieve y3in two
straightforward ways: one is by negating the final element from x, i.e., ¬y3,
and the other is by taking the join of the first three elements from x, i.e.,
22Note that in case n= 1, the function αw
1maps a PCD (viewed as a 1-ladder) onto
itself (but now viewed as a weak α1-structure). In case n=2,αw
2maps a classical square
of opposition (viewed as a 2-ladder) onto a different, but still Aristotelian isomorphic
diagram, viz., onto a different classical square of opposition (but now viewed as a weak
α2-structure). From n≥3 onwards, αw
nstarts producing non-isomorphic diagrams; for
example, αw
3maps an SC hexagon onto a weak JSB hexagon, αw
4maps a Lenzen octagon
onto a weak Moretti octagon, etc.

Alpha-Structures and Ladders in Logical Geometry
y1,¬y1∧y2and ¬y2∧y3. Even though the first option is simpler, the latter
one is more natural in the sense that it extends the way in which we retrieve
the other elements y1and y2.
The previous example suggests the following way of constructing n-ladders
out of strong αn+1-structures.
Theorem 7. Let n≥2. Then we have a well-defined function λs
n:So
n+1 →
Lo
n, which for any D=(F,B)is given by:
λs
n(D, x):=(λs
n(D),λ
s
n(x)),
with n-tuple λs
n(x):=(x1,x
1∨x2,...,n
i=1 xi)and n-ladder λs
n(D):=
(|λs
n(x)|∪¬|λs
n(x)|,B).
Proof. We have the following identities for all 1 ≤j≤n−1thatprove
that there are subalternations between two consecutive elements in λs
n(x):
j

i=1
xi∧¬
j+1

i=1
xi=j

i=1
xi∧j+1

i=1
¬xi
=
j

i=1 xi∧
j+1

k=1
¬xk
=
j

i=1
0=0,
and
¬j

i=1
xi∧j+1

i=1
xi=j

i=1
¬xi∧j+1

i=1
xi
=
j+1

i=1 xi∧
j

k=1
¬xk
=
j+1

i=1
j

k=1
(xi∧¬xk)
=
j

k=1
(xj+1 ∧¬xk)=
j

k=1
xj+1 =xj+1 =0.
In the above, the penultimate identity holds because for all 1 ≤k≤j,
we have C(xk,x
j+1), which implies LI(xj+1,¬xk). Since the elements from
λs
n(x) are clearly distinct from each other, from their negations, and from 0
and 1, we have also shown that λs
n(D)isann-ladder.

A. De Klerck, L. Demey
We now show that λs
nand αs
nare each other’s inverses, as desired.
Theorem 8. Let n≥2.Thenλs
n◦αs
nis the identity function on Lo
nand
αs
n◦λs
nis the identity function on So
n+1.
Proof. For a ny n-tuple y,weseethat(λs
n◦αs
n)(y)=λs
n(αs
n(y))= λs
n((y1,¬y1
∧y2,...,¬yn−1∧yn,¬yn)) = (y1,y
1∨(¬y1∧y2),...,y
1∨n−1
i=1 (¬yi∧yi+1)).
We need to show that this tuple is equal to y. Now, notice that we have the
following identities for all 2 ≤j≤n:
y1∨
j−1

i=1
(¬yi∧yi+1)=y1∨(¬y1∧y2)∨
j−1

i=2
(¬yi∧yi+1)
=(y1∨¬y1)∧(y1∨y2)∨
j−1

i=2
(¬yi∧yi+1)
=(y1∨y2)∨
j−1

i=2
(¬yi∧yi+1)
.
.
.
=y1∨y2∨···∨yj
=yj.
The final equality holds since LI(yk,y
j) for all 1 ≤k≤j−1. This proves
the first part of the theorem.
For the second part, note that for any (n+ 1)-tuple x,wehavethat
(αs
n◦λs
n)(x)=αs
n(λs
n(x)) = αs
n((x1,x
1∨x2,...,n
i=1 xi)) = (x1,¬x1∧(x1∨
x2),...,¬n−1
i=1 xi∧n
i=1 xi,¬n
i=1 xi). We need to show that this
tuple is equal to x. Now notice that we have the following identities for all
1≤j≤n−1:
¬j

i=1
xi∧j+1

i=1
xi=j

i=1
¬xi∧j+1

i=1
xi
=
j+1

i=1 xi∧
j

k=1
¬xk
=xj+1 ∧
j

k=1
¬xk
=xj+1.

Alpha-Structures and Ladders in Logical Geometry
The final equality holds since C(xk,x
j+1) implies that LI(xj+1 ,¬xk), for
all 1 ≤k≤j. The above chains of equalities already prove that (αs
n◦
λs
n)(x)=(x1,...,x
n,¬n
i=1 xi). Since the tuple xcomes from a strong
αn+1-structure, the exact same line of reasoning that appears in the proof
of Theorem 3can be used here to prove that xn+1 =¬n
i=1 xi, and hence,
(αs
n◦λs
n)(x)=x.
So far, we only have a way of constructing ladders out of strong α-
structures. However, Theorem 1gives us a way of going from weak to strong
α-structures. This suggests the following theorem.
Theorem 9. Let n≥1. Then we have a well-defined function λw
n:Wo
n→
Lo
n, which for any D=(F,B)is given by:
λw
n(D, x):=(λw
n(D),λ
w
n(x)),
with n-tuple λw
n(x):=(x1,x
1∨x2,...,n
i=1 xi)and n-ladder λw
n(D):=
(|λw
n(x)|∪¬|λw
n(x)|,B).
Proof. Entirely analogous to the proof of Theorem 7.23
We now show that λw
nand αw
n, too, are each other’s inverses, as desired.
Theorem 10. Let n≥1.Thenλw
n◦αw
nis the identity function on Lo
nand
αw
n◦λw
nis the identity function on Wo
n.
Proof. Analogous to the proof of Theorem 8.
5.3. Putting Everything Together
We now have all the necessary ingredients at our disposal to prove the
following satisfying theorem.
Theorem 11. Let n≥2. Then we have canonical ways to construct weak
αn-structures, strong αn+1-structures and n-ladders out of each other, which
are all compatible with each other. More formally, we have the commutative
diagram in Figure 11.
Proof. It is immediately clear by construction that αw
n=Dropn◦αs
n.
Together with Theorems 3,8and 10, this is enough to ensure commutativity
of the entire diagram.
23Fo r low val u es of n, entirely analogous remarks apply to λw
nas to αw
n;recallFoot-
note 22.

A. De Klerck, L. Demey
Figure 11. The commutative triangle for Boolean complexity n+1≥3
Figure 12. The complete situation for Boolean complexity n+1≥3
It bears repeating that the weak αn-structures, strong αn+1-structures
and n-ladders in this diagram all have the same Boolean complexity, n+1.
For the sake of completeness, we draw the entire picture that covers both
ordered and unordered diagrams in Figure 12. This figure shows canonical
ways (up to choice of tuples) of constructing weak/strong α-structures and
ladders out of each other, which was the main purpose of this paper.
Let us investigate the triangle in Figure11 in some more detail, for dif-
ferent values of n. From n≥3 onwards, the classes Lo
n,Wo
nand So
n+1 are
all non-empty and pairwise distinct. For n= 2, Theorem 4tells us that
the classes Wo
2and Lo
2coincide with each other (both comprise the classical
squares of opposition); however, it bears emphasizing that αw
2and λw
2are
not identity functions (cf. Footnotes 22 and 23). Finally, for n=1,sucha
triangle technically does not exist since, by the proof of Proposition 1,So
2

Alpha-Structures and Ladders in Logical Geometry
is empty (cf. Footnote 9). Therefore, the functions Add1,Drop1,αs
1and λs
1
do not exist either. However, we still have Wo
1and Lo
1, and by Theorem 4,
they even coincide (both comprise the PCDs). In this case, αw
1and λw
1are
simply identity functions (again cf. Footnotes 22 and 23). It is not hard to
link all these triangles together, using the functions fnfrom the following
theorem.
Theorem 12. Let n≥3. Then we have a well-defined function fn:Wo
n→
So
n, which for any D=(F,B)is given by:
fn(D, x):=(fn(D),f
n(x)),
with n-tuple fn(x):=(x1,x
2,...,x
n−1,¬n−1
i=1 xi)and αn-structure fn(D):=
(|fn(x)|∪¬|fn(x)|,B).
Proof. Very similar to the proof of Theorem 1.
Note that in the spirit of this theorem, it would also be possible to treat
the case n= 2 in exactly the same way, except for the fact that the range
of f2is Wo
1instead of So
2. This leads to the chain of triangles in Figure 13.
For the sake of completeness, we have also included the ‘reduced’ triangle,
which includes Wo
1and Lo
1but lacks So
2,asthelatterisempty.Aboveeach
triangle, we have also written the Boolean complexity (BC) shared by all
diagrams mentioned in that triangle.24 Finally, we could of course also have
included all the classes of unordered diagrams, by adding Choose and Forget
arrows everywhere (as we did in Figure 12). However, this would only have
harmed the simplicity of the figure, so we leave these arrows out on purpose.
Our treatment of α-structures and ladders using the functions from The-
orem 11 closely resembles the set-theoretic approach of Pellissier [45]. Given
an n-ladder in some appropriate logic, Pellissier implicitly calculates a set-
theoretic Boolean algebra that is isomorphic to the Boolean closure of the
n-ladder inside the logic’s Lindenbaum-Tarski algebra. He then proceeds to
24Given the tight connection between a diagram’s Boolean complexity and its Boolean
closure (cf. Definition 5), the upper sequence of Boolean complexities in Figure 13 can
equivalently be viewed as a sequence of Boolean closures (or yet equivalently, as a se-
quence of β-structures; cf. Footnote 6). In particular, for n≥1, every ordered diagram
((F,B),x)∈Lo
n∪Wo
n∪So
n+1 has Boolean complexity n+1, so its Boolean closure ClB(F)
is isomorphic to {0,1}n+1 (or equivalently, ClB(F)−{0B,1B}is a βn-structure). It bears
emphasizing, however, that the notion of Boolean closure is far more general than those of
α-structure and ladder. Indeed, we can take the Boolean closure ClB(F) of any Aristotelian
diagram (F,B) whatsoever, and recent category-theoretical work on logical geometry has
shown that this operation of taking the Boolean closure constitutes a (reflective) functor
[6].

A. De Klerck, L. Demey
Figure 13. The chain of triangles
search for all possible α-structures inside this algebra. There is only a single
strong αn+1-structure, which can also be constructed using αs
n. To find all n
weak αn-structures, it is not enough to just use αw
n. Instead, we should first
apply αs
n, then allow for a change of the tuple x(i.e., Forget the old tuple
and Choose a new one), and finally apply Dropn. Anyway, it is clear that
Pellissier’s intuitions were enough to reach the appropriate conclusions, but
that they can be formulated and proven more generally and systematically
in the framework of logical geometry.
6. Conclusion
In this paper we combined previous work in n-opposition theory on α-
structures and ladders with the contemporary research line of logical geome-
try, which provides a systematic mathematical framework for studying Aris-
totelian diagrams. Using this approach, we were able to define α-structures
and ladders in a general way, using Boolean algebra. This setting then al-
lowed us to classify these diagrams into infinite series of Aristotelian families
and Boolean subfamilies, viz., Ln,W
nand Snfor n∈N0. Finally, we showed
how to move back and forth between these various series, by defining math-
ematically elegant ways to construct weak/strong α-structures and ladders
out of each other. Some of these constructions naturally generalize earlier
results by Moretti and Pellissier [43,45].
Acknowledgements. This research was funded by the research project ‘BIT-
SHARE: Bitstring Semantics for Human and Artificial Reasoning’ (IDN-
19-009, Internal Funds KU Leuven) and the ERC Starting Grant ‘START-
DIALOG: Towards a Systematic Theory of Aristotelian Diagrams in Log-
ical Geometry’. Funded by the European Union (ERC, STARTDIALOG,
101040049). Views and opinions expressed are however those of the author(s)

Alpha-Structures and Ladders in Logical Geometry
only and do not necessarily reflect those of the European Union or the Euro-
pean Research Council Executive Agency. Neither the European Union nor
the granting authority can be held responsible for them. The second author
holds a research professorship (BOFZAP) at KU Leuven. We would like to
thank Atahan Erbas, Stef Frijters, Hans Smessaert and two anonymous re-
viewers for extensive discussions and feedback on an earlier version of this
paper.
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References
[1] Ben-Yami, H., The quantified argument calculus, Review of Symbolic Logic 7:120–146,
2014.
[2] Blanch´
e, R., Sur l’opposition des concepts, Theoria 19:89–130, 1953.
[3] Blanch´
e,R.,Structures Intellectuelles, Vrin, Paris, 1966.
[4] Correia, M., Boethius on the square of opposition, in J.-Y. B´eziau and D. Jacquette,
(eds.), Around and Beyond the Square of Opposition, Springer, Basel, 2012, pp. 41–52.
[5] Cze˙
zowski, T., On certain peculiarities of singular propositions, Mind 64:392–395,
1955.
[6] De Klerck,A.,L.Vignero, and L. Demey,The category of Aristotelian diagrams
with Boolean morphisms, under review, 2023.
[7] De Klerck,A.,L.Vignero, and L. Demey, Morphisms between Aristotelian dia-
grams, Logica Universalis, forthcoming.

A. De Klerck, L. Demey
[8] Demey, L., Structures of oppositions for public announcement logic, in J.-Y. B´eziau
and D. Jacquette, (eds.), Around and Beyond the Square of Opposition, Springer,
Basel, 2012, pp. 313–339.
[9] Demey, L., Computing the maximal Boolean complexity of families of Aristotelian
diagrams, Journal of Logic and Computation 28:1323–1339, 2018.
[10] Demey, L., Boolean considerations on John Buridan’s octagons of opposition, History
and Philosophy of Logic 40:116–134, 2019.
[11] Demey, L., Metalogic, metalanguage and logical geometry, Logique et Analyse
248:453–478, 2019.
[12] Demey, L., From Euler diagrams in Schopenhauer to Aristotelian diagrams in logical
geometry, in J. Lemanski, (ed.), Language, Logic, and Mathematics in Schopenhauer,
Springer, Cham, 2020, pp. 181–205.
[13] Demey, L., and A. Erbas, Boolean subtypes of the U4 hexagon of opposition, Axioms
13:1–20, 2024.
[14] Demey, L., and H. Smessaert, Combinatorial bitstring semantics for arbitrary logical
fragments, Journal of Philosophical Logic 47:325–363, 2018.
[15] Demey, L., and H. Smessaert, Geometric and cognitive differences between logical
diagrams for the Boolean algebra B4,Annals of Mathematics and Artificial Intelligence
83(2):185-208, 2018.
[16] Demey, L., and H. Smessaert, From Euler diagrams to Aristotelian diagrams, in V.
Giardino, S. Linker, R. Burns, F. Bellucci, J.-M. Boucheix, and P. Viana, (eds.), Dia-
grammatic Representation and Inference, Lecture Notes in Computer Science 13462,
Springer, Cham, 2022, pp. 279–295.
[17] Demey, L., and H. Smessaert, Aristotelian and Boolean properties of the Keynes-
Johnson octagon of opposition, Journal of Philosophical Logic forthcoming.
[18] Dubois, D., and H. Prade,FromBlanch´e’s hexagonal organization of concepts to
formal concept analysis and possibility theory, Logica Universalis 6:149–169, 2012.
[19] Dunn,J.M.,andG.M.Hardegree,Algebraic Methods in Philosophical Logic, Ox-
ford University Press Oxford, 2001.
[20] Frijters, S., and L. Demey, The modal logic of Aristotelian diagrams, Axioms
12(5):1–26, 2023.
[21] Furs, S. N., Computation of Aristotle’s and Gergonne’s syllogisms, Studia Logica
46:209–225, 1987.
[22] Givant, S., and P. Halmos,Introduction to Boolean Algebras, Springer, New York,
NY, 2009.
[23] Gombocz, W. L., Apuleius is better still: a correction to the square of opposition,
Phronesis 43:124–131, 1990.
[24] Holliday,W.H.,T.Hoshi,andT.F.Icard III, Information dynamics and uniform
substitution, Synthese 190:31–55, 2013.
[25] Hruschka, J., and J. Joerden, Supererogation: Von deontologischen Sechseck zum
deontologischen Zehneck, Archiv f¨ur Rechts- und Sozialphilosophie 73:93–123, 1987.
[26] Jacoby, P., A triangle of opposites for types of propositions in Aristotelian logic, New
Scholasticism 24:32–56, 1950.

Alpha-Structures and Ladders in Logical Geometry
[27] Joerden, J., On the logic of supererogation, Jahrbuch f¨ur Recht und Ethik 6:145–159,
1998.
[28] Joerden, J., Deontological square, hexagon, and decagon: A deontic framework for
supererogation, Logica Universalis 6:201–216, 2012.
[29] Johnson, W.E., Logic. Part I., Cambridge University Press, Cambridge, 1921.
[30] Keynes,J.N.,Studies and Exercises in Formal Logic (Third Edition), MacMillan,
London, 1894.
[31] Khomskii, Y., William of Sherwood, singular propositions and the hexagon of oppo-
sition, in J.-Y. B´eziau and G. Payette, (eds.), The Square of Opposition. A General
Framework for Cognition, Peter Lang, Bern, 2012, pp. 43–60.
[32] Klima, G., (ed.), John Buridan, Summulae de Dialectica, Yale University Press, New
Haven, CT, 2001.
[33] Kraszewski, Z., Logika stosunk´ow zakresowych, Studia Logica 4:63–116, 1956.
[34] Kretzmann,N.,William of Sherwood’s Introduction to Logic, Minnesota Archive
Editions, Minneapolis, MN, 1966.
[35] Lemanski, J., Schopenhauers Logikdiagramme in den Mathematiklehrb¨uchern
Adolph Diesterwegs, Siegener Beitr¨age zur Geschichte und Philosophie der Mathe-
matik 16:101–131, 2022.
[36] Lemanski, J., and L. Demey, Schopenhauer’s partition diagrams and logical geom-
etry, in A. Basu, G. Stapleton, S. Linker, C. Legg, E. Manalo, and P. Viana, (eds.),
Diagrammatic Representation and Inference, Springer, Cham, 2021, pp. 149–165.
[37] Lenzen, W., How to square knowledge and belief, in J.-Y. B´eziau and D. Jacquette,
(eds.), Around and Beyond the Square of Opposition, Springer, Basel, 2012, pp. 305–
311.
[38] Lilje, G. W., Singular statements, Teaching Philosophy 10:219–225, 1987.
[39] Londey, D., and C. Johanson, Apuleius and the square of opposition, Phronesis
29:165–173, 1984.
[40] Mackie, J. L., “This” as a singular quantifier, Mind 67(268):522–526, 1958.
[41] Moktefi, A., and F. Schang, Another side of categorical propositions: The Keynes-
Johnson octagon of oppositions, History and Philosophy of Logic forthcoming, 1–17.
[42] Moretti, A., Geometry of modalities? Yes: Through n-opposition theory, in J.-Y.
B´eziau, A. Costa-Leite, and A. Facchini, (eds.), Aspects of Universal Logic.University
of Neuchˆatel, Neuchˆatel, 2004.
[43] Moretti,A.,The Geometry of Logical Opposition, PhD thesis, Universit´ede
Neuchˆatel, 2009.
[44] Parsons, T., The traditional square of opposition, in E. N. Zalta, (ed.), Stanford
Encyclopedia of Philosophy (Summer 2017 Edition). Stanford, CA, CSLI, 2017.
[45] Pellissier, R., “Setting” n-opposition, Logica Universalis 2:235–263, 2008.
[46] Peterson, P. L., Complexly fractionated syllogistic quantifiers, Journal of Philosoph-
ical Logic 20:287–313, 1991.
[47] Peterson, P. L., Logical negation in English? Logique et Analyse 36(141–142):121–
134, 1993.
[48] Peterson, P. L., Distribution and proportion, Journal of Philosophical Logic 24:193–
225, 1995.

A. De Klerck, L. Demey
[49] Raab,J.,The relationship of QUARC and classical logic, Master’s thesis, Ludwig-
Maximilians-Universit¨at M¨unchen, 2016.
[50] Read, S., John Buridan’s theory of consequence and his octagons of opposition, in
J.-Y. B´eziau and D. Jacquette, (eds.), Around and Beyond the Square of Opposition,
Springer, Basel, 2012, pp. 93–110.
[51] Sesmat,A.,Logique II. Les Raisonnements. La syllogistique, Hermann, Paris, 1951.
[52] Seuren,P.A.M.,andD.Jaspers, Logico-cognitive structure in the lexicon, Lan gu age
90(3):607–643, 2014.
[53] Smessaert, H., On the 3D visualisation of logical relations, Logica Universalis 3:303–
332, 2009.
[54] Smessaert, H., and L. Demey, Logical geometries and information in the square of
oppositions, Journal of Logic, Language and Information 23:527–565, 2014.
[55] Smessaert, H., and L. Demey, The unreasonable effectiveness of bitstrings in logical
geometry, in J.-Y. B´eziau and G. Basti, (eds.), The Square of Opposition: A Corner-
stone of Thought, Springer, Cham, 2017, pp. 197–214.
[56] Smessaert,H.,A.Shimojima, and L. Demey, Free rides in logical space diagrams
versus Aristotelian diagrams, in A.-V. Pietarinen, P. Chapman, L. Bosveld-de Smet,
V. Giardino, J. E. Corter, and S. Linker, (eds.), Diagra mm at ic Re pre sentation and
Inference, vol. 12169 of Lecture Notes in Computer Science, Springer, Cham, 2020,
pp. 419–435.
[57] van Ditmarsch,H.P.,W.van der Hoek,andB.P.Kooi,Dynamic Epistemic
Logic, Springer, Dordrecht, 2007.
A. De Klerck,L. Demey
KU Leuven
Leuven
Belgium
lorenz.demey@kuleuven.be