Analogical Proportions and Square
of Oppositions
Laurent Miclet1 and Henri Prade2
1
2
University of Rennes 1 – Irisa, Lannion, France
CNRS/IRIT – University of Toulouse, Toulouse, France
[email protected], [email protected]
Abstract. The paper discusses analogical proportions in relation with
the square of oppositions, a classical structure in Ancient logic which is
related to the different forms of statements that may be involved in deductive syllogisms. The paper starts with a short reminder on the logical
modeling of analogical proportions, viewed here as Boolean expressions
expressing similarities and possibly differences between four items, as in
the statement “a is to b as c is to d”. The square of oppositions and its
hexagon-based extension is then restated in a knowledge representation
perspective. It is observed that the four vertices of a square of oppositions
form a constrained type of analogical proportion that emphasizes differences. In fact, the different patterns making an analogical proportion true
can be covered by a square of oppositions or by a “square of agreement”,
leading to disjunctive expressions of the analogical proportion. Besides,
an “analogical octagon” is shown to capture the general construction of
an analogical proportion from two sets of properties. Since the square of
oppositions offers a common setting relevant for syllogisms and analogical proportions, it also provides a basis for the discussion of the possible
interplay between deductive arguments and analogical arguments.
1
Introduction
Deductive reasoning and analogical reasoning are two ways of drawing inferences.
Both were already clearly identified in the Antiquity. Syllogistic reasoning was
then the basis for deduction. Related to syllogisms is the square of oppositions
which dates back to the same time. In its original form, the square of oppositions
was associated with universal and existential statements and their negations,
which corresponds to the different statements encountered in syllogisms.
While deductive reasoning may involve both universal and existential statements, analogical reasoning only considers instantiated statements. A basic figure
of analogical reasoning is analogical proportion, i.e. a comparative statement of
the form “a is to b as c is to d”. A logical view of analogical proportion has been
recently proposed [6]. Then the analogical proportion reads “a differs from b as
c differs from d and b differs from a as d differs from c”, which involves negation
in the logical writing. This fact, together with the quaternary nature of such
statements, leads us to establish a connection between the analogical proportion
and the square of oppositions, whose meaning is investigated in the following.
A. Laurent et al. (Eds.): IPMU 2014, Part II, CCIS 443, pp. 324–334, 2014.
c Springer International Publishing Switzerland 2014
Analogical Proportions and Square of Oppositions
325
The paper first briefly recalls the logical view of the analogical proportion in
Section 2, and its relevance to analogical reasoning. Then the square of oppositions and its hexagonal extension to a triple of similar squares is restated in
Section 3, pointing out its interest in the analysis of the relations between categorical statements. Section 4 shows the relevance of the square of oppositions for
discussing analogical proportions as well, while Section 5 discusses the interplay
of arguments based on structures of opposition, before concluding in Section 6.
2
The Propositional View of the Analogical Proportion
An analogical proportion “a is to b as c is to d”, denoted a : b :: c : d, is
supposed to satisfy two characteristic properties: i) a : b :: c : d is equivalent
to c : d :: a : b (symmetry), and ii) a : b :: c : d is equivalent to a : c :: b : d
(central permutation). There are in fact 8 equivalent forms obtained by applying
symmetry and permutation. Moreover, a : b :: a : b always holds (reflexivity).
Viewing a, b, c and d as Boolean variables, and a : b :: c : d as a quaternary
connective forces the analogical proportion to be true for the following 6 patterns
0 : 1 :: 0 : 1, 1 : 0 :: 1 : 0, 1 : 1 :: 0 : 0, 0 : 0 :: 1 : 1, 1 : 1 :: 1 : 1, 0 : 0 :: 0 : 0.
A logical expression, which is true only for these 6 patterns (among 24 = 16
possible entries) has been proposed in [6]:
a : b :: c : d = ((a ∧ ¬b) ≡ (c ∧ ¬d)) ∧ ((¬a ∧ b) ≡ (¬c ∧ d)).
This remarkable expression makes clear that a differs from b as c differs from
d and, conversely, b differs from a as d differs from c. This clearly covers the
patterns 0 : 1 :: 0 : 1, 1 : 0 :: 1 : 0, as well as the 4 remaining patterns where
a and b are identical on the one hand and c and d are also identical on the
other hand. It can be easily checked that under this logical view the analogical
proportion indeed satisfies symmetry, central permutation, and reflexivity. It
is also transitive. Moreover a : b :: ¬b : ¬a also holds. Such a logical view
of an analogical proportion can be advocated as being the genuine symbolic
counterpart of the notion of numerical proportion [10,11].
This view easily extends to Boolean vectors a, b, c, d, whose components
can be thought as binary attribute values, each vector describing a particular
situation. Let xi be a component of a vector x. Then an analogical proportion
a : b :: c : d between such vectors can be defined componentwise, i.e., for each
component i, the analogical proportion ai : bi :: ci : di holds. Then the differences
between a and b pertain to the same attribute(s) as the difference between c
and d and are oriented in the same way (e.g., if from ai to bi , one goes from 1
to 0, it is the same from ci to di ). When there is no difference between ai and
bi , then ci and di are as well identical. This modeling of analogical proportions
has been successfully applied to learning [5] and to the solving of IQ tests [8].
This vector-based view can be easily related to a set-based view originally
proposed in [4], later proved to be equivalent (see [6] for details). In the setbased view each situation is associated with the set of attributes that are true
in it. Let us denote A, B, C, and D the sets thus associated with a, b, c, and
d, respectively. Then we shall write A : B :: C : D if a : b :: c : d holds.
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L. Miclet and H. Prade
Note that in particular, for any pair of sets A and B, A : B :: B : A holds (where
A denotes the set complement of A).
The case of attributes on discrete domains with more than 2 values can be
handled as easily as the binary case. Indeed, consider a finite attribute domain
{v1 , · · · , vm }. This attribute can be binarized by means of the m properties
“having value vi , or not”. Consider the partial description of objects a, b, c,
and d with respect to this attribute. Assume, for instance, that objects a and
c have value v1 , while objects b and d have value v2 . Then it can be checked
that an analogical proportion holds true between the four objects for each of
the m binary property, and in the example, can be more compactly encoded as
an analogical proportion between the attribute values themselves, namely here:
v1 : v2 :: v1 : v2 . More generally, x and y denoting possible values of a considered
attribute, the analogical proportion between objects a, b, c, and d holds for this
attribute iff the 4-tuple of their values wrt this attribute is equal to a 4-tuple
having one of the three forms (s, s, s, s), (s, t, s, t), or (s, s, t, t).
3
The Square of Oppositions
Let us start with a refresher on the classical square of opposition [7]. This square
involves four logically related statements exhibiting universal or existential quantifications: it has been noticed that a statement (A) of the form “every x is p” is
negated by the statement (O) “some x is not p”, while a statement like (E) “no
x is p” is clearly in even stronger opposition to the first statement (A). These
three statements, together with the negation of the last one, namely (I) “some
x is p”, give birth to the Aristotelian square of opposition in terms of quantifiers
A : ∀x p(x), E : ∀x ¬p(x), I : ∃x p(x), O : ∃x ¬p(x), pictured in Figure 1 (where
it is assumed that there are some x in order that the square makes sense).
∀x p(x)
Contraries
A
ad
tr
n
o
C
∃x p(x)
I
s
ie
or
t
ic
s
rie
to
ic
ad
tr
on
C
Subalterns
E
Sub-contraries
∀x ¬p(x)
Subalterns
O
∃x ¬p(x)
Fig. 1. Square of opposition
Such a square is usually denoted by the letters A, I (affirmative half) and E, O
(negative half). The names of the vertices come from a traditional Latin reading:
AffIrmo, nEgO). As can be seen, different relations hold between the vertices:
- (a) A and O are the negation of each other, as well as E and I;
- (b) A entails I, and E entails O ;
Analogical Proportions and Square of Oppositions
327
- (c) A and E cannot be true together, but may be false together;
- (d) I and O cannot be false together, but may be true together.
Viewing the square in a Boolean way, where A, I, E, and O are now associated
with Boolean variables, i.e. A, I, E, and O are the truth values of statements of
the form ∀xS(x) → P (x), ∀xS(x) → ¬P (x), ∃xS(x)∧P (x), and ∃xS(x)∧¬P (x)
respectively. Then, the following can be easily checked.
A
E
I
O
The link between A and E represents the symmetrical relation of contrariety,
whose truth table is given in Figure 2(a).
We recognize the mutual exclusion, i.e. ¬A ∨ ¬E holds.
AEA
E
0 0
1
1
0 1
1
1 0
0
1 1
(a) Contrariety
O
IOI
0 0
0
1
0 1
1
1 0
1
1 1
(b) Subcontrariety
A I A −→ I
0 0
1
1
0 1
0
1 0
1
1 1
(c) Implication
O
AOA
0 0
0
1
0 1
1
1 0
0
1 1
(d) Contradiction
Fig. 2. The four relations involved in the square of oppositions
The link between I and O represents the symmetrical relation of subcontrariety, whose truth table is is given in Figure 2(b).
We recognize the disjunction, i.e. I ∨ O holds.
The vertical arrows represent implication relations A → I and E → O, whose
truth table is given in Figure 2(c).
The diagonal links represent the symmetrical relation of contradiction, whose
truth table is given in Figure 2(d).
We recognize the exclusive or, i.e. ¬(A ≡ O) = (A ∧ ¬O) ∨ (¬ A ∧ O) =
(A ∨ O) ∧ (¬A ∨ ¬O) holds, or if we prefer we have A ≡ ¬O.
Thus, note that ¬A ∨ ¬E and I ∨ O are consequences of A ≡ ¬O, A → I and
E → O in the square, but not E ≡ ¬I. Moreover I ∨ O and E → O are not
enough for entailing E ≡ ¬I.
4
The Analogical Proportion and the Square of
Opposition
Let us first continue to consider the Boolean square of oppositions where A, E,
I and O are binary variables. What are the joint assignments of values for these
variables that are compatible with the logical square of oppositions? It is easy
to prove that there are only 3 valid squares:
328
L. Miclet and H. Prade
A
0
0
E
0
1
1
0
I
1
1
O
0
1
1
0
One can immediately conclude that if A, E, I and O are the (Boolean-valued)
vertices of a square of oppositions, then they form an analogical proportion
when taken in this order, i.e. A : E :: I : O, since 0 : 0 :: 1 : 1, 0 : 1 :: 0 : 1 and
1 : 0 :: 1 : 0 are 3 of the 6 patterns that make an analogical proportion true.
This is fully consistent with an empirical reading of the traditional square
of oppositions. Indeed one can say that “∀xP (x) is to ∀x¬P (x) as ∃xP (x) is
to ∃x¬P (x)”. A less academic example of square of oppositions associated with
the analogical proportion “mice are to shrews as mammals except shrews are to
mammals except mice” is pictured below.
mice
shrews
mammals
except shrews
mamals
except mice
Remark 1. Consistency of analogical proportions with an empirical reading of
the square: Traditionally, the square is based on the empirical notions of quality
(affirmative or negative) and quantity (universal or existential) which can be
used for describing the vertices of the square:
–
–
–
–
A corresponds to a Universal Affirmative statement,
E corresponds to a Universal N egative statement,
I corresponds to a Existential Affirmative statement and
O corresponds to a Existential N egative statement.
Thus in the square of oppositions each vertex can be described by means of a
2-component vector, the first component being the quantity (value 0 if quantity
is existential, 1 if it is universal) the second component being the quality (value
1 if the quality is affirmative, 0 if it is negative). The square thus obtained is the
superposition of the first and the third previous squares, as can be seen.
01
00
11
10
Analogical Proportions and Square of Oppositions
4.1
329
The Analogical Proportion and Its Two Squares
As recalled in Section 2, the analogical proportion a : b :: ¬b : ¬a
always holds. It clearly gives birth to 4 of the 6 patterns that makes an analogical
proportion true, namely 1 : 1 :: 0 : 0, 0 : 0 :: 1 : 1, 0 : 1 :: 0 : 1 and 1 : 0 :: 1 : 0.
However, the first one (1 : 1 :: 0 : 0) is forbidden when an analogical proportion
is stated in terms of a square of oppositions, since in the following square a and b
form a square of oppositions together with their complements if and only if their
conjunction is false (indeed a and b cannot be true in the same time). Thus,
the square of oppositions appears to be a strict restriction of the analogical
proportion.
What about the 3 other patterns making true an analogical proportion that
are left aside, namely 1 : 1 :: 0 : 0, 1 : 1 :: 1 : 1 and 0 : 0 :: 0 : 0 ? Interestingly
enough, they can also be organized in another square that might be called square
of agreement. It is pictured below (the double arrow represents equivalence). In
this square, the following holds a ≡ b, c ≡ d, c → a, c → b, d → a, and d → b. It
corresponds to the pattern a : a :: b : b under the constraint b → a. Under these
constraints, the square has only 3 possible instantiations, namely 1 : 1 :: 0 : 0,
1 : 1 :: 1 : 1 and 0 : 0 :: 0 : 0.
a
b
a
b
¬b
¬a
c
d
Fig. 3. Square of opposition and square of agreement
Moreover it can be noticed that the three patterns involved in the square of
opposition (a : b :: c : d = 0 : 0 :: 1 : 1, 0 : 1 :: 0 : 1 or 1 : 0 :: 1 : 0) satisfy
constraints a → c and b → d, while the three patterns involved in the square of
agreement (a : b :: c : d = 1 : 1 :: 0 : 0, 1 : 1 :: 1 : 1 or 0 : 0 :: 0 : 0) satisfy the
constraints c → a and d → b.
Since analogical proportions are both a matter of dissimilarity and similarity
[11], it should come as no surprise that one “half” of it satisfies the square of
opposition, while the other “half” satisfies a square of agreement. Moreover, we
can see that that (a ≡ d) ∧ (b ≡ c) is true only for the 4 patterns 1 : 1 :: 0 : 0,
0 : 0 :: 1 : 1, 0 : 1 :: 0 : 1 and 1 : 0 :: 1 : 0, and thus ((a ≡ d)∧(b ≡ c)∧(¬a∨¬b)) is
true only for the last 3 patterns, and thus corresponds to the square of opposition.
Similarly, ((a ≡ b)∧(c ≡ d) is true only for the 4 patterns 0 : 0 :: 1 : 1, 1 : 1 :: 0 : 0,
1 : 1 :: 1 : 1 and 0 : 0 :: 0 : 0, while ((a ≡ b) ∧ (c ≡ d) ∧ (c → b) ∧ (d → a)) is true
only for the last 3 and corresponds to the square of agreement (∧(d → a) can
330
L. Miclet and H. Prade
be deleted and is put here only for emphasizing symmetry). This leads to three
noticeable, equivalent, disjunctive expressions of the analogical proportion:
a : b :: c : d = ((a ≡ d) ∧ (b ≡ c)) ∨ ((a ≡ b) ∧ (c ≡ d))
a : b :: c : d = ((a ≡ b) ∧ (c ≡ d)) ∨ ((a ≡ c) ∧ (b ≡ d))
a : b :: c : d = (((a ≡ c ∧ b ≡ d)) ∨ ((a ≡ d) ∧ (b ≡ c))
(since (a ≡ c) ∧ (b ≡ d) is true only for the 4 patterns 1 : 0 :: 1 : 0, 0 : 1 :: 0 : 1,
1 : 1 :: 1 : 1 and 0 : 0 :: 0 : 0). A counterpart of the second expression can be
found in [13] in their factorization-based view of analogical proportion.
4.2
Analogical Proportion and the Hexagon of Oppositions
As proposed and advocated by Blanché [2], it is always possible to complete a
classical square of opposition into a hexagon by adding the vertices Y =def I∧O,
and U =def A ∨ E. It fully exhibits the logical relations inside a structure of
oppositions generated by the three mutually exclusive situations A, E, and Y,
where two vertices linked by a diagonal are contradictories, A and E entail U,
while Y entails both I and O. Moreover I = A ∨ Y and O = E ∨ Y. Conversely, three mutually exclusive situations playing the roles of A, E, and Y
always give birth to a hexagon [3], which is made of three squares of opposition:
AEOI, YAUO, and EYIU, as in the figure below. The interest of this hexagonal construct has been rediscovered and advocated again by Béziau [1] in the
recent years in particular for solving delicate questions in paraconsistent logic
modeling.
When the six vertices are Boolean variables, there is a unique instantiation
that satisfies all the expected relations in the hexagon:
U
0
A
E
0
0
I
O
1
1
Y
1
Thus the hexagon of oppositions is made of three squares of oppositions.
Each of these squares exactly corresponds to one of the three possible analogical
patterns. Indeed, the square AEOI corresponds to the analogical proportion
A : E :: I : O, i.e., 0 : 0 :: 1 : 1, the square YAUO to Y : A :: O : U, i.e.,
1 : 0 :: 1 : 0, and the square EYIU to E : Y :: U : I, i.e., 0 : 1 :: 0 : 1.
Since a structure of oppositions is generated by any three mutually exclusive
situations A, E, and Y, taking any pair of subsets R and S such that R ⊂ S (in
order to insure that one cannot be in R and in ¬S in the same time), one gets the
Analogical Proportions and Square of Oppositions
331
hexagon below where the following analogical proportions hold R : ¬S :: S : ¬R,
¬R ∩ S : R :: ¬R : R ∪ ¬S, and ¬S : ¬R ∩ S :: R ∪ ¬S : S.
It is possible as well to build a more general hexagon from a pair of unconstrained subsets R and S, with proportions R : R ∩ S :: R ∪ S : R, R ∩ S : R ::
R : R ∪ S, and R ∩ S : R ∩ S :: R ∪ S : R ∪ S. An illustration in terms of Boolean
variables may be obtained by taking R as “speak English” and S as “speak
Spanish”, where the hexagon structures the logical relations between 6 possible
epistemic states regarding the competence of an agent wrt these languages.
R∪S
R∪S
R
S
R
R∩S
S
R
R∪S
R
R∩S
R∩S
(a)
(b)
Fig. 4. Two hexagons constructed from a square of oppositions
4.3
From an Hexagon to an Octagon
A way to display the construction of a general analogical proportion from unconstrained subsets R and S is to start from the hexagon of Figure 4(a) and to
R
S
io
n
R̄ ∩ S̄
s
rie
to
ic
ad
tr
on
C
ies
trar
Con
R∩S
R ∩ S̄ Sub
al
A
na terns
lo
gi
S̄
ca
lP
ro
po
rt
R̄
R̄ ∩ S
Fig. 5. The analogical octagon constructed from R and S
332
L. Miclet and H. Prade
add one node R ∩ S between R and S, as well as one node R̄ ∩ S̄ between nodes
R̄ and S̄, and finally to turn the node R ∪ S̄ into the node R ∩ S̄ (see Figure 5).
The following square in this octagon is a complete analogical proportion:
R ∩ S : R ∩ S̄ :: R̄ ∩ S : R̄ ∩ S̄
Note that the nodes R ∩ S and R̄ ∩ S̄, from one side, and R̄ ∩ S and R ∩ S̄, from
the other side, are not contradictories. This new figure is not a full octagon of
oppositions, but could rather be called an analogical octagon, since it captures
the general construction of an analogical proportion from two sets of properties
R and S. Taking R (resp. R̄) as “aerial” (resp. “aquatic”), and S (resp. S̄) as
“move on ground” (resp. “move above ground”), it leads to state that “ants are
to birds as crabs are to fishes” for instance.
5
Mixing Analogical Arguments and Deductive
Arguments
Stated in the setting of first order logic, a basic pattern for analogical reasoning
(see, e.g., e.g. [12]) is then to consider 2 terms s and t, to observe that they share
a property P , and knowing that another property Q also holds for s, to infer
that it holds for t as well. This is known as the “analogical jump” and can be
described with the following inference pattern:
P (s) P (t) Q(s)
Q(t)
A typical instance of this kind of inference would be:
isBird(Coco) isBird(T weety) canF ly(Coco)
canF ly(T weety)
leading (possibly) to a wrong conclusion about Tweety. Making such an inference pattern valid would require the implicit hypothesis that P determines Q
inasmuch as ∃ u P (u) ∧ ¬Q(u). This may be ensured if there exists an underlying
functional dependency.
The above pattern, may be directly related to the idea of analogical proportion. Taking advantage that “P(s) is to P(t) as Q(s) is to Q(t)” (indeed they are
similar changing s into t), the above pattern may be restated as
P (s) : P (t) :: Q(s) : Q(t)
P (s), P (t), Q(s)
Q(t)
which is a logically valid pattern of inference, from an analogical proportion
point of view (since the proposition P (s) : P (t) :: Q(s) : Q(t) holds) [9,11].
Analogical patterns, as well as deductive patterns, may be the basis of arguments. Consider the following illustrative sequence of arguments:
“P ’s are Q’s”
“s is a P ”
Analogical Proportions and Square of Oppositions
333
then “s is a Q”
This is a deductive (syllogistic) argument in favor of Q(s). Assume the input of
the following information
“t is also a Q”
then “t should be a P ”
by virtue of the analogical pattern, as t is a Q, s is a P and a Q. This an
analogical argument in favor of P (t). Now assume the following claim is made
“s is to t as u is to v”
based on the following facts: s, t are P ’s and Q’s, while u, v are P ’s and ¬Q’s
(then, for the 4-tuple (s, t, u, v), we have the following patterns 1 : 1 :: 1 : 1 for
P and 1 : 1 :: 0 : 0 for Q).
Then, an opposition takes place, and
“it is wrong that P ’s are Q’s”.
Indeed we have a new argument that questions the first premise (“P ’s are
Q’s’). This little sketch intends to point out the fact analogy as deduction may
be the basis of arguments as well as counterarguments.
An analogical proportion-based statement may also support a deductive argument: “P ’s are Q’s”, indeed “s is a P and a Q as t is a P and a Q”, then
knowing that “r is a P ”, we conclude “r is a Q”, which is a form of syllogism
called epicherem (i.e. the basic syllogism is enriched by a supportive argument).
6
Concluding Remarks
This discussion paper has shown that squares of opposition may play a role
in the analysis of analogical proportions as much as they are encountered in
various other forms of reasoning. Squares of opposition can indeed be viewed as
a constrained form of analogical proportions (since analogical proportions both
encompass the ideas of disagreement and agreement). The paper has also pointed
out the relevance of other structures of oppositions (hexagon, octagon) in the
analysis of analogical proportions. This may also help classifying various forms
of arguments.
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