Analogical proportions are quaternary relations of the form “ : :: : ”, read as “ is to as is to ”, that are prominently used for analogical reasoning. The special case of continuous analogical proportions “ : :: : ”, where the second and third argument are identified, naturally leads to a (ternary) betweenness relation. In case the inputs of the relations are concepts, continuous analogical proportions prove to be relevant for ideas of concept blending from cognitive science. In this paper we embark on a study on the relation between these two fundamental structures focusing on the question how to define analogical proportions based on betweenness relations. We show how to define prominent analogical proportions discussed in the literature with minimal assumptions on the betweenness relation. Furthermore we show how to extend the results if the definitions are required to be inverse in the sense that the continuous version of the defined analogical proportion is the betweenness relation. Last we give a first result on decomposability, showing that with a sufficiently strong betweenness relation an analogical proportion is definable that can be naturally decomposed in a binary relation denoted by “::” and a binary function denoted by “:”.
analogical proportions can be deduced.
Analogical reasoning has found and continues to attract iflling the basic axioms) is large and quite diverse. And so the interest of researchers in the intersection of philoso- the question arises how to tame this large set of models. phy, logic, cognitive science, and computer science. Espe- Usually, analogical proportions are used in some specific cially in the artificial intelligence (AI) community various reasoning setting. Hence, from a practical point of view, practical as well as theoretical challenges of analogical the inputs , , , of analogical proportions are known reasoning were tackled [1]. Analogical reasoning can to have a specific structure, i.e., are elements of a specific be described as reasoning over so-called analogical pro- domain. For example, Prade and Richard [3] consider the portions, i.e., quaternary relations “ : :: : ” with case where the domain is the set of Boolean values {1, 0}. the intended reading “ is to as is to ”. Roughly, Already in this case, a bunch of models of analogical analogical reasoning consists of deriving one of the ar- proportions, concretely: eight models, can be identified. guments “, , , ” given the others. For example, from (We will describe these models in the section on prelim“uncle:aunt :: man:” one would infer that “” must be inaries). The authors further develop their approach to woman. Analogical reasoning can also be used for deriv- handle the domain of vectors of Boolean values which ing properties of instances. For example, when and allows a fine-grained description of domain objects by share property and has property , it can be derived a bunch of (Boolean) features, each feature being reprethat has property (the so-called analogical jump) [2]. sented by a component of the vector. And even more, the
Analogical proportions have been approached and ana- authors show that if the features are continuous or if conlyzed from various angles and with various mathematical ifdence values are associated with them, then analogical and logical tools [3]. This paper focuses on an equally proportions over vectors of real numbers (in [0,1]) can be simple and general approach, namely an axiomatic treat- handled. ment of analogical proportions as described by Prade and In this work we take a different approach to taming Richard [3]. Treating “ : :: : ” as a quaternary rela- the plethora of models of analogical proportions: We tion, the authors describe three basic axioms in First Order assume that the domain is equipped with a (ternary) beLogic (FOL) that any analogical proportion should fulfill. tweenness relation. The motivation to do so is that we Already from these axioms further relevant properties of can naturally bootstrap an analogical proportion to reach a simple betweenness relation (, , ) by identifying the second and third argument of an analogical proportion: (, , ) = : :: : . Prade and Richard [3] call this relation a continuous analogical proportion.
betweenness. The commitment to betweenness is not due suggests a decomposition of the analogical proportions to a specific application scenario. Rather it can be justified into a binary relation :: and two occurrences of a binary on grounds of the analogical proportions themselves. In function :. The question we tackle is whether analogical fact, this form of bootstrapping can be understood as a proportions that are defined via betweenness relations can transformation known as forgetting in category theory. be read in this way : :: : , i.e., whether we can give
Another reason to consider betweenness for analogical “:” and “::” natural interpretations such that the analogical proportions is that both structures are considered quite of- proportion holds iff : :: : . We give a positive ten in geometrical settings. In particular, many analogical answer to this problem for an analogical proportion for proportions such as those induced by knowledge-graph the special case where the betweenness relation is the one embeddings are defined in a geometric space. A typical induced by the euclidean metric. representative is the classical approach TransE [4], where At the bottom line, the introduction of betweenness to analogical reasoning amounts to applying a translation in the considerations on analogical proportions helps to grasp a vector space. the axioms of analogical proportions also in an algebraic
This paper contributes with foundational considerations sense. Furthermore, the properties of complex analogion the relation between two theoretically fundamental and cal proportions can be determined by reducing them to practically relevant structures mentioned above: analog- their properties regarding betweenness. Last but not least, ical proportions and betweenness relations. In all our betweenness lays the foundation for the definition of anaconsiderations we are interested in defining some given logical proportions exactly with the desired strength by analogical proportion based on a betweenness relation. combining the constraints of the desired axioms of the In the first part of the paper (after the preliminaries) we proportion. work with a very weak notion of betweenness and then The rest of the paper is structured as follows: Section 2 answer the question how to define a given analogical pro- contains preliminaries on the basic axioms of analogical portion with it. In that first part of the paper there are no proportions, the eight models of analogical proportions in restrictions on the allowed constructions for defining an the Boolean case and the basic betweenness axioms. In analogical proportion from the betweenness relation. As Section 3, for each of the conditions on analogical propora result of the fact that the presumed betweenness relation tions, an interpretation based on betweenness is presented, is quite a weak notion of order, it is possible to define leading to the definition of analogical proportions for the nearly any weak analogical proportion—independently eight models for arbitrary domains. After that, in Secof the structure of the space and further assumptions on tion 4, an interpretation of the analogical proportion with the betweenness relation. In particular, we discuss how the help of betweenness relations is shown that has the to define each (of appropriate generalizations; see pre- continuous analogical proportion as a special case. Secliminaries) of the conditions in order to satisfy the eight tion 5 considers a different definition of an analogical models of analogical proportion of Prade and Richard [3]. proportion, though still based on betweenness, and it is In fact, we can show that the chosen betweenness relation shown how a decomposition of this proportion is possiis as weak as possible: it relies only on properties that are ble. The paper ends with a discussion of related work in necessary to ensure the definability of the given analogical Section 6 and a conclusion in Section 7. proportion.
In the second part of the paper we again consider the deifnability of analogical proportions based on betweenness, 2. Preliminaries but now we require the construction used in the definition to act as an inverse of the forgetting operation on The basic axioms of analogical proportions according analogical proportions. That is, the betweenness relation to Prade and Richard [3] are the following: (, , ) is now required to be identical to the continu- ∀∀ ( : :: : ) (reflexivity) ous analogical proportion : :: : induced by the given analogical proportion : :: : . So in the sec- ∀∀∀∀( : :: : → : :: : ) (symmetry) ond half of the paper we describe how to define analogical proportions— on the same foundations—but leading to a ∀∀∀∀( : :: : → : :: : ) generalization of continuous analogical proportions. (central permutation)
In the last part of the paper (before the related work and The authors consider (in the first place) the domain the conclusion) we make even further restrictions on the of Boolean values {0, 1}. In that case : :: : analogical proportion induced by a betweenness relation, represents a Boolean function (, , , ) of arity four. namely requiring the analogical proportion to allow for a As Prade and Richard [3] show, there are eight different compositional semantics in the sense explicated in formal Boolean functions that fulfill the three axioms of analogilinguistics [5]. The motivation for this restriction is due to cal proportions. These functions form a Boolean algebra the observation that the notation of analogical proportions given in Fig. 1. We will identify a Boolean function on propositional variables , , , with the set of truth as- the eight models to arbitrary domains. Hence, in Fig. 1 we signments over {0, 1}4 which are mapped to 1 and write present axiomatizations of the generalization of the eight those assignments as bit vectors of length four. Following models w.r.t. the (new) axioms introduced above. It is the naming convention of Prade and Richard the lattice readily checked that these axiomatizations uniquely idenelements are the following: tify the eight models over the domain —so that, indeed, they can be considered as properties of generalizations of Ω 0 = {0000, 1111, 0101, 1010, 0011, 1100} the eight models to arbitrary domains. The easy part is to = {0000, 1111, 0101, 1010, 0011, 1100} ∪ check (just by going through 4-bit assignments) that the {0110, 1001} eight models fulfill the corresponding axioms. That those axiom sets exclude each other can be verified by checking 3 = {0000, 1111, 0101, 1010, 0011, 1100} ∪ each pair of axiom sets , ′ and noting that there is {1110, 1101, 1011, 0111} always a pair of complementary axioms (over domains 4 = {0000, 1111, 0101, 1010, 0011, 1100} ∪ with at least two elements). For example, in case of Kl and Ω0 we have the axioms of ratio symmetry and ratio {0001, 0010, 0100, 1000} antisymmetry. 5 = 3 ∪ {0110, 1001} = 3 ∪ Continuous analogical proportions as introduced by 6 = 4 ∪ {0110, 1001} = 4 ∪ Prade and Richard [3] are defined as analogical propor7 = 3 ∪ {0001, 0010, 0100, 1000} tions where the second and third arguments are identified.
So, given an analogical proportion (, , , ) = : :: = 3 ∪ 4 : the ternary relation is defined as Ω =
{0, 1}4 = {0000, 0001, 0010, . . . , 1111}
braic structure) is cited here on the left of Fig. 1.
Next to the basic axioms of analogical proportions, we will also consider the following axioms, the first of which is also mentioned by Prade and Richard [3] (there it is called unicity, here we prefer the term universal antineutrality to clarify the connection to other related axioms). The other ones are our additions. The intention of these axioms is to give an individual axiomatization of each of the eight analogical proportions and thereby pave the way for considering analogical proportions for arbitrary domains. In the following, all variables are assumed to be (implicitly) universally quantified.
: :: : → = (universal anti-neutrality) (, , ) iff (, , , )
as betweenness relations. The whole point of this paper is to investigate the relationship between (properties of) analogical proportions and betweenness relations.
betweenness. In the following, the most basic ones are presented. As it will turn out later on, not all of the axioms have to be fulfilled for the intended definitions of analogical proportions. ( ̸= ̸= denotes in the following that , and are pairwise unequal)
e.g, by Huntington and Kline [6] and [7].
If (, , ), then , , are distinct
If (, , ), then (, , ) : :: : : :: : → =
: :: : 0 : 0 :: : → = 1 : 1 :: : → = 0 : 0 :: : 1 : 1 :: : : :: :
(ratio symmetry) (ratio anti-symmetry) (universal neutrality) (0-neutrality)
Prade and Richard [3] consider the case of analogical proportions where the domain of the quantifiers range over = {0, 1} and the generalized case where are -bit vectors. For betweenness considerations the domain is going to be a general structure and so we generalize (0-anti-neutrality) When (B0) is valid, the ternary betweenness relation is called open betweenness [8] or sometimes also (1-neutrality) strict betweenness. Without (B0), the betweenness is (1-anti-neutrality) called closed betweenness. We will focus in the following on closed betweenness. (B1) expresses commutativity of (universality) betweenness w.r.t. the outer points. (B2) expresses the constraint that if a point is between two points it cannot have one of those two points in between itself and the other point. (B1) and (B2) are the two fundamental axioms of betweenness which should be fulfilled. (B3) is sometimes called outer transitivity and (B4) inner transitivity.
If (, , ) and ̸= ̸= , then not (, , ) (B2) If (, , ) and (, , ) and ̸= ,
then (, , ) If (, , ) and (, , ), then (, , ) (B4) (B0) (B1) (B3) { univ } { r. sym, 1-neut, 0-a.-neut } { r. a-sym, 1-neut, 0-a.-neut } 5 3 { u-a-neut, r. a-sym } 6 4 { r. sym, 0-neut, 1-a.-neut } { r. a-sym, 0-neut, 1-a.-neut } { r. sym, u-a-neut }
out of {, , , }. As (B1) and (B2) can be considered as fulfilled by the betweenness relation (as these are the most The eight axiomatizations given in Fig. 1 are motivated by basic betweenness axioms), it remains to consider a definithe Boolean case. Now, we take a different perspective on tion based on arbitrary combinations of (, , ) and these axiomatizations by assuming a betweenness relation (, , ) for some , ∈ {, , , } and , ∈ . which is a sufcfiiently weak and thus general notion. It Obviously, it is also possible to consider more complex allows for a definition of the several axioms as specific connections—which we saved for future work. as possible in the sense that each restriction should be Especially for analogical proportions the interplay of minimal, thus should represent exactly the given axioms the elements is of importance. Hence we focus here for a model. The goal of this section is to define analogical on a definition based on (, , ) for some , ∈ proportions based on betweenness for each of the eight {, , , } and ∈ chosen arbitrarily. models, i.e., to give for each model a definition fulfilling Before going into detail regarding the different models, the three basic axioms of analogical proportions and the we first discuss how definitions of analogical proportions additional axioms mentioned in Fig. 1. based on these betweenness relations could look like in
Assume an arbitrary domain equipped with a be- general. An analogical proportion must fulfill the basic tweenness relation is given and the analogical proportions axioms, independently of its construction. Especially symto be defined are determined on element-level, thus each metry and central permutation enforce restrictions on the input is an element of . As can be an arbitrary space construction, as they change the order of elements in the equipped with a betweenness relation, the following defi- analogical proportion. Thus, postulated conditions need nitions are widely applicable. some sort of symmetric behavior to cover the changed
First of all, we discuss how to define in general an ana- order. We will capture this with the notion of a substilogical proportion from a betweenness relation. However, tution of subformulas. For a FOL formula we denote for a better understanding of the discussion, it is useful by [/, /] the result of substituting in parallel all to think of a concrete betweenness relation such as Eu- occurrences of subformula (if contained in ) with clidean betweenness which is defined with the Euclidean subformula and all occurrences of subformula (if distance as follows. contained in ) with . This notion is generalized canonically to the case of more than two subformulas being (, , ) iff (, ) = (, ) + (, ), substituted in parallel. Here and in the following, for simplicity, we use the following shortcut ((, ), (, )):
it is on the line segment connecting and .
A first idea for defining an analogical proportion (, , , ) is to enforce the inputs , , , to be comparable w.r.t. the betweenness relation by, e.g., enforcing (, , ). But already here it becomes obvious that such a definition is too strong to save as a general template for arbitrary models of analogical proportion because it would enforce , and (and depending on the other constraints also ) to be on a line.
((, ), (, )) iff there is , ∈ with (, , ) and (, , ).
Proposition 1. Assume a quaternary relation ((, , , )) is defined with a first order logic formula over containing atoms of the form (, , ) Ω 7 Ω0 with distinct , ∈ {, , , }, ∈ . Then fulfills symmetry and central permutation iff: 1. for all distinct , , , ∈ {, , , }: for any subformula (, ) of of the form (, , ) or (, , ) with a variable it holds that ⊨ [(, )/(, ), (, )/(, )] and 2. for any subformula (, ) of of the form (, , ) or (, , ) with a variable it holds that ⊨ [(, )/(, ), (, )/(, ), (, )/(, ), (, )/(, )] share the properties relevant for exactly this analogical proportion.
Formally, is the same as regarding the set {, , , } iff for all ∈ the following assertion holds: ((, , ) ↔ (, , )) and ((, , ) ↔ (, , )). This leads to the definition of the condition y-same((, ), (, )) stating that (at least) one element of {, } is equivalent to (at least) one element of {, } (regarding the set {, , , }): y-same((, ), (, )) iff y-all((, ), (, )) & y-all((, ), (, )) for distinct , ∈ {, }, , ∈ {, }, Proof. →: (1.) (i) Assume (, , , ) is defined based with y-all((, ), (, )) stating intuitively that all eleon an arbitrary constraint on (, , ) but not on ments in between and should be also in between (, , ) for arbitrary , ∈ . Then with symme- and (and vice versa): try it follows (, , , ) and thus, the constraint needs to be valid on (, , ) (for arbitrary ∈ ), a con- y-all((, ), (, )) iff ∀((, , ) ↔ (, , )) tradiction. (ii) Assume (, , , ) based on an arbi- Note that for the case of Euclidean betweenness this contrary constraint on (, , ) but not on (, , ). dition reduces to the fact that two elements are equivalent With central permutation, symmetry and central permu- if they are at the same point in the vector space. tation follows: (, , , ),(, , , ), (, , , ), We aim at a general construction of analogical propor(, , , ) and thus (, , ) need to be constrained, tions from a betweenness relation that is easy to grasp a contradiction. (iii) For a constraint on (, , ) and and that works also for analogical proportions outside the all other cases, analogous arguments apply. (2.) Needs to set of eight models. Hence we define fragments of an be the case because of central permutation. analogical proportion based on betweenness where each ← : Symmetry is valid, as (, , , ) leads to con- fragment fulfills exactly the desired constraint. These constraints on ((, ), (, )) and ((, ), (, )) and inde- straints can then be combined to create each of the eight pendently constraints on ((, ), (, )). With (B1) and models, but can also be arbitrarily combined with each (1.) it follows that the constraints can be reformulated other or other constraints and thus can serve as basis for to constraints on ((, ), (, )) and ((, ), (, )) and an arbitrary analogical proportion based on betweenness constraints on ((, ), (, )) and thus (, , , ) iff that fulfills the desired constraints. (, , , ). Central permutation can be shown analo- In general there are two types of constraints, existential gously. constraints and filter constraints. Having a bunch of potential conditions to be used in the definition, the conditions of the first type are connected via a disjunction, thus not interfering with the other constraints, whereas constraints of the latter type are connected via conjunction to existing constraints, as they need to restrict each analogical proportion where they are applicable.
are considered case by case. In this context, we introduce a (weak) new betweenness axiom (B0’) stating that in case of closed betweenness the only element between and itself is again.
As the axioms for the (eight) models of analogical proportion are motivated by the binary case, it is necessary to make some adaptation before using it in the general case. In particular, two questions arise: (i) how can 0 and
equal, e.g., when should : :: : be interpreted as : :: : to apply the specific conditions?
The first question is only relevant for some of the conditions and thus considered later on. So we consider now the second question. As we assume that a general space and a betweenness relation on it, but that no other information is provided, the idea is to interpret two elements of to be the same (or similar) if they behave the same regarding the betweenness relation. In setting up the conditions on compared elements as before (now for similarity) this can be restricted to those mentioned in the inputs. For example, is considered similar to if both behave the same regarding and , i.e., intuitively, if the elements ∀, ∈ ((, , ) & (, , ) → = ) (B0’)
fact, out of these only reflexivity needs to be considered, because the other two are already fulfilled when using the construction principles mentioned in Proposition 1. Though, it would have been possible to create conditions which are minimal in the sense that they allow exactly for the condition mentioned, but in the following, it has been assumed that symmetry and central permutation are such basic axioms that they could (and should) be incorporated into the definition of the constraints. Thus, not only the constraints are fulfilled but also all permutations of it regarding symmetry and central permutation.
straightforwardly, stating that the analogical proportion must be valid in case of : :: : (and permutations of it).
Proposition 2. Reflexivity ( ∀∀( : :: : )) is valid iff there is an arbitrary (possibly empty) constraint such that the following equivalence holds: : :: : iff ∨ (y-same((, ), (, )) & y-all((, ), (, ))).
the ratio of an element to itself is similar to the ratio of arbitrary elements. This can be satisfied in terms of betweenness by stating that the analogical proportion : :: : must be fulfilled when = .
Proposition 5. Universal neutrality (∀∀∀( : :: : )) is valid iff there is an arbitrary (possibly empty) constraint such that the following equivalence holds: : :: : iff ∨ y-same((, ), (, )).
Proof. ← : Let y-same((, ), (, )). Then = or = or = or = . With Proposition 1 and by applying central permutation and symmetry, this leads to the case that ( : :: : ) is valid.
→: Let : :: : be valid and be empty. Then, y-same((, ), (, )) is valid. Proof. →: Let : :: : be valid. Then y-same((, ), (, )) and y-all((, ), (, )) are trivially fulfilled.
← : Let y-same((, ), (, )) and y-all((, ), (, )) and let be empty. Then = , = , = or = are possible. Let = , the other cases follow with Proposition 1, central permutation and symmetry.
Then y-all((, ), (, )) and thus ∀ : (, , ) ↔ (, , ) ↔ (, , ) (with (B1)). Thus, by definition of equality of elements, = .
In the following, different types of neutrality and antineutrality are considered. In contrast to the other conditions, these conditions are not only based on arbitrary elements but also on elements of type 0 and 1. As the model can be represented as conjunction of 5 and 6, it is necessary that the restriction of 0-anti-neutrality and 1-anti-neutrality behave in combination as (general) anti-neutrality. Thus, it is not sufficient to define one 0and one 1-element but to partition the space into 0- and
The next constraint considers ratio symmetry, i.e., 1-elements. As 3 and 5 are analogue to 4 and 6, ∀, ( : :: : ). It states a symmetry condition 0 and 1 could be defined interchangeably. Therefore, in on the analogical proportion, namely that the ratio of the following we assume that all elements of type 0 are and is the same as the ratio of and . universal in the sense that they have a betweenness re
This leads to a definition of the constraint similar to the lation to any other element, thus for all ∈ , there case of reflexivity, only considering different elements. is ∈ with (0, , ). Elements are of type 1 if they do not fulfill this condition. As for the validity of an Proposition 3. Ratio symmetry (∀∀( : :: : )) is analogical proportion only the elements contained in this valid iff there is an arbitrary (possibly empty) constraint proportion are directly considered, an element has type such that the following equivalence holds: : :: : 0 regarding , , , (for : :: : ), if ∈ {, , , } iff ∨ (y-same((, ), (, )) & y-same((, ), (, )) & and is universal regarding {, , , }. As 0 and 1 depict y-all((, ), (, )) & y-all((, ), (, ))). no special element but a type of element, it is possible to Proof. Proof similar to the proof of Proposition 2. have 0 : 0 :: : , thus two different elements of type 0. On such an analogical proportion, e.g., the condition
The direct opposite to ratio symmetry is ratio- of 0-neutrality would not be applicable. antisymmetry, stating that the ratio of and is in no Universal neutrality considered in Proposition 5 accase the same as the ratio of and (except = ). counts for the general case, considering all elements. Beside that, there are models incorporating 0-neutrality and Proposition 4. Ratio-antisymmetry (∀∀( : :: : 1-neutrality, thus neutrality based on a special type of ) → = ) is valid iff there is an arbitrary (possibly elements. tautological) constraint such that the following equiva- First, 0-neutrality is considered. This adds to the genlence holds: : :: : iff & (y-same((, ), (, )) eral case the restriction of ((, ), (, )), stating that & y-same((, ), (, )) & y-all((, ), (, )) & and have a betweenness relation with and thus not y-all((, ), (, )) → y-all((, ), (, ))). only being equal but also of type 0.
Proposition 6. 0-neutrality (∀∀(0 : 0 :: : )) is valid iff there is an arbitrary (possibly empty) constraint such that the following equivalence holds: : :: : iff ∨ (((, ), (, )) & y-same((, ), (, ))). Proof. The general case follows with Proposition 5, it remains to show that the restriction on elements of type 0 is valid.
→: Let 0 : 0 :: : . Then ((0, ), (0, )) by definition of 0.
← : Let be empty and ((, ), (, )) and y-same((, ), (, )). Then with y-same, = (analogue for the other cases). With ((, ), (, )) and (B0’) follows (, 1, ), (, 2, ) and (, 3, ) for 1, 2, 3 ∈ . Thus, (and ) are of type 0 and thus 0 : 0 :: : .
Proposition 7. 1-neutrality (∀∀(1 : 1 :: : )) is valid iff there is an arbitrary (possibly empty) constraint such that the following equivalence holds: : :: : iff ∨ (¬((, ), (, )) & y-same((, ), (, ))).
neutrality: ∀∀∀: ( : :: : → = ). This means intuitively that the ratio of an element to itself can not be similar to the ratio of different elements, thus that two elements are either the same or distinct enough to be not comparable with equal elements.
domain independent generalizations of the eight Boolean models can be easily constructed. This construction is illustrated here for the model Ω 0 but of course analogous constructions for all other models are possible. Definition 2. The minimal analogical proportion Ω0 based on a general betweenness relation is defined as follows: Ω0 (, , , ) iff (y-same((, ), (, )) & y-same((, ), (, )) & y-all((, ), (, )) & y-all((, ), (, )) → y-all((, ), (, ))) & (y-same((, ), (, )) → y-all((, ), (, )))
Proposition 11. Ω0 fulfills universal anti-neutrality and ratio anti-symmetry.
tioned above, it turns out that all of them are quite weak.
Proposition 8. Universal anti-neutrality (∀∀∀: ( : For most of the use cases they allow for too many analog :: : → = )) is valid iff there is an ar- ical proportions. That is true even for the model Ω 0, the bitrary (possibly tautological) constraint such that most restricted model of the eight. However, they show the following equivalence holds: : :: : iff that with betweenness it is possible to define an analog & (y-same((, ), (, )) → y-all((, ), (, ))). ical proportion fulfilling the respective axioms without introducing any unnecessary restrictions. These can be Proof. →: Let : :: : . Then y-same((, ), (, )). used as basis and can be, e.g., combined with other axy-all((, ), (, )) is only valid if = . ioms to define a analogical proportion that fits to the given ← : Let y-same((, ), (, )) be not valid. Then not use-case. : :: : . Let y-same((, ), (, )). Let = (other A stronger analogical proportion based on betweenness cases analog). Then y-all((, ), (, )) only if = . is demonstrated in the next section. There an analogical Thus universal anti-neutrality is fulfilled. proportion having the continuous analogical proportion as special case is defined.
same principles as for neutrality: Proposition 9. 0-anti-neutrality (∀∀(0 : 0 :: : → = )) is valid iff there is an arbitrary (possibly tautological) constraint such that the following equivalence holds: : :: : iff & (((, ), (, )) & y-same((, ), (, )) → y-all((, ), (, ))).
Proposition 10. 1-anti-neutrality (∀, (1 : 1 :: : → = )) is valid iff there is an arbitrary (possibly tautological) constraint such that the following equivalence holds: : :: : iff & (¬((, ), (, )) & y-same((, ), (, )) → y-all((, ), (, ))).
and Richard [3] are defined as analogical proportions where the second and third arguments are identified. They interpret the analogical proportion (, , , ) directly as betweenness relation. To recap:
(, , ) iff (, , , ).
ity, it is necessary that the betweenness relation fulfills
Proposition 12. fulfills the basic axioms . (, , ) for all ∈ . Continuous analogical proportions seem to be a good basis for an analogical proportion, as they are widely used also in practical learning approaches, e.g., for enlarging a data set with new examples Proposition 13. (, , , ) iff (, , ) [9]. Now the question arises whether it is possible to deifne an analogical proportion based on betweenness for the Proof. →: (, , , ) iff y-ex((, ), (, )) or general case of : :: : , having the continuous ana- y-all((, ), (, )) and thus (i) there is ∈ with logical proportion as a special case. To state it differently: (, , ), (, , ). Because of (B0’), = How does an analogical proportion (, , , ) need and thus (, , ). (ii) for all ∈ : (, , ) to look like such that (, , , ) iff (, , )? iff (, , ), but again there is exactly one fulfilling Assume in the following an arbitrary betweenness relation it, namely = and thus (, , ). fulfilling (B1), (B2) and (B0’). Then, a possible definition ← : Let (, , ) be valid. (, , ) is valid by would be definition and thus y-ex ((, ), (, )).
After having showed that is a correct analogical proportion fulfilling the basic axioms, we proceed by showing that the continuous analogical proportion is in fact a special case. (, , , ) iff y-ex((, ), (, )) ∨ y-all((, ), (, )), where y-ex((, ), (, )) states intuitively that there must be at least one element in between and which is also in between and (and vice versa): y-ex((, ), (, )) iff ∃ ∈ ((, , ) & (, , )).
Whereas in Section 3 the definition of an analogical proportion based on a restriction as minimal as possible has been examined, in this section, an analogical proportion based on a stronger restriction is presented, namely (when considering again the Euclidean betweenness) the restriction that : :: : can be only valid if , , and are on the same line and the line is part of or vice versa.
For an intuitive understanding, consider again Euclidean betweenness in a two-dimensional space. The 0(, , , ) iff graph on the upper left of Fig. 2 illustrates the intuition behind y-ex((, ), (, )). Basically, it states that the line (, , ) & (, , ). segment and the line segment must be in a specific orientation to each other, as the lines and have to in- The analogical proportion defined in this way fulfills symtersect. The upper right graph is a counterexample. At the metry and central permutation. To gain the missing rebottom of Fig. 2 the special case of the continuous analog- flexivity, the betweenness is enforced to fulfill a specific ical proportion can be seen. If = , then plays the role constraint, namely, of the element existing in between of both and and of for all , : (, , ) (B6) and (= ) and thus leads for Euclidean betweenness to (, , ). This definition can be brought to practice by consider
The disjunct y-all((, ), (, )) is necessary for the ing a special type of betweenness relation, namely one general case to ensure reeflxivity. It is not necessary which allows for the definition of a line in the usual sense in the Euclidean case, because there is always an ele- (including, but not limited to the Euclidean betweenness ment in between each two other elements and thus also and a line in the Euclidean sense). On the basis of such y-ex((, ), (, )) is valid. a betweenness relation we can define a notion of a line,
First, we show that is a correct analogical propor- collinearity etc. For two points , let ⟨⟩ denote the tion in the sense that it fulfills the basic axioms. closed section between , on the line going through , (, , ) & (, , ) or and containing , . Formally, we can define the closed (, , ) or (, , ) and (, , ) iff section on the basis of the betweenness relation and set 0(, , , ). theoretical operations as follows: Central permutation: 0(, , , ) iff (, , ) and (, , ) or (, , ) and (, , ) iff ⟨⟩ = { ∈ | (, , )} ∪ {, } (, , ) and (, , ) or (, , ) and (, , ) iff 0(, , , ).
Thus, also 1 fulfills the basic axioms of betweenness.
If (, , ) and (, , ) (and ̸= ̸= ̸= ), portions that suggests a decomposition of the analogithen (, , ) or (, , ) (B7) cal proportions into a binary relation :: and two occurrences of a binary function :. The question is whether
Note that though we use the interval notation we do not 1(, , , ) can be read in this way : :: : , i.e., have an orientation so that ⟨⟩ = ⟨⟩. In the following whether we can give “:” and “::” natural interpretations one may first think of a Euclidean space with betweenness such that 1 iff : :: : . Note that this problem induced by the Euclidean distance. is known in the literature on linguistics and model the
We define a quaternary relation 1(, , , ) that ory also as an extension problem for compositionality of says the section ⟨⟩ contains ⟨⟩ or vice versa. In other the first form [ 5, p.16]. A trivial one would be to treat words: There is a line on which all of , , , occur and : as the pair (, ) and allow in :: the use of projeceither the section from to occur in between of , or tions operators, so that : :: : can be reduced to vice versa. ( 1( : ), 2( : ), 1( : ), 2( : )). Here we used the operators of left projection (projection on the first argument) and right projection (projection on the second 1(, , , ) iff ⟨⟩ ⊆ ⟨ ⟩ or ⟨⟩ ⊆ ⟨ ⟩ argument) 1(, ) = and 2(, ) = .We seek for This is in fact a special case of the analogical proportion non-trivial decompositions where : fits to the idea of 0. a difference or quotient that can be used to do analogical reasoning. The next proposition gives an afrfimative anProposition 14. If fulfills (B1)–(B4) and (B7), swer. For this we need the notion of orientation. Each then for all , , , the following equivalence holds: line has exactly one of two orientations [10, Theorem 0(, , , ) iff 1(, , , ) 40] which we call and (for from-left-to-right and from-right-to-left on ). Such an orientation induces a total order < and a dual order on the dual orientation ′ with < iff <′ . Each segment ⟨⟩ has exactly one line going through it. In particular, each orientation positions , w.r.t <. Define (, ) as follows Proof. →: Let 0(, , , ) for arbitrary , , , and thus (, , ) and (, , ) or (, , ) and (, , ). Let (, , ) and (, , ) be valid. If ̸= ̸= ̸= , then with (B7) follows (, , ) or (, , ) and with (B3) follows (, , ) or (, , ). This is possible for all elements in between and and thus ⟨⟩ ⊆ ⟨ ⟩. If there are no such elements not equaling , , , , then the condition is trivially fulfilled. Analog for (, , ) and (, , ) being valid.
← : Follows trivially out of the definition of a line.
(where comes before reading from left to right). So (, ) tells us whether under the left-to-right orientaProposition 15. For any ternary betweenness relation ful- tion the order chosen in the pair (, ) fits the order of the iflling (B1) and (B6) (commutativity of ), the induced induced order < by left-to-right-orientation. The special relation 0 fulfills the basic axioms of analogical pro- case of = is treated separately. Now we define on the portions. basis of this operation the following operation: (, ) = ⎪⎩0, ⎧+, iff < ⎪ ⎨ − , iff < otherwise Proof. Reflexivity: 0(, , , ) holds because it holds if (, , ) and (, , ). This is the case for all , because of (B6).
Symmetry: 0(, , , ) iff (, , ) and (, , ) or (, , ) and (, , ) iff (, , ) and (, , ) or (, , ) and (, , ) (and with (B1) iff (, , ) and : × (, ) →− ↦→ segments × {
+, − , 0} (⟨⟩, (, ))
(1)
difference: If for example comes before then ⟨⟩ = [, ] can be considered as the halfline starting from to the right minus the half line to the right starting at . For 1(, , , ) which means ⟨⟩ ⊆ ⟨ ⟩ or ⟨⟩ ⊆ any two closed proper segments on a line we consider the ⟨⟩ which amounts to saying that the following holds: following eight jointly exhaustive and mutually disjoint ⟨⟩{ , , }⟨⟩. relations known from the region connection calculus RCC Case 2: ̸= . a) Subcase = . Then (, ) = 0. 8 [11], applied to the 1-dimensional case of a line: Now 1(, , , ) means in this case 1(, , , ) which means ⟨⟩ ⊆ ⟨ ⟩ or ⟨⟩ ⊆ ⟨ ⟩ which amounts {, , , , , , , } to ⟨⟩{ , }⟨⟩. b) Subcase ̸= . That 1(, , , ) means that Let be given closed proper sections = ⟨⟩ and = ⟨⟩ define ⟨othe⟩r ⊆ d⟨isjun⟩ctoirs⟨tre⟩at⊆e⟨d sym⟩m.Aetsriscuamlley)⟨.If⟩⊆⟨(,⟩()t h̸=e iff ∩ = ∅ (“disconnected”) (, ) were the case, this would e.g. mean from reading right to left that < and < . But due iff ∖ {, } ∩ ∖ {, } = ∅ to ⟨⟩ ⊆ ⟨ ⟩ we must have ≤ and ≤ , i.e. ∩ ̸= ∅ < . Now ⟨⟩ ⊆ ⟨ ⟩ also means ≤ , giving a (“externally connected”) contradiction. So (, ) = (, ) holds. Now, iff ⊊ and ̸⊆ ∖ {, } in the left orientation we have intervals [, ] and [, ] or intervals [, ] and [, ]. Consider the first case (the (“tangential proper part”); other is treated similarly). We have to treat the case of iff ⊊ ∖ {, } proper and improper intervals. (i) Case = and = . and all of , , , are different. Here ⟨⟩ ⊆ ⟨ ⟩ becomes ⟨⟩ ⊆ ⟨ ⟩ which in turn (“ nontangential proper part”) means = . So ⟨⟩ = ⟨⟩ and ⟨⟩ = ⟨⟩, so ⟨⟩⟨⟩. iff ∖ {, } ∩ ∖ {, } ̸= ∅ (ii) Case = and ̸= . Then ⟨⟩ ⊆ ⟨ ⟩ means and ̸⊆ and ̸⊆ ∈ ⟨⟩ which means ⟨⟩⟨⟩. If ̸= , then iff = (“equals”) ⟨⟩ ⊆ ⟨ ⟩ means that ⟨⟩{, }⟨⟩. (iii) Case ̸= and = is not possible: ⟨⟩ ⊆ ⟨ ⟩ would iff (“contains tangentially”) mean ⟨⟩ ⊆ ⟨ ⟩, so = , contradiction. (iv) Case iff ̸= and ̸= . So all of , , , are pairwise different. (“contains non-tangentially”) ⟨⟩ ⊆ ⟨ ⟩ means under the orientation with intervals [, ] and [, ] that we can have only ⟨⟩⟨⟩.
Note that we have denfied those relations such that they ← : Assume that all , , , are different. (The other can be applied also to the case where or is improper. special cases are treated similarly as above) and all of In particular, if or is improper then it can never be the the condition 1.–3. are true. In particular and case that or holds. have the same orientation. Between ⟨⟩ and ⟨⟩ one
We use the usual set theoretical abbreviation to ex- of the eight relations must hold. As all , , , are difpress a relationship out of a given set of relations. For ferent the relation cannot be , , , . If example ⟨⟩{, }⟨⟩ stands for: ⟨⟩⟨⟩ or 1(, , , ) would not hold then none of the con⟨⟩⟨⟩. ifguration (i)–(viii) could hold which means that also
The following proposition is going to show that we , would have to be excluded. We are left have to exclude the relations of non-tangential proper part , , but this contradicts our assumption and its inverse between ⟨⟩ and ⟨⟩ that the third condition holds. Similarly one argues in the for the analogical proportion 1 to hold and vice versa. case where we have improper segments. Proposition 16. 1(, , , ) iff 1. the lines through ⟨⟩ and ⟨⟩ is the same line
and 2. (, ) = 0 or (, ) (, ) = (, ) and
= 3. ⟨⟩{, , , , , }⟨⟩.
Proof. →: Assume 1(, , , ). Then , , , are on the same line. We prove the result considering case-wise. Case 1: = . Then (, ) = 0. Now 1(, , , ) in this case is
The proposition in particular shows that we have a non-trivial solution for the extension problem of compositionality: Choose : to be defined according to (1); define the relation :: on left and right each of which is a pair of 0 or the form (, ) where stands for a closed section and ∈ {+, − , 0}. Last define (1, 1) :: (2, 2) iff 1 and 2 make up the same line and 1 = 0 or 2 = 0 or 1 = 2 and 1{, , , , }2.
non-trivial operation different from simple set operations such as intersection or union. If we consider continuEarly work on axiomatic treatments of analogies ous analogical proportions over the domain of concepts, goes back to Yves Lepage [12]. He considers lin- then concepts in between two others can be considered guistic issues related to analogies of words as in as their conceptual blending. Also the example discussed “walk:walked::search:searched”. He accounts for simi- by Prade and Richard [3] (the concept of centaur as lylarities between words in the discussion of analogies but ing in between the concept of man and horse) indicates betweenness is not mentioned at all. Nonetheless, for at least that work on continuous analogical proportions future work plan to investigate the relations between his might profit from work on concept blending and vice versa. approach and our approach because a recent article [13] Continuous analogical proportions do not directly provide shows that there are relevant connections between similar- an operator of concept blending, but at least they describe ity relations and betweenness relations. a set of concepts in between two given concepts and hence
Schockaert et al. [14] incorporate analogical propor- a set of potential blending concepts, which fits to the idea tions into a description logic framework. They enrich of a “blended space” [17]. For the sake of completeness, each concept with features and are able to define analogi- we mention here also work on hyperdimensional comcal proportions based on these features. For one variant of puting and, more generally, vector symbolic approaches analogical proportions described by Schockaert and col- [19, 20] which provide vector operations corresponding leagues [14], some form of betweenness is used. However, to conceptual blending. the whole approach is based on analogical proportions over the domain of concepts and their features (and thus on sets). In contrast, our approach presumes an arbitrary 7. Conclusion domain of objects—as long as this domain is equipped with a betweenness relation. The idea of continuous analogical proportions motivated
Continuous analogical proportions are widely used also our investigation on the definability of analogical proporin practical approaches, e.g., for the interpolation of new tions via betweenness relations. As a main result we could training examples [9], however, to the best of our knowl- show that such a definition is possible in the representative edge, by now not extended to the general case. Several microcosm of the eight models of analogical proportions ways have been proposed to enhance the analogical pro- [3]– and it was even possible to consider a definition being portion of the Boolean case to multiple values. On the the inverse of the forgetting operation (leading from anaone hand, there are many practice-oriented approaches logical proportions to continuous analogical proportions). such as TransE [4] being focused on a specific (most times In future work we plan, first, to widen the perspecgeometrical) operation and its usability for classification. tive from the microcosm of eight models to other anaOn the other hand, there are multiple approaches starting logical proportions, second, to generalize the decomposwith the Boolean case and generalizing it to the general ability questions from Euclidean betweenness to general case. One approach is based on a graded extension of the betweenness relations, and, third, to investigate the conBoolean case thus allowing values in [0, 1], it results in sequences of our results for frameworks (such as concept graded analogical proportions not being true or false but blending) dealing with analogies and betweenness of conhaving a truth value in [0, 1][15]. cepts.
A general form of an analogical proportion based on
groups has been proposed by Stroppa and Yvon [
Also related to the approach of this paper is a computational framework [17] on conceptual blending [18]. Con- [1] H. Prade, G. Richard, Analogical proportions: Why ceptual blending is founded in cognition science, in partic- they are useful in AI, in: Z.-H. Zhou (Ed.), Proular, in that part dealing with the mechanisms underlying ceedings of the Thirtieth International Joint Conferhuman creativity in inventing concepts, e.g., inventing the ence on Artificial Intelligence, IJCAI-21, Internaconcept of houseboat from concept house and concept tional Joint Conferences on Artificial Intelligence boat [17]. The authors stress that concept blending is a