Analogy

Analogy Charles Tijus Abstract Among cognitive processes, although literally based on logically false propositions, analogical and metaphorical reasoning are the most used and useful thinking for communicating, understanding, discovering, problem-solving and learning. The topic of this chapter about analogy and metaphor, is to address the kind of computation linking a target category to a source category that belongs to another domain that might be able to support reasoning properties based on the fallacy of the falsity of propositions, on imperfection, imprecision and approximation, gradualness, vagueness, fuzziness, uncertainty and implicit plausibility of likeness. Because notable advances in the computation of analogies are from Bernadette Bouchon-Meunier’s work with her team: the fuzzy logic computation of analogical reasoning and schemes, we examine how such modeling of the hu-man computation of analogies can be used in turn to model the machine computa-tion of analogies. 1 Introducing Analogy as One of Two Main Ways of Thinking Cognitive processes of both human and machine are for understanding and for decision-making. These processes are cognitive investigations about external physical things, including people and oneself; about who they are, what they are (structure and functioning), about the events and actions in which they participate, as well as ascribing intention to other agents [1, 2]. Although a cognitive system requires information from the external physical world, investigations about external things are firstly based on what we already know about them: on their internal representation, including the kind of things they are and the categories to which they belong. This inference-making about what things are and about how they behave are made for interaction, for decision-making about our judgments about them, about actions C. Tijus (B) CHArt, University Paris 8, 2, rue de la Liberté 93526 St Denis Cedex 02, Paris, France e-mail: tijus@univ-paris8.fr © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M.-J. Lesot and C. Marsala (eds.), Fuzzy Approaches for Soft Computing and Approximate Reasoning: Theories and Applications, Studies in Fuzziness and Soft Computing 394, https://doi.org/10.1007/978-3-030-54341-9_6 61 62 C. Tijus we might perform on them or with them. Classically, there are four kinds of inferences of these cognitive processes—deduction, induction, analogy and metaphor [3, 4]—that we propose to be reduced to two main kinds: deduction and induction as one kind and analogy and metaphor as another kind. Inference-making of the first kind is literal, which means that the target domain of investigation is domain specific constrained: restricted to the categories of this domain, restricted to the intension and extension of these categories and to the relations among these categories. In that case, things under investigation are supposedly known; as in computer programming for which a literal value is a value given explicitly in the source code of the algorithm for a given variable. Such literal inferencemaking classically stands for both deduction and induction: either from category to exemplar (i.e. deduction: what is known about a category can be attributed to subordinate categories and finally to exemplars of these categories): Cats meow, Maine Coon is a kind of cat, Felix is a Maine Coon, Felix should meow. It stands also from exemplars to category (i.e. induction: what is known about exemplars of a category could be attributed to this category): Felix is a cat and meows, Tom is also a cat and meows: cats might meow. In that case, the top-down (from category to exemplars) or bottom-up (from exemplars to category) processes are about things known to belong to the same given category or, for more complex thinking and reasoning, to the same specific domain. Contrary to deduction and induction that provide “true” sentences, Inferencemaking of the second kind is unliteral (i.e. not literal; not literally comparable). Things under the cognitive investigation are cross-domain constrained: they are of different categories, usually of exclusive domains: inference-making is from singular to singular, from an exemplar of a given category to exemplar of another category. Such unliteral inference-making is found with analogical propositions based on comparison (e.g. the nucleus of the atom is the sun of the solar system) but also with metaphorical propositions [5] based on attributive categorization (e.g. John is a cat): John, a troubadour, sings to beg for a little love. Felix, a cat, meows to beg for food. John’s voice sounds like meowing: propositions such as “John is meowing” or “John is a cat” are analogies. Such associations, like “Juliet is my sun” (i.e. she brings me joy and light) in from Romeo’s diary of Shakespeare’s “Romeo and Juliet,” are based on some fuzzy resemblance, on analogy or metaphor (upon, according to/so to speak). According to Plato, analogies are founded on a reasoning based on “an argument from the similarity of things in some ways inferring their similarity in others” and on a computation based on “partial agreement, likeness or proportion between things”. Although such analogically-based reasoning provides sentences that do not have evidential support and are logically “false”, this kind of reasoning appears to be the most prominent kind of human of thinking [6] and maybe the one that machines will be using in the very near future for understanding, thinking and communicating: • For analogical reasoning that is based on comparison: [A: the nucleus] is to [B: the Atom] what [C: the sun] is to [C: the solar system]. If one knows the relation Analogy 63 between the sun and the solar system, s/he can infer the relation between the nucleus and the atom. • For metaphorical reasoning that is based on attributive categorization [B: singing] is to [A: John] what [D: meowing] is to [C: Felix]. If one knows that Felix is meowing to beg, the conclusion is that John is begging. Thus, Felix is the source of the analogy, while John is the target of the analogy. Note that to produce an analogy or a metaphor, the computation is from-targetto-source. For example, to produce “Juliet is my sun” from Juliet Shakespeare had to find a likeness source. In order to understand the likeness of the source, the cognitive investigation of the listener is a from-source-to-target computation. Thus, to understand “Juliet is my sun” from “sun,” one must find the likeness to attribute to Juliet. Note also that because analogy is based on comparison (x is equivalent to), source and target can permute. For example, “the nucleus of the atom is the sun of the solar system” is equivalent to “the sun of the solar system is the nucleus of the atom.” Conversely, because metaphor is based on implication (x is a kind of y), source and target cannot permute. “John is a cat,” the reverse is not true: “Cats are not as John” [C]. As a matter of fact, there is a very challenging scientific and technological issue to discover and model the what and how of the thinking processes that are able to produce and understand analogies. The kind of computation linking a target category to a source category that belongs to another domain might be able to support reasoning properties such as imperfection, imprecision and approximation, gradualness, vagueness, fuzziness, uncertainty and implicit plausibility of likeness. Notable advances in the computation of analogies are from Bernadette Bouchon-Meunier’s work with her team: the fuzzy logic computation of analogical reasoning and schemes [7–16]. 2 The Necessity of Fuzzy Logic Computation of Analogical Reasoning and Schemes The first main advance of Bernadette Bouchon-Meunier (BBM) about analogical reasoning is due to the hint of using the main principle of fuzzy logic: i.e. the gradualness of membership function. So, as crisp sets representing precise and certain descriptions of objects might be regarded as particular cases of fuzzy sets [10], tautology as identity (the sun is the sun), analogy as comparison (the nucleus of the atom is the sun of the solar system) and antilogy as metaphor (Juliet is the sun of Romeo) are fuzzy sets of special kinds. Identity is computed as a particular case of analogy. Analogy is computed as a particular case of metaphor, with transitivity, asymmetry and irreflexivity relations. The first main advance of BBM on analogical reasoning is due to her insight that the main principle of fuzzy logic might be the central core of analogical thinking 64 C. Tijus and reasoning. There is a graduality of membership function that can be used both for “John is a man” and “John is a cat”; certainty (John is a man) being a special case of uncertainty (John is a cat). So, as crisp sets that represent precise and certain descriptions of objects, they are to be regarded as particular cases of fuzzy sets [10]. Tautology as identity (the sun is the sun), analogy as comparison (the nucleus of the atom is as the sun of the solar system) and antilogy as metaphorical contradiction (Juliet is the sun although Juliet is not the sun) are all fuzzy sets of special kinds. Identity is computed as a particular case of analogy. Analogy is computed as a particular case of metaphor, with transitivity, asymmetry and irreflexivity relations. The second advance of BBM’s team about analogical and metaphorical reasoning is embedded by the first one. This advance relates to tautology that appears to be very useful in daily life activities,—as well as antilogy (In London, even when it’s not raining, it’s raining!) -, to provide useful information. However, according to classic logic, tautologies such as “the sun is the sun” and “Paris is Paris” are per se uninformative. Similarly, according to the Grice’s maxims of pragmatics [17], although they respect quality (truthful, supported by evidence) and manner (avoiding obscurity and ambiguity), tautologies violate two other maxims: quantity (to be as informative as possible) and relation (be pertinent). Here again with the BBM approach, tautologies and antilogies can be seen as special cases of graduality of certainty-uncertainty, in contradt to a full or null membership. As will be seen in the next section, the same categorization-based cognitive process appears to be a good candidate for the computing all of these forms of metaphorical reasoning [18]. The third advance is that analogical reasoning and metaphorical reasoning are cross-domains: a target object T (Tenor), is investigated from the point of view of a source that is used as a V (vehicle) for transmission of meanings. Having similes among the same category (tautology), among different categories of the same domain (analogy), among different categories across domains, or among inter-domains categories (metaphor), T and V can be computed according to the evaluation of their “closeness” through fuzzy modifiers in order to measure their similarity [16]. The advance is that closeness of two objects (i.e. moon and sun) as a semantic distance (very close, close, far, very far) can be described through fuzzy sets that can manage approximation. Closeness approximation depends on to T and V role (the moon-sun closeness in “the moon is a sun” being of a different value in “the sun is a moon”), according to context (the moon-sun closeness in “tonight the moon is a sun” being of a different value in “today the moon is a sun”), and motive (the moon-sun closeness in “in your drawing, the moon is a sun” being of a different value in “in the sky, the moon is a sun”). Moving on, we go further to develop (i) what analogy is and what analogy is not, (ii) analogical reasoning as being metaphorical reasoning, (iii) the powerful use of approximation and imprecision by the brain using analogies and metaphors, (iv) the categorical human resolution of analogies and metaphors through fuzzy inference-making and finally (v) models of solving analogies and metaphors; a section on artificial cognition could mimic human cognition for producing and understanding analogical and metaphorical thinking. Analogy 65 3 What Analogy is and What Analogy is Not Analogy might play an important role in epistemology, history of art, scientific discoveries and innovation, but also in the methods of doing art, science and techniques. There are many historical narratives about the emergence of new ideas, of discoveries and of problem solving. Analogy is generally described in the form “A is to B as C is to D” (A:B::C:D); where the source is some kind of substitute for the target for thinking and reasoning [19]. For instance, a well-known narrative is about how Archimedes found a solution to know whether the crown of King Hiero of Athens was really made out of pure gold, or if it was contaminated with cheap silver. After a long day of worrying, he decided to relax with a warm bath. When he entered the tub, he noticed the water level rising. This was something he knew, but now he suddenly realized that the water displacement was proportionate to the volume of the immersed part of his body. Then he put a weight of gold equal to the crown in a bowl filled with water. Next, the gold was removed and replaced by the crown. Silver would increase the bulk of the crown and cause the bowl to overflow. Thus, his body was a cognitive substitute for the crown to solve the problem. In everyday life, analogies are used and solved either for symbolic representation of things with verbalization, or when acting with physical things, or both. For instance, a verbal analogical problem solving in the form of “The railway is to the train as the sea is to the boat” (railway: train:: sea: boat) while the corresponding physical analogical problem solving is having the engineer in the restless train. Problem solving of verbal scholar analogies such as “The railway is to the train as the sea is to boat” (railway: train:: sea: boat). Problems have three terms, to solve for the fourth (e.g. A: train:: sea: boat) or have one set of two terms and their relation (railway: train:: C: D) to find the one that has the simile relation among other sets of objects (wheels: car; sea: boat; passengers: bus). Note first that, in contrast to the real world problems, these scholar problems have simile relations that are academic (e.g. synonym, antonym, part-to-whole, category/type, object-to-function, performer to related action, cause and effect, degree of intensity, and symbol and representation) and that the relation is given. For instance, in a real world problem, an engineer is asked, to correct and assure passengers comfort because “this train is a boat” making the relation under investigation implicit. It could be in that case that the train behaves as a boat because the train’s rails are to the passengers what the rough seas would be for the passengers of a boat. However, the relation could include other elements (e.g. due to the wind, due to the mountain shape as waves, and due to the wheels). Thus, the analogy is not in the form “A is to B as C is to D” but rather in the form “A is like C”. Secondly, as a matter of facts, not a single relationship but rather many are candidates for concluding the analogical solution. In addition, the conclusive solution can be made of a, set of relations with their interactions. Thirdly, analogy is supposedly done based on literal similarity comparisons [13, 20, 21]. However, between two natural knowledge domain, A and B, it is hard for individuals to evaluate the similarity between A and B. It is hard to evaluate the size 66 C. Tijus of the intersection (A ∩ B) of their feature sets as well as the alignment of these features: which feature in A matches a given feature in B. Such correspondence may also be at a given level of the decomposition tree of attributes but not in others (e.g. while red in A and blue in B do not match at the value level; they match as being a color). Such correspondence may also be at one of the dimensions of the domain description (e.g. dimension of surface, of structure, of function, of procedure, of dynamic behaviour). Fourthly, based on the comparison of A and C, the analogy in the form “A is to B as C is to D” can be permuted as “C is to D as A is to B.” We reasoned that such inter-domain comparisons require precise structural alignment and mapping [16] that must be hard to find. The equivalence between “A is to B” and “C is to B” should be rare, making analogical relations much more oriented from target to source than from source to target. This also means that analogy is from complex to simple. From this point of view, analogy is simplexity [22], given that meanings in everyday life language provide instructions to build understandable points of view [23]. Thus, many analogies appear to be assertions that have a metaphorical form, which is to say, not reversible: if New York is a big apple, a big apple is not New York. This metaphor asymmetry is found with classical analogical problem solving, for instance between two isomorphic sub-domains of algebra and physics [24]. When students, who learned one of these subtopics and are familiar with that source, are presented with a target problem based on the unfamiliar but analogous domain (as in metaphor) the source-to-target transfer is asymmetric. Students who had learned arithmetic (source) were very likely to transfer to physics (target). In contrast, students who had learned physics (source) almost never exhibited transfer to the isomorphic algebra problem (target). If physics is recognized as algebra, algebra is not recognized as physics. For studying the underlying cognitive processes of analogy resolution, an alternative of using already known natural domains is to build up unknown experimental isomorphic micro domains. The building up of experimental domains assert the analogical match of the two A and B descriptions, while minimizing the noise of complement sets of A’s features that are not in B (A–B) and the reverse (B–A). The prominent Cognitive science work of Herbert Simon and collaborators [25], as well as followers [26], is based on using domain-free puzzles such as the Tower of Hanoi (TOH) and the build up of term-to-term isomorphs. Thus, there is full similarity between two A and B TOH isomorphs that are done that way. The same space problem, the same transitions from state to state, each state in A is having its analog in B and the same minimum number of moves for reaching the goal state. Thus, the two TOH isomorphs have the same deep structure, but a different surface appearance. A persistent experimental result in the literature is that much exploration is often involved in solving some of the isomorphs, whose problem spaces are identical, but are packaged differently according to their surface properties. For instance, the classical 3-disks-TOH (e.g. A) is made of disks of permanent size that change place while a possible isomorphic problem (e.g. B) is made of disks of permanent place that change size. Although 7 moves are enough to solve both Analogy 67 problems, some of these are easy to solve with 11 moves on average (A) while others require up to 120 moves (B). Some isomorphs (as B) of the same problem space take 16 times as long to solve as other isomorphs. The difficulty varies by a factor of 16, depending on the surface characteristics. Studies of analogical transfer between isomorphs of these well defined puzzles [27, 28] show that solving a particular problem does not help solving an isomorphic one, except when conditions of transfer are based on generalization and internalization. On the contrary, from externalization according to surface properties and context, learning cannot be transferred when the problem content provides information that can be perceived and used without being explicitly interpreted and formulated. For instance, there are problems for which the solution requires discovering that each winning state is made of an even number. What are the conditions for transferring this solution to another isomorphic target problem? How much of the “even number” concept can be generalized to the same problem but have different surface properties? Here again, it is found that the analogical gain of solving isomorphic problems is asymmetric. Let’s examine two problems. The first one is about numbers. The second one is about tokens. When the problem that is solved first is solely with numbers, the discovery that number 4 is a winning state can be learned and used as a source for the isomorphic target problem with a number of tokens because four tokens is recognized as number 4, it is a winning state. Conversely, when the source problem is the problem with tokens, it is not helpful to solve the analogous target problem solely with numbers. The state number 4 in the target problem is not recognized as equivalent to four tokens in the source problem. Thus, because learning is based on generalization and deep structure, a problem that favors generalization and internalization, such as the problem with solely numbers, is a profitable source while a problem based on externalization through its surface properties, such as the problem with tokens, will impair transfer; except for other problems of same surface resemblance [28]. As physics is recognized as algebra, four tokens is recognized as number 4. As algebra is not recognized as physics, number 4 is not recognized as the equivalent of four tokens (one can externally manipulate four tokens without explicitly counting them). In summary, as noticed by BBM [9], suppose that the price of a house is to be determined. This can be done according to several criteria such as its size, its state, or its location. But because each criterion cannot be independently evaluated, we need references to evaluate the co-occurrences or correlation of attribute values. The needs of references could be satisfied with a large amount of data about home prices and values. A more simple and appropriate reasoning to evaluate the particular target house is to look for a known particular house that can be used as an intra-domain source for this case-based reasoning: “this house is a large villa: let’s compare it with a large villa we know”. The question under investigation is therefore the resemblance of source and target that might be based on comparing their corresponding features. Thus, the target house H1 (T) can be compared to the source house H2 (S) on properties that can be aligned and matched [21]. Within this intra domain comparison 68 C. Tijus of houses, most of the features, and therefore criteria, can be found in the (T ∩ S) intersection of their feature sets. It could be that some features (e.g. a swimming pool) belong to the (T-not S) set while others (e.g. a tennis court) belong to the (not T-S) set and that alignment might be difficult, but most of the reasoning is among the (T ∩ S) set of features that can be compared. In addition, target house H1 might serve as a source for house H2 and vice-versa. Unlike case-based reasoning, analogy is inter-domains. Someone could be searching for the “Rolls-Royce of houses” meaning a house that would be among houses what a Rolls-Royce is among cars. Someone else could have said “I found a house that is a Rolls Royce.” Since the two domains are different, the resemblance relations are fuzzy and the features and criteria are hard to be aligned and matched. Contrary to case-based reasoning, if a Rolls-Royce can serve as a source for a large villa, a large villa will hardly serve as a source for a car. The main reason (unlike case-based reasoning) the source is not a particular thing, but a category of things that serves to make the target inherit the properties of the category. If a Rolls Royce is luxury, opulence, very expensive, distinctive, solid, well known, then the house that is in the category of Rolls Royce will have these features. It will inherit these properties as being an attributive category just as a particular living being will inherit the properties of being a mammal [5]. In summary, we argue that analogy and metaphor are two faces of the same coin. 4 Models of Solving Analogies and Metaphors for Fuzzy Inference Making There are many cognitive models of analogy. See [29] for a review. As for other kinds of cognitive processes, there are two different approaches of modeling and simulating the resolution of analogies. The first approach is based on a bottom-up decentralized process such as the model named AMBR that stands for “Associative Memory-Based Reasoning.” It goes from local to global, blending episodic-contextual and long term memories. For solving analogies, memory and reasoning are highly integrated in Neural Nets and high level features are built starting from the local level. There is an initial distribution of activation resulting from previously solved problems as sources that can prime the relevant features of the target. The functioning of the model can be seen as a collection of basic units that are domain specific cognitive agents that collaborate according to the declarative (what) and procedural (how/why) knowledge they encapsulate. Thus, as agents (e.g. “railway” and “sea”) that entail what (support of train or boat) and how/why (for moving) collaborate to produce analogies such as “The railway is to the train as the sea is to the boat.” Similarly, “Juliet” and “sun” entail what (important) and how/why (for living) collaborate to produce metaphors such as “Juliet is my sun.” Analogy 69 The second approach is somewhat based on a top-down centralized process such the one used in the model named SMT that stands for “Structure Mapping Theory” [21, 30, 31]. In SMT, the similarity between the target and the source is evaluated by computing commonalities and differences. The former provides generalization, abstraction and schemas while the latter provides alignable differences, having “some expression in the base and some corresponding but different expression in the target.” The computation of commonalities and differences strengthens the structural alignment of features that guide the analogical process. These two approaches of analogy are based on the similarity computation of T and S; the comparison being symmetrical and reversible: T can be compared to S and S can be compared to T. When two objects, situations or domains are comparable, either one or the other can serve as a source or as target. However, this is a particular case of analogy. Most source-target analogies are oriented: a target can imply a source, while the reverse is not true. We assert that there is a clear distinction between physical world objects and the categories humans use to represent them, in order to think, talk and communicate about them [32]. Categories as sources for understanding can be used literally as in deduction and induction, but also unliterally in analogies and metaphors in a fuzzy way. As [33], we maintain that in both cases, categorizing a target as a source type yields unseen features. If someone reads that in a fictitious country “Xs are birds,” then one can infer not only that “Xs are animals,” but also that “Xs fly.” As semantic relations, analogies and metaphors are based on categorization. They activate a category and its attached features. They also activate the super-ordinate categories and their respective attached features. Our categorization approach of analogies is the one that has already been proposed for metaphors understanding [5, 18, 34] in which a source (e.g. sun) is a cognitive vehicle to transfer meanings to the target topic (e.g. Juliet) with the notion that vehicles in metaphors are attributive categories. In the past, most of the cognitive models of metaphor understanding have adopted the approach according to which metaphor is an implicit comparison: understanding a metaphor “X (topic) is Y (vehicle)” consists in converting it into a simile “X (the topic) is like Y (the vehicle)”. This comparison-based model of metaphor understanding is a mechanism of property matching. This is the reason these models are confronted with the problem of measuring the similarity of properties as well as with the problem of calculating the distance between properties, which makes a simile literal or unliteral—metaphoric. More recently, an alternative attributive categorization based approach is the Glucksberg’s class inclusion model: a metaphoric statement of the type “X is Y” is solved by looking for the category, represented by the term Y, which furnishes source properties that are potentially relevant for the target topic X. The general hypothesis is that metaphor understanding consists of including the topic in the category of the source-vehicle and attributing to it the properties of that category that are compatible with what is already known about the topic. We assume that interpretation is constructed on-line and that knowledge about the topic intervenes at an early stage in processing by constraining the selection of features. 70 C. Tijus BBM [10] noticed that crisp sets that represent precise and certain descriptions of objects might be regarded as particular cases of fuzzy sets. Similarly, we argue that deduction, induction, abduction, induction, analogy and metaphor are particular cases of metaphor, from the less fuzzy-certain, to the more fuzzy uncertain, analogy a particular case of metaphor. According to BBM [11], gradual reasoning can be obtained by using linguistic modifiers such as in [35, 36], the link between gradual reasoning and analogical reasoning corresponding to the utilization of a relationship between variations of X and variations of Y expressed in gradual knowledge to infer a value of Y from a given value of X. Thus BBM and collaborators introduced a general framework that represents analogy, on the basis of a link between variables and measures of comparison between values of variables. This analogical scheme is a common description of several forms of reasoning used in fuzzy control or in the management of knowledge-based systems, such as deductive reasoning, inductive reasoning or prototypical reasoning and gradual reasoning. A general model for the simulation of those modes of inference-making is a model based on a fuzzy semantic network [37]. Making hierarchies of categories of the semantic network with Galois Lattices [38] allows partonomy, which is the decomposition of an object into its physical parts (the what), and meronomy, which is the decomposition of a category description into its cognitive parts (the how and why as well as conceptual features). Unlike classical Galois Lattices, the inheritance of properties and membership link (e.g. “is a” for category; “is a kind of” for subordinate category to super-ordinate category), can be interval-valued for fuzzy inclusion all along the path from instances to subordinate categories, then to the highest general super-ordinate categories. Another fuzzy measure is the extent a given feature can possibly be the attached feature of a given category. Within the lineage of categories, it is important for the structure of a category to distinguish among levels of categorization [39]: among subordinate (siamese), basic level (cat) and super-ordinate (animal) categories. A distinction that can be made with the partonomy and meronomy decomposition of descriptions; allowing gradual evaluation of concreteness of categories as well as of the domain of comparisons. These fuzzy and categorical approaches of analogy differ from those that are based on similarity computed from features comparisons. These are two different cognitive approaches since psychological studies and in cognitive science show that similarity does not match categorization [20]. For instance, the similarity score (Russia, Cuba) is 7 and (Cuba, Jamaica) is 8, similarity (Russia, Jamaica) should be around 7, but the similarity score is 1! People categorize things differently from the way they evaluate things to be very similar: similar objects can belong to different categories while dissimilar objects can belong to the same category. BBM [11] proposed a fuzzy prototype-based reasoning for making and solving analogies. A fuzzy prototype of a category enables one to generate typicality and the set of relevant objects and therefore can be used for matching source to target, as Tverski’s proposal. The degree of typicality depends on both the resemblance to other objects in the same category and on the dissimilarity to objects in other categories. Thus, the analogical question at hand is: “does the target gradually satisfy Analogy 71 the prototype of the source category?” These are solutions for the different modes of reasoning, including analogy and metaphor. 5 Discussion For cognitive purposes, objects are psychologically grouped in categories. Once a category exists, it has an extension that includes all the instances of the category (even innumerable) and an intension that includes the properties (even innumerable) shared by the objects (e.g. unseen). Also for cognitive purposes, categories entail categories as well as forms of categorical hierarchies. The importance of categorization can be noticed from the following points of view: • Reasoning: As categorization corresponds often to an abductive process. When putting an object in an existing category, we provide it the “rule” or “set” of properties of the category; and because the “is-a-kind-of” relation entails modus ponens (Socrate is mortal because Socrate is a kind of person that is a kind of mortal), but not modus tollens (which is not based on categories; i.e. things that would be a non-person that are non-mortal); • Comprehension by inference: Since two things, or two categories, are put together, common properties of the super-ordinate category, act as a filter, indicating the “what-is-about” in terms of structure, functionality and usability. For instance, a piano and a guitar are put in “music and band playing music,” while a piano and a fridge are put in “large heavy objects” and “how to carry large heavy objects”; and • Comprehension of the world structure: Since a category in a hierarchy of categories factorize different kinds of properties and provide the causal links between procedure-function and structure because the “how-to-use” the object as well as the “in-order-to” will be based on other features of the object, such as structural properties. For instance, notice that in folk taxonomy “to have wings” and “flying” are properties of birds. Analogies and metaphors are usual modes of thinking and reasoning although based on false categorization: “electricity is like water,” or “this lawyer is really a shark.” Thus, there is a powerful use of approximation and imprecision by the brain using analogies and metaphors through fuzzy inference-making. In their prominent paper on the fuzzy approach to analogical reasoning, BBM and Valverde [12] address the problem of the representation of resemblances involved in analogical reasoning and use fuzzy relations to compare situations. As fuzziness entails the diverse forms of reasoning, from true literal sentences to false unliteral sentences, it is a powerful computation mode for human thinking and reasoning that is mainly metaphor and analogy-based reasoning. In addition, analogical and metaphorical sentences often include modifiers that BBM and Marsala put as the core of interpretable Fuzzy Systems [16]. For instance, when describing electricity, a common analogy is a water tank, where charge stands for the water amount, voltage 72 C. Tijus for the water pressure, and current for the water flow. They are said to be equivalent, but electricity is not used to explain water flow. There are even situations where the water analogy is rather misleading. Electricity is like water but they cannot be mixed. Water is largely used to produce electricity, not the reverse. One might think that a metaphor such as “this is truly a gem” means a true literal sentence although this is metaphorical “image-based language” that strengthens the metaphor. According to BBM [12], we use fuzzy relations to compare situations that can be used to model a natural analogy: resemblance relations can be used to define a kind of analogical scheme compatible with approximate reasoning in fuzzy logic, with measures of satisfiability, resemblance and inclusion. These fuzzy relations can be regarded as measures of a categorization process devoted to analogy and metaphor with the purpose of transmitting knowledge from the source to the target. References 1. Zibetti, E., Tijus, C.: Understanding actions: contextual dimensions and heuristics. In International and Interdisciplinary Conference on Modeling and Using Context, pp. 542–555. 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