The Problem of Context for Similarity: An Insight from Analogical Cognition

Round 1

Reviewer 1 Report

Authors revisit Goodman’s view on similarity and try to rescue similarity by invoking work on analogical cognition and salience. 

The paper has a clear aim and is well written. The argumentation is ok, but at some point the connections could be made a bit more explicit. 

I have one major concern about the paper, which can and should be taken up. Authors mention Goodman’s work on similarity and also Medin, Goldstone, and Gentner. Some important distinctions are not discussed and these are most relevant for the paper. After Goodman, several different theories about similarity have been proposed and elaborated:

geometrical models: similarity as distance in a similarity space (or in a space obtained by multi-dimensional scaling). Similarity is typically a decreasing exponential function of the distance between two points. This is useful for comparing color shades in color space.

featural models: the comparison of the number of properties seems to be the one used by the authors; strangely enough, the most famous featural model (Tversky 1977 Psychological Review) is not discussed

alignment models: structure is taken into account; this may play a role in authors’ account; a clear comparison with this view in the paper would be helpful

transformational models

graph-theoretic model (Carnap’s Aufbau and neo-Carnapians Mormann and Leitgeb)

Overviews are available in the literature, see, among others, “Similarity” by Goldstone and Son in the Cambridge Handbook of Thinking and Reasoning, eds. K. Holyoak & R. Morrison, 2005.

I have the impression that authors jump from one characterization to another, in particular from the geometrical model to the featural model. Moreover, the relation to the alignment model is not totally clear to me.

Minor remarks:

line 86: importance instead of important

line 130 objections plural?

line 189 the problem being …: awkward phrasing

Author Response

Reply Review 1

Comments and Suggestions for Authors

Authors revisit Goodman’s view on similarity and try to rescue similarity by invoking work on analogical cognition and salience. 

The paper has a clear aim and is well written. The argumentation is ok, but at some point the connections could be made a bit more explicit. 

We have made extensive corrections to improve the transitions between the different parts of the paper and clarify the connections between them.

In section 2.1, we have added a summary of Goodman’s objections at the end of the argumentation (p.4):

Therefore, we claim that this definition of similarity defeats most of Goodman’s objections against similarity. Similarity can’t be restricted to one shared property or a list of properties because it also integrates differing degrees of importance to each of those properties. Similarity sorts and weights properties and determines the most important in a context. A good approximation of similarity can be obtained by computing the few most important properties while keeping the open aspect of similarity stressed by Medin et al.[10] as a one of the most important characteristics of similarity.

We have also clarified some aspects of the example to link it with the overall argumentation (p.4):

However, adding a new property, like the presence of a small chink on the green glass and one of the blue glass, which is quite irrelevant to our judgement, with a weight of 0.05, we have a similarity of respectively 2.5, 1.75 and 1.8, which does not significantly change our representation of the three glasses. We could even say that the attributed similarity of the three glasses is the same with or without the added characteristic of the chink. Therefore, if the computation of similarity is restricted to the few most important properties, a good approximation of the similarity value is obtained, even though it is open to potential new properties.

This definition also supports the flexibility of context. Suppose we are not in a neutral context but in a goal-oriented one, where we want to choose which glass to throw away. Then we ascribe a weight of 0.9 to any crack or chink in the glass, which gives us a similarity of 2.5 between A and B, 1.75 between A and C and 2.65 between B and C. The glasses B and C are then considered more similar than A and B because we have changed the value of p for one element in the new context. The new property which was unimportant before becomes central in this context.

In section 3, we summarized the problem tackled by the analogical process before giving the actual description of the process (p.5).

The analogical process is therefore dealing with the problem of determining which properties are the more important to understand the analogy. A subject who would only see the common spherical shape of all the elements would miss the most interesting aspects of the analogy. They would therefore be unable to learn the causal relation between the attraction of the nucleus on the electrons and the revolution of the electrons around the nucleus.

We reorganized some parts of the article to clarify the link between them. The description of the interactions with several aspects of our cognitive system is switched under the label “Overview of the analogical process” (section 3.1, p.5-6).

The analogical process aims at describing the cognitive steps through which one can reach a proper understanding of the analogy.

The analogical process interacts with four major parts of our cognitive system: perception, long-term memory, working memory and goals.

•            Perception handles the stimuli given by the sensory system and feeds this information to the working memory, where the analogical process takes place.

•            The analogical process seeks a source situation from the long-term memory.

•            The long-term memory gives back a source situation or a category, that is, a constructed representation of some acquired knowledge.

•            The analogical process compares the two situations in the working memory, mapping the various parts together.

It attributes some evaluation to the mapping and makes inferences from the source situation and applies them to the target situation. The second representation of the target situation is richer and more complex. It contains some knowledge from previous experiences.

And the summary of how the analogical process can be considered a sorting function is moved just before the problems it encounters (section 3.2, p.9):

The goal of the analogical reasoning is to identify the shared properties and dismiss the irrelevant differences. It must also identify the most relevant properties the two situations have in common [14]: the atom and the solar system are not similar because they have spherical elements for example but because the causal relation between their centres and peripheric elements is the same. Not only the analogical process dismisses irrelevant differences, it also emphasizes the most important common properties, among all properties. Therefore, the analogical process can be considered as a sorting function.

We clarified with an example what role structural alignment plays in the analogical process in order to help one understand what is our position towards Gentner’s theory (p.9).

Structural alignment measures the number of connexions created among the predicates. The analogical process aims at maximizing the number of connexions and more importantly for Gentner, of higher-level connexion. For example, the spherical shapes of the elements of the atom and solar system is an isolated property and will be dismissed for having a very low structural power. On the contrary, the relations of attraction and revolution are connected by a causal link. Therefore, the analogical process will evaluate those relations as more important, because they join more properties and reveal a common causal structure between the two systems.

I have one major concern about the paper, which can and should be taken up. Authors mention Goodman’s work on similarity and also Medin, Goldstone, and Gentner. Some important distinctions are not discussed and these are most relevant for the paper. After Goodman, several different theories about similarity have been proposed and elaborated:

geometrical models: similarity as distance in a similarity space (or in a space obtained by multi-dimensional scaling). Similarity is typically a decreasing exponential function of the distance between two points. This is useful for comparing color shades in color space.

featural models: the comparison of the number of properties seems to be the one used by the authors; strangely enough, the most famous featural model (Tversky 1977 Psychological Review) is not discussed

alignment models: structure is taken into account; this may play a role in authors’ account; a clear comparison with this view in the paper would be helpful

transformational models

graph-theoretic model (Carnap’s Aufbau and neo-Carnapians Mormann and Leitgeb)

Overviews are available in the literature, see, among others, “Similarity” by Goldstone and Son in the Cambridge Handbook of Thinking and Reasoning, eds. K. Holyoak & R. Morrison, 2005.

I have the impression that authors jump from one characterization to another, in particular from the geometrical model to the featural model. Moreover, the relation to the alignment model is not totally clear to me.

Thank you very much for the reference to Goldstone and Son’s paper. We have clarified our position in the introduction (p.2):

Goldstone and Son [5] survey several theories of similarity. They distinguish four major families of models: geometric models, featural models, structural alignment models and transformational models. We can unite, as suggested in their article, the two first families of models into one [3,6], which focuses on unstructured representations: a list of unrelated features or dimensions. The authors also mention some attempts [7] to link the last two families, both of which highlight the underlying structure of the representation and relations between several properties.

Mainly it is the first two families of models that have been taken into account in discussion of similarity [8]. They propose to consider similarity as a function of the features or dimensions shared by two objects. However, they don’t solve the problem of context as proposed by Goodman, because they integrate features or dimensions without considering any order or structure between them. They don’t tackle the problem of the respects of similarity, without which similarity is meaningless according to Goodman.

Rather than giving up on similarity, we would like to examine some advantage of the structural alignment models, which have been mainly overlooked in the recent philosophical discussions[9]. This family of models rests upon works on analogical cognition. We would like to examine the particularity of the analogical process to consider it as a potential complement for featural/dimensional accounts of similarity. We suggest that the two approaches (featural and structural) are not opposed but complementary. Similarity can integrate both featural properties and relational properties.

We have also given some historical background on the discussion of similarity between Carnap, Goodman and Tversky in the beginning of section 2.1 (p. 2-3). This paragraph also disentangles the geometrical and the featural account of similarity.

In 1972 [4], Goodman presents several arguments against similarity which leads him to dismiss the notion entirely. His rebuttal is mainly addressed to Carnap’s geometrical model [10] of similarity. Nevertheless, some of his objections are also problematic for featural models of similarity and any other account of similarity, particularly his focus on the respects in which two things are similar.

Carnap [10] proposed a geometrical account of similarity which has been central to discussion of similarity in the twentieth century [8]. The geometrical model describes similarity in a geometrical space characterized by its properties: minimality, symmetry and triangle inequality. Similarity is defined as the inverse distance between two objects: the closer two objects are in the space, the more similar they are. Goodman criticizes this approach on several points. One of his major points is the insensitivity to context: the appreciation of similarity depends highly on who makes the judgement and in which circumstances. Several studies in psychology [6,11] have shown that the properties of the geometrical space (minimality, symmetry and triangle inequality) are often transgressed when people make judgement of similarity.

Tversky [11] suggests another model to account for the context-sensitivity of similarity: the featural model. Similarity is defined as a function of shared and distinctive features, weighted with parameters which are influenced by context. Those parameters emphasize the shared features in some context or the distinctive features in some other. This model explains why judgements of similarity are not symmetric: North Korea is more similar to China than China is similar to North Korea because the features of the subject of the comparison are weighted more heavily. As China has more distinctive features than North Korea, the distinctive features overweight the shared features when it is the subject of the comparison. On the contrary, North Korea has very few distinctive features and most of its features are shared with China, therefore amplifying their judged similarity.

However, if this account resolves the context-sensitivity of similarity, it doesn’t answer Goodman’s other objection against Carnap’s account of similarity: the aspect in which two things are similar. Attributing a distance between two objects as a measure of similarity treats all features as having equal weight in assigning similarity. This is a characteristic of Tversky's model. Goodman, by contrast, argues that different properties vary in their importance for similarity. Goodman proposes for consideration, but dismisses just afterwards, a qualitative definition of similarity in place of a quantitative definition: similarity corresponds to the respects in which two things are similar.

Minor remarks:

line 86: importance instead of important

line 130 objections plural?

line 189 the problem being …: awkward phrasing

Corrections have been made.

Reviewer 2 Report

In my opinion, the revision of this manuscript has been successful. What we have now is an interesting and credible response to Goodman, 1972. The role of similarity in cognition can now be constructively viewed as depending on analogical reasoning and its heuristic inferences.

Author Response

Reply Review 2

Comments and Suggestions for Authors

In my opinion, the revision of this manuscript has been successful. What we have now is an interesting and credible response to Goodman, 1972. The role of similarity in cognition can now be constructively viewed as depending on analogical reasoning and its heuristic inferences.

Thank to another reviewer’s the reference to Goldstone and Son’s paper which survey the different theories of similarity, we have clarified our position among those theories in the introduction (p.2):

Goldstone and Son [5] survey several theories of similarity. They distinguish four major families of models: geometric models, featural models, structural alignment models and transformational models. We can unite, as suggested in their article, the two first families of models into one [3,6], which focuses on unstructured representations: a list of unrelated features or dimensions. The authors also mention some attempts [7] to link the last two families, both of which highlight the underlying structure of the representation and relations between several properties.

Mainly it is the first two families of models that have been taken into account in discussion of similarity [8]. They propose to consider similarity as a function of the features or dimensions shared by two objects. However, they don’t solve the problem of context as proposed by Goodman, because they integrate features or dimensions without considering any order or structure between them. They don’t tackle the problem of the respects of similarity, without which similarity is meaningless according to Goodman.

Rather than giving up on similarity, we would like to examine some advantage of the structural alignment models, which have been mainly overlooked in the recent philosophical discussions[9]. This family of models rests upon works on analogical cognition. We would like to examine the particularity of the analogical process to consider it as a potential complement for featural/dimensional accounts of similarity. We suggest that the two approaches (featural and structural) are not opposed but complementary. Similarity can integrate both featural properties and relational properties.

We have also given some historical background on the discussion of similarity between Carnap, Goodman and Tversky in the beginning of section 2.1 (p. 2-3). This paragraph also disentangles the geometrical and the featural account of similarity.

In 1972 [4], Goodman presents several arguments against similarity which leads him to dismiss the notion entirely. His rebuttal is mainly addressed to Carnap’s geometrical model [10] of similarity. Nevertheless, some of his objections are also problematic for featural models of similarity and any other account of similarity, particularly his focus on the respects in which two things are similar.

Carnap [10] proposed a geometrical account of similarity which has been central to discussion of similarity in the twentieth century [8]. The geometrical model describes similarity in a geometrical space characterized by its properties: minimality, symmetry and triangle inequality. Similarity is defined as the inverse distance between two objects: the closer two objects are in the space, the more similar they are. Goodman criticizes this approach on several points. One of his major points is the insensitivity to context: the appreciation of similarity depends highly on who makes the judgement and in which circumstances. Several studies in psychology [6,11] have shown that the properties of the geometrical space (minimality, symmetry and triangle inequality) are often transgressed when people make judgement of similarity.

Tversky [11] suggests another model to account for the context-sensitivity of similarity: the featural model. Similarity is defined as a function of shared and distinctive features, weighted with parameters which are influenced by context. Those parameters emphasize the shared features in some context or the distinctive features in some other. This model explains why judgements of similarity are not symmetric: North Korea is more similar to China than China is similar to North Korea because the features of the subject of the comparison are weighted more heavily. As China has more distinctive features than North Korea, the distinctive features overweight the shared features when it is the subject of the comparison. On the contrary, North Korea has very few distinctive features and most of its features are shared with China, therefore amplifying their judged similarity.

However, if this account resolves the context-sensitivity of similarity, it doesn’t answer Goodman’s other objection against Carnap’s account of similarity: the aspect in which two things are similar. Attributing a distance between two objects as a measure of similarity treats all features as having equal weight in assigning similarity. This is a characteristic of Tversky's model. Goodman, by contrast, argues that different properties vary in their importance for similarity. Goodman proposes for consideration, but dismisses just afterwards, a qualitative definition of similarity in place of a quantitative definition: similarity corresponds to the respects in which two things are similar.

We have made extensive corrections to improve the transitions between the different parts of the paper and clarify the connections between them.

In section 2.1, we have added a summary of Goodman’s objections at the end of the argumentation (p.4):

Therefore, we claim that this definition of similarity defeats most of Goodman’s objections against similarity. Similarity can’t be restricted to one shared property or a list of properties because it also integrates differing degrees of importance to each of those properties. Similarity sorts and weights properties and determines the most important in a context. A good approximation of similarity can be obtained by computing the few most important properties while keeping the open aspect of similarity stressed by Medin et al.[10] as a one of the most important characteristics of similarity.

We have also clarified some aspects of the example to link it with the overall argumentation (p.4):

However, adding a new property, like the presence of a small chink on the green glass and one of the blue glass, which is quite irrelevant to our judgement, with a weight of 0.05, we have a similarity of respectively 2.5, 1.75 and 1.8, which does not significantly change our representation of the three glasses. We could even say that the attributed similarity of the three glasses is the same with or without the added characteristic of the chink. Therefore, if the computation of similarity is restricted to the few most important properties, a good approximation of the similarity value is obtained, even though it is open to potential new properties.

This definition also supports the flexibility of context. Suppose we are not in a neutral context but in a goal-oriented one, where we want to choose which glass to throw away. Then we ascribe a weight of 0.9 to any crack or chink in the glass, which gives us a similarity of 2.5 between A and B, 1.75 between A and C and 2.65 between B and C. The glasses B and C are then considered more similar than A and B because we have changed the value of p for one element in the new context. The new property which was unimportant before becomes central in this context.

In section 3, we summarized the problem tackled by the analogical process before giving the actual description of the process (p.5).

The analogical process is therefore dealing with the problem of determining which properties are the more important to understand the analogy. A subject who would only see the common spherical shape of all the elements would miss the most interesting aspects of the analogy. They would therefore be unable to learn the causal relation between the attraction of the nucleus on the electrons and the revolution of the electrons around the nucleus.

We reorganized some parts of the article to clarify the link between them. The description of the interactions with several aspects of our cognitive system is switched under the label “Overview of the analogical process” (section 3.1, p.5-6).

The analogical process aims at describing the cognitive steps through which one can reach a proper understanding of the analogy.

The analogical process interacts with four major parts of our cognitive system: perception, long-term memory, working memory and goals.

•            Perception handles the stimuli given by the sensory system and feeds this information to the working memory, where the analogical process takes place.

•            The analogical process seeks a source situation from the long-term memory.

•            The long-term memory gives back a source situation or a category, that is, a constructed representation of some acquired knowledge.

•            The analogical process compares the two situations in the working memory, mapping the various parts together.

It attributes some evaluation to the mapping and makes inferences from the source situation and applies them to the target situation. The second representation of the target situation is richer and more complex. It contains some knowledge from previous experiences.

And the summary of how the analogical process can be considered a sorting function is moved just before the problems it encounters (section 3.2, p.9):

The goal of the analogical reasoning is to identify the shared properties and dismiss the irrelevant differences. It must also identify the most relevant properties the two situations have in common [14]: the atom and the solar system are not similar because they have spherical elements for example but because the causal relation between their centres and peripheric elements is the same. Not only the analogical process dismisses irrelevant differences, it also emphasizes the most important common properties, among all properties. Therefore, the analogical process can be considered as a sorting function.

We clarified with an example what role structural alignment plays in the analogical process in order to help one understand what is our position towards Gentner’s theory (p.9).

Structural alignment measures the number of connexions created among the predicates. The analogical process aims at maximizing the number of connexions and more importantly for Gentner, of higher-level connexion. For example, the spherical shapes of the elements of the atom and solar system is an isolated property and will be dismissed for having a very low structural power. On the contrary, the relations of attraction and revolution are connected by a causal link. Therefore, the analogical process will evaluate those relations as more important, because they join more properties and reveal a common causal structure between the two systems.

Round 2

Reviewer 1 Report

Remarks from the previous review round have been addressed.

The overall structure could be a bit tighter.