One of Machery's central arguments for his Heterogeneity Hypothesis in Doing without Concepts (Machery Reference Machery2009) is that, for each different theory of concepts, there are reliable data-sets – behavioral, neuroscientific, and dissociation – that are best (or only) explained by that theory. That is, prototype-based, exemplar-based, and theory-based kinds of concepts are all required in order to explain all of the data. Thus, Machery concludes, we have heterogeneity: for many of the cognitive items thought to be single concepts, people have cognitively distinct prototype-like, exemplar-like, and theory-like kinds of concepts.
This argument requires that these three theories/kinds of concepts be qualitatively different, not special cases of a more general framework. If there were a unifying account that encompassed all three theories of concepts, then that theory would (by hypothesis) be able to explain all the same data as the heterogeneity hypothesis that Machery proposes in his book. Hybrid theories of concepts are the most prominent instances of such a unifying account, and Machery provides numerous arguments against them precisely because they have the potential to undermine his data-based argument for the heterogeneity hypothesis. Hybrid theories are not, however, the only way to unify the three theories of concepts.
The behavioral data used in Machery's argument can all, I contend, be explained by a single theory of concepts based on graphical models. At a (very) high level, a graphical model encodes a set of relationships, whether informational/probabilistic, causal, communication, taxonomic, or other. More formally, a graphical model has two components: (1) a graph composed of nodes/vertices and edges (directed and/or undirected) that encode the qualitative relationships; and (2) a representation of the quantitative relationships. The precise quantitative component depends on model-specific features; joint probability distributions and structural/linear equations are typical. Bayesian networks (causal or probabilistic), structural equation models, and Markov random fields/networks are probably the most common graphical models, though there are many other types. (Lauritzen [Reference Lauritzen1996] provides a comprehensive formal overview of graphical models.)
One instance of the theory-theory (causal model theory) is already explicitly formulated in the language of graphical models (specifically, causal Bayesian networks). A deeper connection is suggested by the fact that all three theories of concepts ultimately understand concepts as structured relations among features, components, causes and effects, and so on, though there are between-theory differences in the nature of the objects and relations.
The suggested deep connection exists: All three theories of concepts can provably be represented in the graphical models framework (Danks Reference Danks2004; Reference Danks, Gopnik and Schulz2007). More precisely, for each particular prototype-based, exemplar-based, or theory-basedFootnote
1
theory of concepts, there is a corresponding class of graphical models such that (A) there is a one-to-one mapping between (i) a concept in the “traditional” psychological theory, and (ii) a particular graphical model in that class; and (B) inference/reasoning using that graphical model is behaviorally indistinguishable from using the corresponding prototype-based, exemplar-based, or theory-based concept. As a concrete example, multiplicative prototype concepts with second-order features are isomorphic to Markov random fields with restricted clique potential functions. (Danks [2004; 2007] proved this result for categorization; the results have since been extended to all of the activities that Machery discusses.) These three “kinds” of concepts are (at least, formally) each representable as a different set of restrictions within the shared graphical models framework, specifically the framework of so-called chain graphs.
These formal results open the door for homogeneity to reemerge as a live possibility. Of course, this homogeneity occurs at the level of graphical models, rather than exemplars versus prototypes versus theories. An individual can have what appear to be both prototype-based concepts and exemplar-based concepts simply by having different specific graphical models. The force of the argument for heterogeneity is thus blunted: There is a single theory that can account for all of the (behavioral) data. Importantly, this proposed theory is not a hybrid theory: It does not hold that a single concept is composed of different graphical models corresponding to the different types of concepts. Rather, the proposal is that any particular concept corresponds to only one graphical model, and diversity in our graphical models leads to apparent diversity in types of concepts.
This account focuses on the formal/computational structure of the different theories of concepts, and one might object that it ignores other, more “metaphysical” claims made by their proponents. This concern goes to the hard question of how to define or characterize a theory. At one extreme, we could define a theory (of concepts) by its behavioral predictions, and ignore all other assertions made about the theory. At the other extreme, we could say that a theory is given by a maximal set of consistent claims made by proponents of that theory, regardless of the nature of those claims. The graphical models account takes the intermediate view that a theory should be interpreted relatively minimally as the set of claims necessary to explain the phenomena that purportedly fall within its reach (similar in spirit to the approach in Strevens Reference Strevens2000). I contend that (though I do not have room to argue for) the proper “minimal” understanding of theories of concepts is in terms of the formal/computational structure that they attribute to particular concepts. Although there is a clear rhetorical difference between saying a concept is based on a “prototype” versus an “exemplar,” I contend that the difference matters only when it leads to a difference in the concept's formal/computational structure.
Machery provides an admirable survey of the vast behavioral/psychological literature on concepts, and his conclusion – there are at least three distinct kinds of concepts – is reasonable if there is no unifying framework for those “kinds.” But all of the behavioral data that Machery discusses can in fact be explained as bodies of knowledge encoded as graphical models: The differences in behavior can be explained as different graphical structures – that is, different special cases – rather than truly distinct kinds.
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One of Machery's central arguments for his Heterogeneity Hypothesis in Doing without Concepts (Machery Reference Machery2009) is that, for each different theory of concepts, there are reliable data-sets – behavioral, neuroscientific, and dissociation – that are best (or only) explained by that theory. That is, prototype-based, exemplar-based, and theory-based kinds of concepts are all required in order to explain all of the data. Thus, Machery concludes, we have heterogeneity: for many of the cognitive items thought to be single concepts, people have cognitively distinct prototype-like, exemplar-like, and theory-like kinds of concepts.
This argument requires that these three theories/kinds of concepts be qualitatively different, not special cases of a more general framework. If there were a unifying account that encompassed all three theories of concepts, then that theory would (by hypothesis) be able to explain all the same data as the heterogeneity hypothesis that Machery proposes in his book. Hybrid theories of concepts are the most prominent instances of such a unifying account, and Machery provides numerous arguments against them precisely because they have the potential to undermine his data-based argument for the heterogeneity hypothesis. Hybrid theories are not, however, the only way to unify the three theories of concepts.
The behavioral data used in Machery's argument can all, I contend, be explained by a single theory of concepts based on graphical models. At a (very) high level, a graphical model encodes a set of relationships, whether informational/probabilistic, causal, communication, taxonomic, or other. More formally, a graphical model has two components: (1) a graph composed of nodes/vertices and edges (directed and/or undirected) that encode the qualitative relationships; and (2) a representation of the quantitative relationships. The precise quantitative component depends on model-specific features; joint probability distributions and structural/linear equations are typical. Bayesian networks (causal or probabilistic), structural equation models, and Markov random fields/networks are probably the most common graphical models, though there are many other types. (Lauritzen [Reference Lauritzen1996] provides a comprehensive formal overview of graphical models.)
One instance of the theory-theory (causal model theory) is already explicitly formulated in the language of graphical models (specifically, causal Bayesian networks). A deeper connection is suggested by the fact that all three theories of concepts ultimately understand concepts as structured relations among features, components, causes and effects, and so on, though there are between-theory differences in the nature of the objects and relations.
The suggested deep connection exists: All three theories of concepts can provably be represented in the graphical models framework (Danks Reference Danks2004; Reference Danks, Gopnik and Schulz2007). More precisely, for each particular prototype-based, exemplar-based, or theory-basedFootnote 1 theory of concepts, there is a corresponding class of graphical models such that (A) there is a one-to-one mapping between (i) a concept in the “traditional” psychological theory, and (ii) a particular graphical model in that class; and (B) inference/reasoning using that graphical model is behaviorally indistinguishable from using the corresponding prototype-based, exemplar-based, or theory-based concept. As a concrete example, multiplicative prototype concepts with second-order features are isomorphic to Markov random fields with restricted clique potential functions. (Danks [2004; 2007] proved this result for categorization; the results have since been extended to all of the activities that Machery discusses.) These three “kinds” of concepts are (at least, formally) each representable as a different set of restrictions within the shared graphical models framework, specifically the framework of so-called chain graphs.
These formal results open the door for homogeneity to reemerge as a live possibility. Of course, this homogeneity occurs at the level of graphical models, rather than exemplars versus prototypes versus theories. An individual can have what appear to be both prototype-based concepts and exemplar-based concepts simply by having different specific graphical models. The force of the argument for heterogeneity is thus blunted: There is a single theory that can account for all of the (behavioral) data. Importantly, this proposed theory is not a hybrid theory: It does not hold that a single concept is composed of different graphical models corresponding to the different types of concepts. Rather, the proposal is that any particular concept corresponds to only one graphical model, and diversity in our graphical models leads to apparent diversity in types of concepts.
This account focuses on the formal/computational structure of the different theories of concepts, and one might object that it ignores other, more “metaphysical” claims made by their proponents. This concern goes to the hard question of how to define or characterize a theory. At one extreme, we could define a theory (of concepts) by its behavioral predictions, and ignore all other assertions made about the theory. At the other extreme, we could say that a theory is given by a maximal set of consistent claims made by proponents of that theory, regardless of the nature of those claims. The graphical models account takes the intermediate view that a theory should be interpreted relatively minimally as the set of claims necessary to explain the phenomena that purportedly fall within its reach (similar in spirit to the approach in Strevens Reference Strevens2000). I contend that (though I do not have room to argue for) the proper “minimal” understanding of theories of concepts is in terms of the formal/computational structure that they attribute to particular concepts. Although there is a clear rhetorical difference between saying a concept is based on a “prototype” versus an “exemplar,” I contend that the difference matters only when it leads to a difference in the concept's formal/computational structure.
Machery provides an admirable survey of the vast behavioral/psychological literature on concepts, and his conclusion – there are at least three distinct kinds of concepts – is reasonable if there is no unifying framework for those “kinds.” But all of the behavioral data that Machery discusses can in fact be explained as bodies of knowledge encoded as graphical models: The differences in behavior can be explained as different graphical structures – that is, different special cases – rather than truly distinct kinds.
ACKNOWLEDGMENT
The author is partially supported by a James S. McDonnell Foundation Scholar Award.