{"title": "Quantification and the language of thought", "book": "Advances in Neural Information Processing Systems", "page_first": 943, "page_last": 951, "abstract": "Many researchers have suggested that the psychological complexity of a concept is related to the length of its representation in a language of thought.  As yet, however, there are few concrete proposals about the nature of this language. This paper makes one such proposal: the language of thought allows first order quantification (quantification over objects) more readily than second-order quantification (quantification over features). To support this proposal we present behavioral results from a concept learning study inspired by the work of Shepard, Hovland and Jenkins.\"", "full_text": "Quanti\ufb01cation and the language of thought\n\nCharles Kemp\n\nDepartment of Psychology\nCarnegie Mellon University\n\nckemp@cmu.edu\n\nAbstract\n\nMany researchers have suggested that the psychological complexity of a concept\nis related to the length of its representation in a language of thought. As yet,\nhowever, there are few concrete proposals about the nature of this language. This\npaper makes one such proposal: the language of thought allows \ufb01rst order quanti\ufb01-\ncation (quanti\ufb01cation over objects) more readily than second-order quanti\ufb01cation\n(quanti\ufb01cation over features). To support this proposal we present behavioral re-\nsults from a concept learning study inspired by the work of Shepard, Hovland and\nJenkins.\n\nHumans can learn and think about many kinds of concepts, including natural kinds such as elephant\nand water and nominal kinds such as grandmother and prime number. Understanding the mental\nrepresentations that support these abilities is a central challenge for cognitive science. This paper\nproposes that quanti\ufb01cation plays a role in conceptual representation\u2014for example, an animal X\nquali\ufb01es as a predator if there is some animal Y such that X hunts Y . The concepts we consider\nare much simpler than real-world examples such as predator, but even simple laboratory studies can\nprovide important clues about the nature of mental representation.\nOur approach to mental representation is based on the language of thought hypothesis [1]. As\npursued here, the hypothesis proposes that mental representations are constructed in a compositional\nlanguage of some kind, and that the psychological complexity of a concept is closely related to\nthe length of its representation in this language [2, 3, 4]. Following previous researchers [2, 4],\nwe operationalize the psychological complexity of a concept in terms of the ease with which it is\nlearned and remembered. Given these working assumptions, the remaining challenge is to specify\nthe representational resources provided by the language of thought. Some previous studies have\nrelied on propositional logic as a representation language [2, 5], but we believe that the resources\nof predicate logic are needed to capture the structure of many human concepts. In particular, we\nsuggest that the language of thought can accommodate relations, functions, and quanti\ufb01cation, and\nfocus here on the role of quanti\ufb01cation.\nOur primary proposal is that quanti\ufb01cation is supported by the language of thought, but that quan-\nti\ufb01cation over objects is psychologically more natural than quanti\ufb01cation over features. To test this\nidea we compare concept learning in two domains which are very similar except for one critical\ndifference: one domain allows quanti\ufb01cation over objects, and the other allows quanti\ufb01cation over\nfeatures. We consider several logical languages that can be used to formulate concepts in both do-\nmains, and \ufb01nd that learning times are best predicted by a language that supports quanti\ufb01cation over\nobjects but not features.\nOur work illustrates how theories of mental representation can be informed by comparing concept\nlearning across two or more domains. Existing studies work with a range of domains, and it is useful\nto consider a \u201cconceptual universe\u201d that includes these possibilities along with many others that have\nnot yet been studied. Table 1 charts a small fragment of this universe, and the penultimate column\nshows example stimuli that will be familiar from previous studies of concept learning. Previous\nstudies have made important contributions by choosing a single domain in Table 1 and explaining\n\n1\n\n\fwhy some concepts within this domain are easier to learn than others [2, 4, 6, 7, 8, 9]. Comparisons\nacross domains can also provide important information about learning and mental representation,\nand we illustrate this claim by comparing learning times across Domains 3 and 4.\nThe next section introduces the conceptual universe in Table 1 in more detail. We then present a\nformal approach to concept learning that relies on a logical language and compare three candidate\nlanguages. Language OQ (for object quanti\ufb01cation) supports quanti\ufb01cation over objects but not fea-\ntures, language F Q (for feature quanti\ufb01cation) supports quanti\ufb01cation over features but not objects,\nand language OQ + F Q supports quanti\ufb01cation over both objects and features. We use these lan-\nguages to predict learning times across Domains 3 and 4, and present an experiment which suggests\nthat language OQ comes closest to the language of thought.\n\n1 The conceptual universe\n\nTable 1 provides an organizing framework for thinking about the many domains in which learning\ncan occur. The table includes 8 domains, each of which is de\ufb01ned by specifying some number of\nobjects, features, and relations, and by specifying the range of each feature and each relation. We\nrefer to the elements in each domain as items, and the penultimate column of Table 1 shows items\nfrom each domain. The \ufb01rst row shows a domain commonly used by studies of Boolean concept\nlearning. Each item in this domain includes a single object a and speci\ufb01es whether that object\nhas value v1 (small) or v2 (large) on feature F (size), value v3 (white) or v4 (gray) on feature G\n(color), and value v5 (vertical) or v6 (horizontal) on feature H (texture). Domain 2 also includes\nthree features, but now each item includes three objects and each feature applies to only one of the\nobjects. For example, feature H (texture) applies to only the third object in the domain (i.e. the third\nsquare on each card). Domain 3 is similar to Domain 1, but now the three features can be aligned\u2014\nfor any given item each feature will be absent (value 0) or present. The example in Table 1 uses three\nfeatures (boundary, dots, and slash) that can each be added to an unadorned gray square. Domain 4\nis similar to Domain 2, but again the feature values can be aligned, and the feature for each object\nwill be absent (value 0) or present. Domains 5 and 6 are similar to domains 2 and 4 respectively, but\neach one includes relations rather than features. In Domain 6, for example, the relation R assigns\nvalue 0 (absent) or value 1 (present) to each undirected pair of objects.\nThe \ufb01rst six domains in Table 1 are all variants of Domain 1, which is the domain typically used by\nstudies of Boolean concept learning. Focusing on six related domains helps to establish some of the\ndimensions along which domains can differ, but the \ufb01nal two domains in Table 1 show some of the\nmany alternative possibilities. Domain 7 includes two categorical features, each of which takes three\nrather than two values. Domain 8 is similar to Domain 6, but now the number of objects is 6 rather\nthan 3 and relation R is directed rather than undirected. To mention just a handful of possibilities\nwhich do not appear in Table 1, domains may also have categorical features that are ordered (e.g.\na size feature that takes values small, medium, and large), continuous valued features or relations,\nrelations with more than two places, and objects that contain sub-objects or parts.\nSeveral learning problems can be formulated within any given domain. The most basic is to learn a\nsingle item\u2014for example, a single item from Domain 8 [4]. A second problem is to learn a class of\nitems\u2014for example, a class that includes four of the items in Domain 1 and excludes the remaining\nfour [6]. Learning an item class can be formalized as learning a unary predicate de\ufb01ned over items,\nand a natural extension is to consider predicates with two or more arguments. For example, problems\nof the form A is to B as C is to ? can be formulated as problems where the task is to learn a binary\nrelation analogous(\u00b7, \u00b7) given the single example analogous(A, B). Here, however, we focus on the\ntask of learning item classes or unary predicates.\nSince we focus on the role of quanti\ufb01cation, we will work with domains where quanti\ufb01cation is\nappropriate. Quanti\ufb01cation over objects is natural in cases like Domain 4 where the feature values\nfor all objects can be aligned. Note, for example, that the statement \u201cevery object has its feature\u201d\npicks out the \ufb01nal example item in Domain 4 but that no such statement is possible in Domain 2.\nQuanti\ufb01cation over features is natural in cases like Domain 3 where the ranges of each feature can be\naligned. For example, \u201cobject a has all three features\u201d picks out the \ufb01nal example item in Domain 3\nbut no such statement is possible in Domain 1. We therefore focus on Domains 3 and 4, and explore\nthe problem of learning item classes in each domain.\n\n2\n\n\f.\nf\ne\nR\n\n]\n1\n1\n\n,\n0\n1\n,\n7\n\n,\n6\n\n,\n2\n[\n\n]\n6\n[\n\n]\n2\n1\n[\n\n]\n6\n[\n\n]\n3\n1\n[\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n.\n\n.\n\n.\n\n,\n\n.\n\n.\n\n.\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n]\n9\n\n,\n8\n[\n\n,\n\n]\n4\n[\n\n.\n\n.\n\n.\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\ns\n\nm\ne\nt\nI\n\ne\nl\np\nm\na\nx\nE\n\ns\nn\no\ni\nt\na\nl\ne\nR\n\ns\ne\nr\nu\nt\na\ne\nF\n\nO\ns\nt\nc\ne\nj\nb\nO\n\nn\no\ni\nt\na\nc\n\ufb01\n\ni\nc\ne\np\ns\n\nn\ni\na\nm\no\nD\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n}\n2\nv\n,\n1\nv\n{\n\n\u2192\n\n)\nb\n,\na\n(\n\n}\n4\nv\n,\n3\nv\n{\n\n\u2192\n\n)\nc\n,\na\n(\n\n}\n6\nv\n,\n5\nv\n{\n\n\u2192\n\n)\nc\n,\nb\n(\n\n:\n\n:\n\n:\n\nR\n\nS\n\nT\n\n}\n1\n,\n0\n{\n\n\u2192\nO\n\u00d7\nO\n\n:\n\nR\n\n\u2014\n\n\u2014\n\n\u2014\n\n\u2014\n\n\u2014\n\n\u2014\n\n}\n2\nv\n,\n1\nv\n{\n\u2192\nO\n\n}\n4\nv\n,\n3\nv\n{\n\u2192\nO\n\n:\n\n:\n\n}\n6\nv\n,\n5\nv\n{\n\u2192\nO\n\n:\n\nF\n\nG\n\nH\n\n}\n2\nv\n,\n1\nv\n{\n\u2192\na\n\n}\n4\nv\n,\n3\nv\n{\n\u2192\n\nb\n\n}\n6\nv\n,\n5\nv\n{\n\u2192\n\nc\n\n}\n1\nv\n,\n0\n{\n\u2192\nO\n\n}\n2\nv\n,\n0\n{\n\u2192\nO\n\n:\n\n:\n\n:\n\n:\n\n:\n\n}\n3\nv\n,\n0\n{\n\u2192\nO\n\n:\n\n}\n1\nv\n,\n0\n{\n\u2192\na\n\n}\n2\nv\n,\n0\n{\n\u2192\n\nb\n\n:\n\n:\n\n}\n3\nv\n,\n0\n{\n\u2192\n\nc\n\n:\n\nF\n\nG\n\nH\n\nF\n\nG\n\nH\n\nF\n\nG\n\nH\n\n}\na\n{\n\n}\nc\n,\nb\n,\na\n{\n\n}\na\n{\n\n}\nc\n,\nb\n,\na\n{\n\n}\nc\n,\nb\n,\na\n{\n\n}\nc\n,\nb\n,\na\n{\n\n\u2014\n\n}\n3\nv\n,\n2\nv\n,\n1\nv\n{\n\u2192\nO\n\n}\n6\nv\n,\n5\nv\n,\n4\nv\n{\n\u2192\nO\n\n:\n\n:\n\nF\n\nG\n\n}\na\n{\n\n}\n1\n,\n0\n{\n\n\u2192\nO\n\u00d7\nO\n\n:\n\nR\n\n\u2014\n\n}\nf\n,\ne\n,\nd\n,\nc\n,\nb\n,\na\n{\n\ne\nr\na\n\ns\ne\nn\ni\nl\n\ne\nl\nb\nu\no\nd\n\ne\nh\nt\n\ns\nr\ne\nb\nm\ne\nm\ne\nh\nt\n\nl\nl\na\nc\n\ne\n\nW\n\nm\no\nr\nf\n\ne\ng\nr\ne\nm\ne\n\n,\nr\ne\nv\ne\nw\no\nh\n\ne\nv\no\nb\na\n\ns\nn\ni\na\nm\no\nd\n\nx\ni\ns\n\ne\nh\nT\n\n,\ns\nn\no\ni\ns\nn\ne\nm\ni\nd\n\ne\ns\ne\nh\nT\n\n.\ns\nn\no\ni\ns\nn\ne\nm\ni\nd\n\n.\nn\ni\na\nm\no\nd\n\ne\nh\nt\n\n.\ns\nn\no\ni\nt\na\nl\ne\nr\n\nf\no\nt\ne\ns\n\na\nd\nn\na\n\n,\ns\ne\nr\nu\nt\na\ne\nf\n\nf\no\nt\ne\ns\n\ne\ne\nr\nh\nt\n\ng\nn\no\nl\na\n\nr\ne\nf\nf\ni\nd\n\nh\nc\ni\nh\nw\ns\n\nm\ne\nt\ni\n\nt\nh\ng\ni\ne\n\ns\ne\nd\nu\nl\nc\nn\ni\n\ne\nn\no\n\nh\nc\na\nE\n\na\n\nn\ni\n\n,\ns\nt\nc\ne\nj\nb\no\nf\no\nt\ne\ns\n\na\ny\nb\nd\ne\nn\n\ufb01\ne\nd\ns\ni\n\ne\nn\no\nh\nc\na\ne\nd\nn\na\n\nn\no\ni\nt\na\nl\ne\nr\n\nd\nn\na\n\ne\nr\nu\nt\na\ne\nf\n\nh\nc\na\ne\n\nf\no\n\nn\no\ni\ns\nn\ne\nt\nx\ne\n\ne\nh\nt\n\n,\n\nn\nw\no\nh\ns\n\ne\nr\na\n\ng\nn\ni\ny\nf\ni\nc\ne\np\ns\n\ny\nb\n\n.\n]\n6\n[\n\n.\ns\ne\ns\na\nc\n\nx\ni\ns\n\n.\nl\na\n\ns\nn\ni\na\nm\no\nd\nt\nh\ng\ni\nE\n\n.\ne\ns\nr\ne\nv\ni\nn\nu\nl\na\nu\nt\np\ne\nc\nn\no\nc\n\ne\nh\nT\n\nd\ne\nt\na\ne\nr\nc\n\nt\ne\n\ne\nh\nt\nn\ni\n\ns\nn\no\ni\nt\na\nt\nn\ne\ns\ne\nr\np\ne\nr\n\ng\nn\ni\ny\nl\nr\ne\nd\nn\nu\n\nd\nr\na\np\ne\nh\nS\n\ns\ni\n\nm\ne\nt\ni\n\nn\na\n\nf\no\n\nd\nn\na\n\n,\ns\nm\ne\nt\ni\n\nk\nr\no\nw\ne\nh\nt\n\no\nt\n\nn\ni\na\nm\no\nd\n\nd\ne\nt\na\nl\ne\nr\n\n#\n\n1\n\n2\n\n3\n\n4\n\n5\n\n6\n\n7\n\n8\n\n3\n\n:\n1\ne\nl\nb\na\nT\n\nh\nc\na\ne\n\nf\no\n\nt\nn\ne\nr\ne\nf\nf\ni\nd\n\ny\nl\ne\ns\no\nl\nc\n\n\f(a)\n\n111\n\n(b)\n\n1 (I)\n\n2 (II)\n\n3 (III)\n\n4 (III)\n\n5 (IV)\n\n110\n\n101\n\n011\n\n100\n\n010\n\n001\n\n000\n\n6 (IV)\n\n7 (V)\n\n8 (V)\n\n9 (V)\n\n10 (VI)\n\nFigure 1: (a) A stimulus lattice for domains (e.g. Domains 3, 4, and 6) that can be encoded as a\ntriple of binary values where 0 represents \u201cabsent\u201d and 1 represents \u201cpresent.\u201d (b) If the order of\nthe values in the triple is not signi\ufb01cant, there are 10 distinct ways to partition the lattice into two\nclasses of four items. The SHJ type for each partition is shown in parentheses.\n\nDomains 3 and 4 both include 8 items each and we will consider classes that include exactly four\nof these items. Each item in these domains can be represented as a triple of binary values, where 0\nindicates that a feature is absent and value 1 indicates that a feature is present. Each triple represents\nthe values of the three features (Domain 3) or the feature values for the three objects (Domain 4).\nBy representing each domain in this way, we have effectively adopted domain speci\ufb01cations that\nare simpli\ufb01cations of those shown in Table 1. Domain 3 is represented using three features of\nthe form F, G, H : O \u2192 {0, 1}, and Domain 4 is represented using a single feature of the form\nF : O \u2192 {0, 1}. Simpli\ufb01cations of this kind are possible because the features in each domain can\nbe aligned\u2014notice that no corresponding simpli\ufb01cations are possible for Domains 1 and 2.\nThe eight binary triples in each domain can be organized into the lattice shown in Figure 1a. Here\nwe consider all ways to partition the vertices of the lattice into two groups of four. If partitions that\ndiffer only up to a permutation of the features (Domain 3) or objects (Domain 4) are grouped into\nequivalence classes, there are ten of these classes, and a representative of each is shown in Figure 1b.\nPrevious researchers [6] have pointed out that the stimuli in Domain 1 can be organized into a cube\nsimilar to Figure 1a, and that there are six ways to partition these stimuli into two groups of four\nup to permutations of the features and permutations of the range of each feature. We refer to these\nequivalence classes as the six Shepard-Hovland-Jenkins types (or SHJ types), and each partition in\nFigure 1b is labeled with its corresponding SHJ type label. Note, for example, that partitions 3 and 4\nare both examples of SHJ type III. For us, partitions 3 and 4 are distinct since items 000 (all absent)\nand 111 (all present) are uniquely identi\ufb01able, and partition 3 assigns these items to different classes\nbut partition 4 does not.\nPrevious researchers have considered differences between some of the \ufb01rst six domains in Table 1.\nShepard et al. [6] ran experiments using compact stimuli (Domain 1) and distributed stimuli (Do-\nmains 2 and 4), and observed the same dif\ufb01culty ranking of the six SHJ types in all cases. Their\nwork, however, does not acknowledge that Domain 4 leads to 10 distinct types rather than 6, and\ntherefore fails to address issues such as the relative complexities of concepts 5 and 6 in Figure 1.\nSocial psychologists [13, 14] have studied Domain 6 and found that learning patterns depart from\nthe standard SHJ order\u2014in particular, that SHJ type VI (Concept 10 in Figure 1) is simpler than\ntypes III, IV and V. This \ufb01nding has been used to support the claim that social learning relies on\na domain-speci\ufb01c principle of structural balance [14]. We will see, however, that the relative sim-\nplicity of type VI in domains like 4 and 6 is consistent with a domain-general account based on\nrepresentational economy.\n\n2 A representation length approach to concept learning\n\nThe conceptual universe in Table 1 calls for an account of learning that can apply across many\ndomains. One candidate is the representation length approach, which proposes that concepts are\nmentally represented in a language of thought, and that the subjective complexity of a concept is\n\n4\n\n\fdetermined by the length of its representation in this language [4]. We consider the case where\na concept corresponds to a class of items, and explore the idea that these concepts are mentally\nrepresented in a logical language. More formally, a concept is represented as a logical sentence, and\nthe concept includes all models of this sentence, or all items that make the sentence true.\nThe predictions of this representation length approach depend critically on the language chosen.\nHere we consider three languages\u2014an object quanti\ufb01cation language OQ that supports quanti\ufb01ca-\ntion over objects, a feature quanti\ufb01cation language F Q that supports quanti\ufb01cation over features,\nand a language OQ + F Q that supports quanti\ufb01cation over both objects and features. Language\nOQ is based on a standard logical language known as predicate logic with equality. The language\nincludes symbols representing objects (e.g. a and b), and features (e.g. F and G) and these symbols\ncan be combined to create literals that indicate that an object does (Fa) or does not have a certain\nfeature (Fa\n\u2032). Literals can be combined using two connectives: AND (FaGa) and OR (Fa + Ga). The\nlanguage includes two quanti\ufb01ers\u2014for all (\u2200) and there exists (\u2203)\u2014and allows quanti\ufb01cation over\nobjects (e.g. \u2200xFx, where x is a variable that ranges over all objects in the domain). Finally, language\nOQ includes equality and inequality relations (= and 6=) which can be used to compare objects and\nobject variables (e.g. =xa or 6=xy).\nTable 2 shows several sentences formulated in language OQ. Suppose that the OQ complexity of\neach sentence is de\ufb01ned as the number of basic propositions it contains, where a basic proposition\ncan be a positive or negative literal (Fa or Fa\n\u2032) or an equality or inequality statement (=xa or 6=xy).\nEquivalently, the complexity of a sentence is the total number of ANDs plus the total number of\nORs plus one. This measure is equivalent by design to Feldman\u2019s [2] notion of Boolean complexity\nwhen applied to a sentence without quanti\ufb01cation. The complexity values in Table 2 show minimal\ncomplexity values for each concept in Domains 3 and 4. Table 2 also shows a single sentence\nthat achieves each of these complexity values, although some concepts admit multiple sentences of\nminimal complexity.\nThe complexity values in Table 2 were computed using an \u201cenumerate then combine\u201d approach. We\nbegan by enumerating a set of sentences according to criteria described in the next paragraph. Each\nsentence has an extension that speci\ufb01es which items in the domain are consistent with the sentence.\nGiven the extensions of all sentences generated during the enumeration phase, the combination\nphase considered all possible ways to combine these extensions using conjunctions or disjunctions.\nThe procedure terminated once extensions corresponding to all of the concepts in the domain had\nbeen found. Although the number of possible sentences grows rapidly as the complexity of these\nsentences increases, the number of extensions is \ufb01xed and relatively small (28 for domains of size\n8). The combination phase is tractable since sentences with the same extension can be grouped into\na single equivalence class.\nThe enumeration phase considered all formulae which had at most two quanti\ufb01ers and which\nhad a complexity value lower than four. For example, this phase did not include the formula\nxFyFz (too many quanti\ufb01ers) or the formula \u2200x\u2203y 6=xy Fy(Fx + Gx + Hx) (complexity\n\u2203x\u2203y\u2203z 6=yz F\u2032\ntoo high). Despite these restrictions, we believe that the complexity values in Table 2 are identical\nto the values that would be obtained if we had considered all possible sentences.\nLanguage F Q is similar to OQ but allows quanti\ufb01cation over features rather than objects. For\nexample, F Q includes the statement \u2200Q Qa, where Q is a variable that ranges over all features in\nthe domain. Language F Q also allows features and feature variables to be compared for equality\nor inequality (e.g. =QF or 6=QR). Since F Q and OQ are closely related, it follows that the F Q\ncomplexity values for Domains 3 and 4 are identical to the OQ complexity values for Domains 4\nand 3. For example, F Q can express concept 5 in Domain 3 as \u2200Q\u2203R 6=QR Ra.\nWe can combine OQ and F Q to create a language OQ + F Q that allows quanti\ufb01cation over both\nobjects and features. Allowing both kinds of quanti\ufb01cation leads to identical complexity values for\nDomains 3 and 4. Language OQ + F Q can express each of the formulae for Domain 4 in Table 2,\nand these formulae can be converted into corresponding formulae for Domain 3 by translating each\ninstance of object quanti\ufb01cation into an instance of feature quanti\ufb01cation.\nLogicians distinguish between \ufb01rst-order logic, which allows quanti\ufb01cation over objects but not\npredicates, and second-order logic, which allows quanti\ufb01cation over objects and predicates. The\ndifference between languages OQ and OQ + F Q is super\ufb01cially similar to the difference between\n\ufb01rst-order and second-order logic, but does not cut to the heart of this matter. Since language\n\n5\n\n\f#\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n\nDomain 3\n\nGa\n\nFa\n\n\u2032Ha\n\n\u2032 + FaHa\n\nFa\n\n\u2032Ga + FaHa\n\nFa\n\n\u2032Ga\n\n\u2032 + FaHa\n\nGa(Fa + Ha) + FaHa\n\nGa\n\n\u2032(Fa + Ha) + FaHa\n\nGa\n\n\u2032(Fa + Ha) + FaGaHa\n\nHa(Fa\n\n\u2032 + Ga) + FaGa\n\n\u2032Ha\n\n\u2032\n\nFa(Ga + Ha) + Fa\n\n\u2032Ga\n\n\u2032Ha\n\n\u2032\n\nGa\n\n\u2032(FaHa\n\n\u2032 + Fa\n\n\u2032Ha) + Ga(Fa\n\n\u2032Ha\n\n\u2032 + FaHa)\n\nC\n1\n4\n4\n4\n5\n5\n6\n6\n6\n10\n\nDomain 4\n\nFb\n\nFa\n\n\u2032Fc\n\n\u2032 + FaFc\n\nFa\n\n\u2032Fb + FaFc\n\nFa\n\n\u2032Fb\n\n\u2032 + FaFc\n\n\u2200x\u2203y 6=xy Fy\n\n(\u2200xFx) + Fb\n\n\u2032\u2203yFy\n\n(\u2200xFx) + Fb\n\n\u2032(Fa\n\n\u2032 + Fc\n\n\u2032)\n\nFc(Fa\n\n\u2032 + Fb) + FaFb\n\n\u2032Fc\n\n\u2032\n\n(\u2200xFx\n\n\u2032) + Fa(Fb + Fc)\n\n(\u2200xFx) + \u2203y\u2200zFy(=zy +Fz\n\n\u2032)\n\nC\n1\n4\n4\n4\n2\n3\n4\n6\n4\n4\n\nTable 2: Complexity values C and corresponding formulae for language OQ. Boolean complexity\npredicts complexity values for both domains that are identical to the OQ complexity values shown\nhere for Domain 3. Language F Q predicts complexity values for Domains 3 and 4 that are identical\nto the OQ values for Domains 4 and 3 respectively. Language OQ + F Q predicts complexity values\nfor both domains that are identical to the OQ complexity values for Domain 4.\n\nOQ + F Q only supports quanti\ufb01cation over a pre-speci\ufb01ed set of features, it is equivalent to a\ntyped \ufb01rst order logic that includes types for objects and features [15]. Future studies, however, can\nexplore the cognitive relevance of higher-order logic as developed by logicians.\n\n3 Experiment\n\nNow that we have introduced languages OQ, F Q and OQ + F Q our theoretical proposals can be\nsharply formulated. We suggest that quanti\ufb01cation over objects plays an important role in mental\nrepresentations, and predict that OQ complexity will account better for human learning than Boolean\ncomplexity. We also propose that quanti\ufb01cation over objects is more natural than quanti\ufb01cation over\nfeatures, and predict that OQ complexity will account better for human learning than both F Q\ncomplexity and OQ + F Q complexity. We tested these predictions by designing an experiment\nwhere participants learned concepts from Domains 3 and 4.\nMethod. 20 adults participated for course credit. Each participant was assigned to Domain 3 or\nDomain 4 and learned all ten concepts from that domain. The items used for each domain were the\ncards shown in Table 1. Note, for example, that each Domain 3 card showed one square, and that\neach Domain 4 card showed three squares. These items are based on stimuli developed by Sakamoto\nand Love [12].\nThe experiment was carried out using a custom built graphical interface. For each learning problem\nin each domain, all eight items were simultaneously presented on the screen, and participants were\nable to drag them around and organize them however they liked. Each problem had three phases.\nDuring the learning phase, the four items belonging to the current concept had red boundaries, and\nthe remaining four items had blue boundaries. During the memory phase, these colored boundaries\nwere removed, and participants were asked to sort the items into the red group and the blue group.\nIf they made an error they returned to the learning phase, and could retake the test whenever they\nwere ready. During the description phase, participants were asked to provide a written description of\nthe two groups of cards. The color assignments (red or blue) were randomized across participants\u2014\nin other words, the \u201cred groups\u201d learned by some participants were identical to the \u201cblue groups\u201d\nlearned by others. The order in which participants learned the 10 concepts was also randomized.\nModel predictions. The OQ complexity values for the ten concepts in each domain are shown in\nTable 2 and plotted in Figure 2a. The complexity values in Figure 2a have been normalized so that\nthey sum to one within each domain, and the differences of these normalized scores are shown in\nthe \ufb01nal row of Figure 2a. The two largest bars in the difference plot indicate that Concepts 10\nand 5 are predicted to be easier to learn in Domain 4 than in Domain 3. Language OQ can express\n\n6\n\n\fOQ complexity\n\n1 2 3 4 5 6 7 8 9 10\n\n1 2 3 4 5 6 7 8 9 10\n\na)\n\n3\n\n \n\ni\n\nn\na\nm\no\nD\n\n4\n\n \n\ni\n\nn\na\nm\no\nD\n\n0.2\n\n0.1\n\n0\n\n0.2\n\n0.1\n\n0\n\ne\nc\nn\ne\nr\ne\n\nf\nf\ni\n\nD\n\n0.1\n0.05\n0\n\u22120.05\n\nLearning time\n\n1 2 3 4 5 6 7 8 9 10\n\n1 2 3 4 5 6 7 8 9 10\n\nb)\n\n0.2\n\n0.1\n\n0\n\n0.2\n\n0.1\n\n0\n\n0.1\n0.05\n0\n\u22120.05\n\n1 2 3 4 5 6 7 8 9 10\n\n1 2 3 4 5 6 7 8 9 10\n\nFigure 2: Normalized OQ complexity values and normalized learning times for the 10 concepts in\nDomains 3 and 4.\n\nstatements like \u201ceither 1 or 3 objects have F \u201d (Concept 10 in Domain 4), or \u201c2 or more objects have\nF \u201d (Concept 5 in Domain 4). Since quanti\ufb01cation over features is not permitted, however, analogous\nstatements (e.g. \u201cobject a has either 1 or 3 features\u201d) cannot be formulated in Domain 3.\nConcept 10 corresponds to SHJ type VI, which often emerges as the most dif\ufb01cult concept in studies\nof Boolean concept learning. Our model therefore predicts that the standard ordering of the SHJ\ntypes will not apply in Domain 4. Our model also predicts that concepts assigned to the same SHJ\ntype will have different complexities. In Domain 4 the model predicts that Concept 6 will be harder\nto learn than Concept 5 (both are examples of SHJ type IV), and that Concept 8 will be harder to\nlearn than Concepts 7 or 9 (all three are examples of SHJ type V).\nResults. The computer interface recorded the amount of time participants spent on the learning\nphase for each concept. Domain 3 was a little more dif\ufb01cult than Domain 4 overall: on average,\nDomain 3 participants took 557 seconds and Domain 4 participants took 467 seconds to learn the\n10 concepts. For all remaining analyses, we consider learning times that are normalized to sum to 1\nfor each participant. Figure 2b shows the mean values for these normalized times, and indicates the\nrelative dif\ufb01culties of the concepts within each condition.\nThe difference plot in Figure 2b supports the two main predictions identi\ufb01ed previously. Concepts\n10 and 5 are the cases that differ most across the domains, and both concepts are easier to learn in\nDomain 3 than Domain 4. As predicted, Concept 5 is substantially easier than Concept 6 in Domain\n4 even though both correspond to the same SHJ type. Concepts 7 through 9 also correspond to the\nsame SHJ type, and the data for Domain 4 suggest that Concept 8 is the most dif\ufb01cult of the three,\nalthough the difference between Concepts 8 and 7 is not especially large.\nFour sets of complexity predictions are plotted against the human data in Figure 3. Boolean com-\nplexity and OQ complexity make identical predictions about Domain 3, and OQ complexity and\nOQ + F Q complexity make identical predictions about Domain 4. Only OQ complexity, however,\naccounts for the results observed in both domains.\nThe concept descriptions generated by participants provide additional evidence that there are psy-\nchologically important differences between Domains 3 and 4. If the descriptions for concepts 5 and\n10 are combined, 18 out of 20 responses in Domain 4 referred to quanti\ufb01cation or counting. One\nrepresentative description of Concept 5 stated that \u201cred has multiple \ufb01lled\u201d and that \u201cblue has one\n\ufb01lled or none.\u201d Only 3 of 20 responses in Domain 3 mentioned quanti\ufb01cation. One representative\ndescription of Concept 5 stated that \u201cred = multiple features\u201d and that \u201cblue = only one feature.\u201d\n\n7\n\n\f0.2\n\n0.1\n\n0.2\n\n0.1\n\nLearning time\n(Domain 3)\n\nLearning time\n(Domain 4)\n\nr=0.84\n\nr=0.84\n\n0.2\n\n0.1\n\nr=0.26\n\n0.2\n\n0.1\n\n0.2\n\n0.1\n\nr=0.26\n\n0\n\n0.1\n\n0.2\n\n0\n\n0.1\n\n0.2\n\n0\n\n0.1\n\n0.2\n\n0\n\n0.1\n\n0.2\n\nr=0.27\n\nr=0.83\n\n0.2\n\n0.1\n\n0.2\n\n0.1\n\nr=0.27\n\n0.2\n\n0.1\n\nr=0.83\n\n0\n\n0.1\n\n0.2\nBoolean complexity\n\n0.1\n\n0\n0.2\nOQ complexity\n\n0.1\n\n0\n0.2\nF Q complexity\n\n0\n\n0.1\n\n0.2\n\nOQ + F Q complexity\n\nFigure 3: Normalized learning times for each domain plotted against normalized complexity values\npredicted by four languages: Boolean logic, OQ, F Q and OQ + F Q.\n\nThese results suggest that people can count or quantify over features, but that it is psychologically\nmore natural to quantify over objects rather than features.\nAlthough we have focused on three speci\ufb01c languages, the results in Figure 2b can be used to\nevaluate alternative proposals about the language of thought. One such alternative is an extension\nof Language OQ that allows feature values to be compared for equality. This extended language\nsupports concise representations of Concept 2 in both Domain 3 (Fa = Ha) and Domain 4 (Fa = Fc),\nand predicts that Concept 2 will be easier to learn than all other concepts except Concept 1. Note,\nhowever, that this prediction is not compatible with the data in Figure 2b. Other languages might\nalso be considered, but we know of no simple language that will account for our data better than\nOQ.\n\n4 Conclusion\nComparing concept learning across qualitatively different domains can provide valuable information\nabout the nature of mental representation. We compared two domains that that are similar in many\nrespects, but that differ according to whether they include a single object (Domain 3) or multiple\nobjects (Domain 4). Quanti\ufb01cation over objects is possible in Domain 4 but not Domain 3, and this\ndifference helps to explain the different learning patterns we observed across the two domains. Our\nresults suggest that concept representations can incorporate quanti\ufb01cation, and that quantifying over\nobjects is more natural than quantifying over features.\nThe model predictions we reported are based on a language (OQ) that is a generic version of \ufb01rst\norder logic with equality. Our results therefore suggest that some of the languages commonly con-\nsidered by logicians (e.g. \ufb01rst order logic with equality) may indeed capture some aspects of the\n\u201claws of thought\u201d [16]. A simple language like OQ offers a convenient way to explore the role of\nquanti\ufb01cation, but this language will need to be re\ufb01ned and extended in order to provide a more\naccurate account of mental representation. For example, a comprehensive account of the language\nof thought will need to support quanti\ufb01cation over features in some cases, but might be formulated\nso that quanti\ufb01cation over features is typically more costly than quanti\ufb01cation over objects.\nMany possible representation languages can be imagined and a large amount of empirical data will\nbe needed to identify the language that comes closest to the language of thought. Many relevant\nstudies have already been conducted [2, 6, 8, 9, 13, 17], but there are vast regions of the conceptual\nuniverse (Table 1) that remain to be explored. Navigating this universe is likely to involve several\nchallenges, but web-based experiments [18, 19] may allow it to be explored at a depth and scale\nthat are currently unprecedented. Characterizing the language of thought is undoubtedly a long term\nproject, but modern methods of data collection may support rapid progress towards this goal.\nAcknowledgments I thank Maureen Satyshur for running the experiment. This work was supported in part by\nNSF grant CDI-0835797.\n\n8\n\n\fReferences\n[1] J. A. Fodor. The language of thought. Harvard University Press, Cambridge, 1975.\n[2] J. Feldman. Minimization of Boolean complexity in human concept learning. Nature, 407:\n\n630\u2013633, 2000.\n\n[3] D. Fass and J. Feldman. Categorization under complexity: A uni\ufb01ed MDL account of human\nlearning of regular and irregular categories. In S. Thrun S. Becker and K. Obermayer, editors,\nAdvances in Neural Information Processing Systems 15, pages 35\u201334. MIT Press, Cambridge,\nMA, 2003.\n\n[4] C. Kemp, N. D. Goodman, and J. B. Tenenbaum. Learning and using relational theories. In J.C.\nPlatt, D. Koller, Y. Singer, and S. 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Heider\u2019s structural balance principle as a conceptual rule. Journal of Personality\n\nand Social Psychology, 31(4):713\u2013720, 1975.\n\n[15] H. B. Enderton. A mathematical introduction to logic. Academic Press, New York, 1972.\n[16] G. Boole. An investigation of the laws of thought on which are founded the mathematical\n\ntheories of logic and probabilities. 1854.\n\n[17] B. C. Love and A. B. Markman. The nonindependence of stimulus properties in human cate-\n\ngory learning. Memory and Cognition, 31(5):790\u2013799, 2003.\n\n[18] L. von Ahn. Games with a purpose. Computer, 39(6):92\u201394, 2006.\n[19] R. Snow, B. O\u2019Connor, D. Jurafsky, and A. Ng. Cheap and fast\u2013but is it good? Evaluating\nnon-expert annotations for natural language tasks. In Proceedings of the 2008 Conference on\nempirical methods in natural language processing, pages 254\u2013263. Association for Computa-\ntional Linguistics, 2008.\n\n9\n\n\f", "award": [], "sourceid": 175, "authors": [{"given_name": "Charles", "family_name": "Kemp", "institution": null}]}