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{"title"=>"What do transitive inference and class inclusion have in common? Categorical (co)products and cognitive development", "type"=>"journal", "authors"=>[{"first_name"=>"Steven", "last_name"=>"Phillips", "scopus_author_id"=>"15027040200"}, {"first_name"=>"William H.", "last_name"=>"Wilson", "scopus_author_id"=>"8657857100"}, {"first_name"=>"Graeme S.", "last_name"=>"Halford", "scopus_author_id"=>"7006362836"}], "year"=>2009, "source"=>"PLoS Computational Biology", "identifiers"=>{"sgr"=>"74549205195", "doi"=>"10.1371/journal.pcbi.1000599", "pui"=>"358137772", "pmid"=>"20011111", "scopus"=>"2-s2.0-74549205195", "issn"=>"1553734X", "isbn"=>"1553-734X"}, "id"=>"5139f1e7-41b0-348a-bb91-1ac056f71844", "abstract"=>"Transitive inference, class inclusion and a variety of other inferential abilities have strikingly similar developmental profiles-all are acquired around the age of five. Yet, little is known about the reasons for this correspondence. Category theory was invented as a formal means of establishing commonalities between various mathematical structures. We use category theory to show that transitive inference and class inclusion involve dual mathematical structures, called product and coproduct. Other inferential tasks with similar developmental profiles, including matrix completion, cardinality, dimensional changed card sorting, balance-scale (weight-distance integration), and Theory of Mind also involve these structures. By contrast, (co)products are not involved in the behaviours exhibited by younger children on these tasks, or simplified versions that are within their ability. These results point to a fundamental cognitive principle under development during childhood that is the capacity to compute (co)products in the categorical sense.", "link"=>"http://www.mendeley.com/research/transitive-inference-class-inclusion-common-categorical-coproducts-cognitive-development", "reader_count"=>53, "reader_count_by_academic_status"=>{"Unspecified"=>1, "Professor > Associate Professor"=>4, "Researcher"=>10, "Student > Doctoral Student"=>2, "Student > Ph. D. Student"=>11, "Student > Postgraduate"=>3, "Student > Master"=>6, "Other"=>5, "Student > Bachelor"=>1, "Lecturer"=>1, "Professor"=>9}, "reader_count_by_user_role"=>{"Unspecified"=>1, "Professor > Associate Professor"=>4, "Researcher"=>10, "Student > Doctoral Student"=>2, "Student > Ph. D. Student"=>11, "Student > Postgraduate"=>3, "Student > Master"=>6, "Other"=>5, "Student > Bachelor"=>1, "Lecturer"=>1, "Professor"=>9}, "reader_count_by_subject_area"=>{"Unspecified"=>3, "Engineering"=>2, "Mathematics"=>2, "Agricultural and Biological Sciences"=>8, "Medicine and Dentistry"=>2, "Philosophy"=>2, "Physics and Astronomy"=>3, "Psychology"=>18, "Social Sciences"=>3, "Computer Science"=>10}, "reader_count_by_subdiscipline"=>{"Engineering"=>{"Engineering"=>2}, "Medicine and Dentistry"=>{"Medicine and Dentistry"=>2}, "Social Sciences"=>{"Social Sciences"=>3}, "Physics and Astronomy"=>{"Physics and Astronomy"=>3}, "Psychology"=>{"Psychology"=>18}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>8}, "Computer Science"=>{"Computer Science"=>10}, "Mathematics"=>{"Mathematics"=>2}, "Unspecified"=>{"Unspecified"=>3}, "Philosophy"=>{"Philosophy"=>2}}, "reader_count_by_country"=>{"Canada"=>1, "Sweden"=>1, "Netherlands"=>1, "United States"=>5, "Serbia and Montenegro"=>1, "Italy"=>2, "United Kingdom"=>1, "Australia"=>1, "Germany"=>1, "Spain"=>2}, "group_count"=>3}

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• http://doi.org/10.1371/journal.pone.0089152
• http://doi.org/10.1371/journal.pone.0035028
• http://doi.org/10.3389/fpsyg.2017.01943
• http://doi.org/10.1016/j.tics.2010.08.005
• http://doi.org/10.3389/fpsyg.2018.01926
• http://doi.org/10.1371/journal.pcbi.1005683
• http://doi.org/10.1371/journal.pcbi.1000858
• http://doi.org/10.1016/j.pbiomolbio.2013.06.001
• http://doi.org/10.3389/fpsyg.2016.01139
• http://doi.org/10.1371/journal.pone.0032502
• http://doi.org/10.3389/fpsyg.2016.01310
• http://doi.org/10.1371/journal.pcbi.1002102
• http://doi.org/10.1016/j.cogdev.2012.08.003

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• {"type"=>"COMMENT", "annotationUri"=>"info:doi/10.1371/annotation/cb7b202b-64b3-4217-ba02-4fe1e90d9b32", "title"=>"Maps, Mappings and Animal Models", "body"=>"In Category Theory (see e.g., Lawvere & Schanuel 1997), the most important issue is maps and mappings (as corroborated by these authors as well).\r\n\r\nHowever, for being able to correlate the ongoing mapping processes in the human mind with real neural maps (i.e., neural activity patterns in vivo), we may turn to smaller animals first (e.g., zebrafish, fruit flies), where we can really see the synaptic networks in action and in real-time and during the whole ontogenetic development.\r\n\r\nHence, it would be interesting to have a cross-talk between these authors and zebrafish (or fruit fly) experts, because transitive inference and class inclusion could also be studied in these (more transparent and better investigable) model animals. \r\n\r\nBut when these mapping processes will be demonstrated (or \"unveiled\") by experimental mappings, then a big step in modern neuroscience will have been done indeed...", "isRemoved"=>false, "created"=>"2009-12-17T20:11:55Z", "lastModified"=>"2009-12-17T20:11:55Z", "creator"=>{"userId"=>"81273"}, "highlightedText"=>"", "competingInterestStatement"=>{"creatorWasPrompted"=>true, "hasCompetingInterests"=>false}, "parentArticle"=>{"doi"=>"info:doi/10.1371/journal.pcbi.1000599", "state"=>"published", "journals"=>{"PLoSCompBiol"=>{"journalKey"=>"PLoSCompBiol", "eIssn"=>"1553-7358", "title"=>"PLOS Computational Biology"}}}, "replyTreeSize"=>1, "mostRecentActivity"=>"2010-01-14T08:36:23Z", "replies"=>[{"type"=>"REPLY", "parentID"=>18119, "annotationUri"=>"info:doi/10.1371/reply/39e14f67-c193-4211-9a38-957b088b6cce", "title"=>"RE: Maps, Mappings and Animal Models", "body"=>"Though we agree that much is to be gained by integrating animal and human studies, we think that the (implicit) form of Transitive Inference demonstrated by non-humans is fundamentally different from the (explicit) form acquired by older children and adults. Adults also perform transitive inference without being explicitly aware of the structure (see Frank et al 2005), and one characteristic difference between the two is the many more trials required to achieve criterion in the implicit form.\n\nCategory theory also provides another perspective on this difference, alluded to in the third and second last paragraphs of our paper. That is, Transitive Inference is modeled as a product in the category Set, but more succinctly as a composition in the category Rel. However, Class Inclusion cannot be modeled as a composition in Rel. Hence, if the non-human and infant examples of Transitive Inference are indeed the same as the explicit form demonstrated by older children and adults, then these groups should also be capable of Class Inclusion.\n\nReference\n\nFrank MJ, Rudy JW, Levy WB, O'Reilly RC. (2005). When logic fails: Implicit transitive inference in humans. Memory and Cognition, 33(4), 742-750.", "isRemoved"=>false, "created"=>"2010-01-14T08:36:23Z", "lastModified"=>"2010-01-14T08:36:23Z", "creator"=>{"userId"=>"60869"}, "highlightedText"=>"", "competingInterestStatement"=>{"creatorWasPrompted"=>true, "hasCompetingInterests"=>false}, "parentArticle"=>{"doi"=>"info:doi/10.1371/journal.pcbi.1000599", "state"=>"published", "journals"=>{"PLoSCompBiol"=>{"journalKey"=>"PLoSCompBiol", "eIssn"=>"1553-7358", "title"=>"PLOS Computational Biology"}}}, "replyTreeSize"=>0, "mostRecentActivity"=>"2010-01-14T08:36:23Z", "replies"=>[]}]}
• {"type"=>"COMMENT", "annotationUri"=>"info:doi/10.1371/annotation/5e857104-da10-4467-a73c-6aae69a725c0", "title"=>"transitive inference versus class inclusion", "body"=>"The following comment was posted on behalf of Steven A. Sloman:\n\nThe proposed framework is inspired, elegant, and compelling in many ways. It is sufficiently powerful to deserve more detailed empirical testing. One question that arises is how it deals with differences between transitive inference and class inclusion among adults. Transitive inference is easy for adults, in fact so easy that it is sometimes overgeneralized (e.g., Goldvarg & Johnson-Laird, 2001, found that adults concluded from A prevents B and B prevents C that A prevents C). In contrast, class inclusion is often neglected by adults (e.g., Sloman, 1998, found that people did not conclude that every individual rose must have a property when told that every individual flower had the property; see also Hampton, 1982). I don't see how the proposed theory accounts for this rather massive difference in people's sensitivity to the inference rules.\n\nGoldvarg, E., & Johnson-Laird, P. (2001). Naive causality: A mental model theory of causal meaning and reasoning. Cognitive Science, 25, 565-610.\n\nHampton, J. A. (1982). A demonstration of intransitivity in natural categories. Cognition, 12, 151-164.\n\nSloman, S. A.. (1998). Categorical inference is not a tree: The myth of inheritance hierarchies. Cognitive Psychology, 35, 1-33. ", "isRemoved"=>false, "created"=>"2010-01-21T12:18:58Z", "lastModified"=>"2010-01-21T12:18:58Z", "creator"=>{"userId"=>"60869"}, "highlightedText"=>"", "competingInterestStatement"=>{"creatorWasPrompted"=>true, "hasCompetingInterests"=>false}, "parentArticle"=>{"doi"=>"info:doi/10.1371/journal.pcbi.1000599", "state"=>"published", "journals"=>{"PLoSCompBiol"=>{"journalKey"=>"PLoSCompBiol", "eIssn"=>"1553-7358", "title"=>"PLOS Computational Biology"}}}, "replyTreeSize"=>0, "mostRecentActivity"=>"2010-01-21T12:18:58Z", "replies"=>[]}

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Figshare

• {"files"=>["https://ndownloader.figshare.com/files/432688"], "description"=>"<div><p>Transitive inference, class inclusion and a variety of other inferential abilities have strikingly similar developmental profiles—all are acquired around the age of five. Yet, little is known about the reasons for this correspondence. Category theory was invented as a formal means of establishing commonalities between various mathematical structures. We use category theory to show that transitive inference and class inclusion involve <em>dual</em> mathematical structures, called product and coproduct. Other inferential tasks with similar developmental profiles, including matrix completion, cardinality, dimensional changed card sorting, balance-scale (weight-distance integration), and Theory of Mind also involve these structures. By contrast, (co)products are not involved in the behaviours exhibited by younger children on these tasks, or simplified versions that are within their ability. These results point to a fundamental cognitive principle under development during childhood that is the capacity to compute (co)products in the categorical sense.</p></div>", "links"=>[], "tags"=>["transitive", "inference", "inclusion", "categorical"], "article_id"=>145377, "categories"=>["Mathematics", "Neuroscience"], "users"=>["Steven Phillips", "William H. Wilson", "Graeme S. Halford"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1000599", "stats"=>{"downloads"=>1, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/What_Do_Transitive_Inference_and_Class_Inclusion_Have_in_Common_Categorical_Co_Products_and_Cognitive_Development/145377", "title"=>"What Do Transitive Inference and Class Inclusion Have in Common? Categorical (Co)Products and Cognitive Development", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2009-12-11 01:29:37"}

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F1000Prime