{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T10:05:29Z","timestamp":1775037929427,"version":"3.50.1"},"reference-count":54,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2015,2,26]],"date-time":"2015-02-26T00:00:00Z","timestamp":1424908800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2015,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory (Parsons, 1990; Parsons, 2008, sec. 49; McGee, 1997; Lavine, 1999; V\u00e4\u00e4n\u00e4nen &amp; Wang, 2014). Another great enterprise in contemporary philosophy of mathematics has been Wright\u2019s and Hale\u2019s project of founding mathematics on abstraction principles (Hale &amp; Wright, 2001; Cook, 2007). In Walsh (2012), it was noted that one traditional abstraction principle, namely Hume\u2019s Principle, had a certain relative categoricity property, which here we term<jats:italic>natural relative categoricity<\/jats:italic>. In this paper, we show that most other abstraction principles are<jats:italic>not<\/jats:italic>naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to (i) stability-like acceptability criteria on abstraction principles (cf. Cook, 2012), (ii) the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli (2010b) and Fine (2002), and (iii) supervaluational ideas coming out of the work of Hodes (1984, 1990, 1991).<\/jats:p>","DOI":"10.1017\/s1755020315000052","type":"journal-article","created":{"date-parts":[[2015,2,26]],"date-time":"2015-02-26T10:57:30Z","timestamp":1424948250000},"page":"572-606","source":"Crossref","is-referenced-by-count":11,"title":["RELATIVE CATEGORICITY AND ABSTRACTION PRINCIPLES"],"prefix":"10.1017","volume":"8","author":[{"given":"SEAN","family":"WALSH","sequence":"first","affiliation":[]},{"given":"SEAN","family":"EBELS-DUGGAN","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2015,2,26]]},"reference":[{"key":"S1755020315000052_ref53","first-page":"393","volume-title":"Philosophy of Mathematics Today","author":"Wright","year":"1998"},{"key":"S1755020315000052_ref52","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1093\/oso\/9780198239208.003.0008","volume-title":"Language, Thought, and Logic: Essays in Honour of Michael Dummett","author":"Wright","year":"1997"},{"key":"S1755020315000052_ref51","volume-title":"Frege\u2019s Conception of Numbers as Objects","author":"Wright","year":"1983"},{"key":"S1755020315000052_ref46","doi-asserted-by":"publisher","DOI":"10.1080\/01445348608837096"},{"key":"S1755020315000052_ref44","volume-title":"The Bounds of Logic: A Generalized Viewpoint","author":"Sher","year":"1991"},{"key":"S1755020315000052_ref42","doi-asserted-by":"publisher","DOI":"10.1093\/0195148770.001.0001"},{"key":"S1755020315000052_ref38","volume-title":"Mathematical Thought and its Objects","author":"Parsons","year":"2008"},{"key":"S1755020315000052_ref36","doi-asserted-by":"publisher","DOI":"10.2307\/2998341"},{"key":"S1755020315000052_ref35","volume-title":"Model Theory: An Introduction","author":"Marker","year":"2002"},{"key":"S1755020315000052_ref33","doi-asserted-by":"publisher","DOI":"10.1215\/00318108-111-1-25"},{"key":"S1755020315000052_ref31","unstructured":"Lavine S . (1999). Skolem was wrong. Unpublished."},{"key":"S1755020315000052_ref54","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1305\/ndjfl\/1039096303","article-title":"Is Hume\u2019s Principle analytic?","volume":"40","author":"Wright","year":"1999","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S1755020315000052_ref30","volume-title":"Set Theory","author":"Kunen","year":"2011"},{"key":"S1755020315000052_ref29","volume-title":"Set Theory","author":"Kunen","year":"1980"},{"key":"S1755020315000052_ref48","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2011.12.016"},{"key":"S1755020315000052_ref28","first-page":"73","volume-title":"Generalized Quantifiers in Natural Language","author":"Keenan","year":"1985"},{"key":"S1755020315000052_ref26","volume-title":"Introduction to Set Theory","author":"Hrbacek","year":"1999"},{"key":"S1755020315000052_ref25","doi-asserted-by":"publisher","DOI":"10.2307\/2274911"},{"key":"S1755020315000052_ref24","doi-asserted-by":"publisher","DOI":"10.1007\/BF00485187"},{"key":"S1755020315000052_ref22","doi-asserted-by":"publisher","DOI":"10.2307\/2216025"},{"key":"S1755020315000052_ref21","doi-asserted-by":"publisher","DOI":"10.1093\/0198236395.001.0001"},{"key":"S1755020315000052_ref18","volume-title":"The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number","author":"Frege","year":"1980"},{"key":"S1755020315000052_ref17","volume-title":"Die Grundlagen der Arithmetik","author":"Frege","year":"1884"},{"key":"S1755020315000052_ref14","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780199246182.001.0001","volume-title":"The Limits of Abstraction","author":"Fine","year":"2002"},{"key":"S1755020315000052_ref13","volume-title":"A Mathematical Introduction to Logic","author":"Enderton","year":"2001"},{"key":"S1755020315000052_ref12","first-page":"25","volume-title":"Model-Theoretic Logics","author":"Ebbinghaus","year":"1985"},{"key":"S1755020315000052_ref10","doi-asserted-by":"publisher","DOI":"10.1093\/bjps\/axr047"},{"key":"S1755020315000052_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-4265-2"},{"key":"S1755020315000052_ref6","volume-title":"Logic, Logic, and Logic","author":"Boolos","year":"1998"},{"key":"S1755020315000052_ref8","unstructured":"Button T. , & Walsh S . (2015). Ideas and results in model theory: Reference, realism, structure and categoricity. arXiv:1501.00472."},{"key":"S1755020315000052_ref3","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-2010-010"},{"key":"S1755020315000052_ref1","doi-asserted-by":"publisher","DOI":"10.5840\/jphil2010107415"},{"key":"S1755020315000052_ref7","doi-asserted-by":"crossref","DOI":"10.1515\/9780691187068","volume-title":"Fixing Frege","author":"Burgess","year":"2005"},{"key":"S1755020315000052_ref16","first-page":"89","volume-title":"Oxford Studies in Epistemology","volume":"1","author":"Fine","year":"2006"},{"key":"S1755020315000052_ref47","article-title":"Internal categoricity in arithmetic and set theory","author":"V\u00e4\u00e4n\u00e4nen","year":"2014","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S1755020315000052_ref27","doi-asserted-by":"publisher","DOI":"10.1023\/B:LOGI.0000046129.22633.96"},{"key":"S1755020315000052_ref32","first-page":"105","volume-title":"The Continuum Companion to Philosophical Logic","author":"Linnebo","year":"2011"},{"key":"S1755020315000052_ref23","doi-asserted-by":"publisher","DOI":"10.2307\/2026440"},{"key":"S1755020315000052_ref43","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/7.3.293"},{"key":"S1755020315000052_ref11","volume-title":"Frege\u2019s Philosophy of Mathematics","author":"Demopoulos","year":"1995"},{"key":"S1755020315000052_ref20","doi-asserted-by":"publisher","DOI":"10.1093\/0199241279.003.0012"},{"key":"S1755020315000052_ref4","doi-asserted-by":"publisher","DOI":"10.5840\/philtopics19891721"},{"key":"S1755020315000052_ref49","unstructured":"Walsh S . (2014a). Fragments of Frege\u2019s Grundgesetze and G\u00f6del\u2019s constructible universe. arXiv:1407.3861."},{"key":"S1755020315000052_ref41","doi-asserted-by":"publisher","DOI":"10.1093\/0195139305.001.0001"},{"key":"S1755020315000052_ref34","volume-title":"Extensions of First Order Logic","author":"Manzano","year":"1996"},{"key":"S1755020315000052_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s11098-004-4922-x"},{"key":"S1755020315000052_ref37","first-page":"13","article-title":"The uniqueness of the natural numbers","volume":"39","author":"Parsons","year":"1990","journal-title":"Iyyun"},{"key":"S1755020315000052_ref2","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/nkq010"},{"key":"S1755020315000052_ref40","volume-title":"Foundations without Foundationalism: A Case for Second-Order Logic","author":"Shapiro","year":"1991"},{"key":"S1755020315000052_ref39","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199291267.001.0001"},{"key":"S1755020315000052_ref45","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-59971-2"},{"key":"S1755020315000052_ref5","first-page":"261","volume-title":"Meaning and Method: Essays in Honor of Hilary Putnam","author":"Boolos","year":"1990"},{"key":"S1755020315000052_ref19","volume-title":"Abstract Objects","author":"Hale","year":"1987"},{"key":"S1755020315000052_ref50","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020313000397"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020315000052","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,7]],"date-time":"2024-06-07T12:21:24Z","timestamp":1717762884000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020315000052\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,26]]},"references-count":54,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2015,9]]}},"alternative-id":["S1755020315000052"],"URL":"https:\/\/doi.org\/10.1017\/s1755020315000052","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,2,26]]}}}