{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T12:39:40Z","timestamp":1777639180440,"version":"3.51.4"},"reference-count":77,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,2,14]],"date-time":"2014-02-14T00:00:00Z","timestamp":1392336000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Rev. symb. logic"],"published-print":{"date-parts":[[2014,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here, an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics.<\/jats:p>","DOI":"10.1017\/s1755020313000397","type":"journal-article","created":{"date-parts":[[2014,2,14]],"date-time":"2014-02-14T08:32:57Z","timestamp":1392366777000},"page":"84-119","source":"Crossref","is-referenced-by-count":8,"title":["LOGICISM, INTERPRETABILITY, AND KNOWLEDGE OF ARITHMETIC"],"prefix":"10.1017","volume":"7","author":[{"given":"SEAN","family":"WALSH","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,2,14]]},"reference":[{"key":"S1755020313000397_ref72","doi-asserted-by":"publisher","DOI":"10.1002\/9780470696675"},{"key":"S1755020313000397_ref77","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1305\/ndjfl\/1038336879","article-title":"Neo-Fregean foundations for real analysis: Some reflections on Frege\u2019s constraint","volume":"41","author":"Wright","year":"2000","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S1755020313000397_ref60","doi-asserted-by":"publisher","DOI":"10.2307\/2023482"},{"key":"S1755020313000397_ref68","volume-title":"Mathematical Knowledge","author":"Steiner","year":"1975"},{"key":"S1755020313000397_ref56","first-page":"180","volume-title":"Philosophy in America","author":"Parsons","year":"1965"},{"key":"S1755020313000397_ref66","doi-asserted-by":"publisher","DOI":"10.1093\/0195139305.001.0001"},{"key":"S1755020313000397_ref61","volume-title":"Ways of Paradox and Other Essays","author":"Quine","year":"1976"},{"key":"S1755020313000397_ref54","doi-asserted-by":"publisher","DOI":"10.2178\/bsl\/1186666149"},{"key":"S1755020313000397_ref53","doi-asserted-by":"publisher","DOI":"10.1515\/9781400858927"},{"key":"S1755020313000397_ref75","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1093\/oso\/9780198236542.003.0016","volume-title":"Philosophy of Mathematics Today","author":"Wright","year":"1998"},{"key":"S1755020313000397_ref73","volume-title":"Frege\u2019s Conception of Numbers as Objects","volume":"2","author":"Wright","year":"1983"},{"key":"S1755020313000397_ref52","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9452-5"},{"key":"S1755020313000397_ref9","doi-asserted-by":"crossref","DOI":"10.1515\/9780691187068","volume-title":"Fixing Frege","author":"Burgess","year":"2005"},{"key":"S1755020313000397_ref74","volume-title":"Philosophy of Mathematics Today","author":"Wright","year":"1998"},{"key":"S1755020313000397_ref10","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1515\/9781400865253.183","volume-title":"Diffusion, Quantum Theory, and Radically Elementary Mathematics","volume":"47","author":"Buss","year":"2006"},{"key":"S1755020313000397_ref28","volume-title":"Metamathematics of First-Order Arithmetic","author":"H\u00e1jek","year":"1998"},{"key":"S1755020313000397_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-0087-8"},{"key":"S1755020313000397_ref29","volume-title":"Abstract Objects","author":"Hale","year":"1987"},{"key":"S1755020313000397_ref51","volume-title":"Model Theory: An Introduction","volume":"217","author":"Marker","year":"2002"},{"key":"S1755020313000397_ref30","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/8.2.100"},{"key":"S1755020313000397_ref70","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020310000316"},{"key":"S1755020313000397_ref25","volume-title":"The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number","author":"Frege","year":"1980"},{"key":"S1755020313000397_ref62","doi-asserted-by":"publisher","DOI":"10.2307\/2214851"},{"key":"S1755020313000397_ref65","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1305\/ndjfl\/1038336880","article-title":"Frege meets Dedekind: A neo-logicist treatment of real analysis","volume":"41","author":"Shapiro","year":"2000","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S1755020313000397_ref63","volume-title":"Mathematics as a Science of Patterns","author":"Resnik","year":"1997"},{"key":"S1755020313000397_ref23","first-page":"357","volume-title":"The Analytical Way. Proceedings of the 6th European Congress of Analytic Philosophy","author":"Fischer","year":"2010"},{"key":"S1755020313000397_ref76","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":"S1755020313000397_ref38","first-page":"396","volume-title":"Essays on Bertrand Russell","author":"Hochberg","year":"1970"},{"key":"S1755020313000397_ref20","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511910616.004"},{"key":"S1755020313000397_ref34","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1305\/ndjfl\/1038336841","article-title":"Cardinality, counting, and equinumerosity","volume":"41","author":"Heck","year":"2000","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S1755020313000397_ref2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1475-4975.1981.tb00426.x"},{"key":"S1755020313000397_ref4","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199891610.001.0001"},{"key":"S1755020313000397_ref37","doi-asserted-by":"publisher","DOI":"10.1093\/analys\/16.5.118"},{"key":"S1755020313000397_ref14","first-page":"402","volume-title":"The Theory of Models","author":"de Bouv\u00e8re","year":"1965"},{"key":"S1755020313000397_ref46","volume-title":"Introduction to Metamathematics","volume":"1","author":"Kleene","year":"1952"},{"key":"S1755020313000397_ref71","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8349.2006.00136.x"},{"key":"S1755020313000397_ref31","doi-asserted-by":"publisher","DOI":"10.1093\/0198236395.001.0001"},{"key":"S1755020313000397_ref44","doi-asserted-by":"publisher","DOI":"10.1023\/A:1005651027553"},{"key":"S1755020313000397_ref59","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1387\/theoria.442","article-title":"Goodman\u2019s extensional isomorphism and syntactical interpretations","volume":"65","author":"Pol\u00e1nski","year":"2009","journal-title":"Theoria. An International Journal for Theory, History and Foundations of Science"},{"key":"S1755020313000397_ref48","doi-asserted-by":"publisher","DOI":"10.1057\/9780230245198_13"},{"key":"S1755020313000397_ref50","volume-title":"Extensions of First Order Logic","volume":"19","author":"Manzano","year":"1996"},{"key":"S1755020313000397_ref36","doi-asserted-by":"publisher","DOI":"10.1007\/BF00172227"},{"key":"S1755020313000397_ref21","volume-title":"A Mathematical Introduction to Logic","author":"Enderton","year":"2001"},{"key":"S1755020313000397_ref49","volume-title":"Aspects of Incompleteness","volume":"10","author":"Lindstr\u00f6m","year":"2003"},{"key":"S1755020313000397_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-4265-2"},{"key":"S1755020313000397_ref42","doi-asserted-by":"publisher","DOI":"10.2307\/2274223"},{"key":"S1755020313000397_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(86)90037-0"},{"key":"S1755020313000397_ref19","first-page":"25","volume-title":"Model-Theoretic Logics","author":"Ebbinghaus","year":"1985"},{"key":"S1755020313000397_ref57","volume-title":"Mathematics in Philosophy: Selected Essays","author":"Parsons","year":"1983"},{"key":"S1755020313000397_ref3","doi-asserted-by":"publisher","DOI":"10.1080\/01445349408837226"},{"key":"S1755020313000397_ref22","doi-asserted-by":"publisher","DOI":"10.4064\/fm-49-1-35-92"},{"key":"S1755020313000397_ref47","doi-asserted-by":"publisher","DOI":"10.1057\/9780230245198_5"},{"key":"S1755020313000397_ref45","unstructured":"Kalsbeek M . (1989). An Orey Sentence for Predicative Arithmetic. Unpublished. Master\u2019s Thesis, Institute for Language, Logic, and Information. ITLI Prepublication Series X-89\u201301."},{"key":"S1755020313000397_ref43","doi-asserted-by":"publisher","DOI":"10.7551\/mitpress\/9780262015868.001.0001"},{"key":"S1755020313000397_ref69","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1201\/9781439865873-16","volume-title":"Logic in Tehran","volume":"26","author":"Visser","year":"2006"},{"key":"S1755020313000397_ref7","volume-title":"Logic, Logic, and Logic","author":"Boolos","year":"1998"},{"key":"S1755020313000397_ref24","unstructured":"Frege G . (1967). Kleine Schriften. Hildesheim: Olms. Edited by Ignacio Angelelli."},{"key":"S1755020313000397_ref17","doi-asserted-by":"publisher","DOI":"10.1093\/0195148770.003.0005"},{"key":"S1755020313000397_ref39","doi-asserted-by":"crossref","DOI":"10.2307\/j.ctv2x8v901","volume-title":"Logic, Ontology, and Language: Essays on Truth and Reality","author":"Hochberg","year":"1984"},{"key":"S1755020313000397_ref40","doi-asserted-by":"publisher","DOI":"10.1007\/BF00485187"},{"key":"S1755020313000397_ref67","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581007"},{"key":"S1755020313000397_ref18","doi-asserted-by":"publisher","DOI":"10.1093\/019823628X.003.0002"},{"key":"S1755020313000397_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01048481"},{"key":"S1755020313000397_ref41","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S1755020313000397_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(65)50063-9"},{"key":"S1755020313000397_ref35","volume-title":"Frege\u2019s Theorem","author":"Heck","year":"2011"},{"key":"S1755020313000397_ref26","volume-title":"Introduction to Boolean Algebras","author":"Givant","year":"2009"},{"key":"S1755020313000397_ref6","first-page":"143","volume-title":"Benacerraf and His Critics","author":"Boolos","year":"1996"},{"key":"S1755020313000397_ref5","first-page":"469","volume-title":"Proceedings of the Biennial Meeting of the Philosophy of Science Association","volume":"2","author":"Boolos","year":"1984"},{"key":"S1755020313000397_ref16","volume-title":"Frege\u2019s Philosophy of Mathematics","author":"Demopoulos","year":"1995"},{"key":"S1755020313000397_ref58","volume-title":"Mathematical Thought and Its Objects","author":"Parsons","year":"2008"},{"key":"S1755020313000397_ref33","doi-asserted-by":"publisher","DOI":"10.5840\/harvardreview1999717"},{"key":"S1755020313000397_ref27","volume-title":"The Structure of Appearance","author":"Goodman","year":"1951"},{"key":"S1755020313000397_ref32","doi-asserted-by":"publisher","DOI":"10.1023\/A:1004299720847"},{"key":"S1755020313000397_ref12","first-page":"231","article-title":"On definitional equivalence and related topics","volume":"1","author":"Corcoran","year":"1980","journal-title":"History and Philosophy of Logic"},{"key":"S1755020313000397_ref55","first-page":"97","volume-title":"Probl\u00e8mes de la construction du nombre","volume":"11","author":"Papert","year":"1960"},{"key":"S1755020313000397_ref64","volume-title":"Foundations without Foundationalism: A Case for Second-Order Logic","volume":"17","author":"Shapiro","year":"1991"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020313000397","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,24]],"date-time":"2024-05-24T07:19:32Z","timestamp":1716535172000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020313000397\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,14]]},"references-count":77,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014,3]]}},"alternative-id":["S1755020313000397"],"URL":"https:\/\/doi.org\/10.1017\/s1755020313000397","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,2,14]]}}}