{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T15:11:27Z","timestamp":1773673887134,"version":"3.50.1"},"reference-count":35,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2022,4,18]],"date-time":"2022-04-18T00:00:00Z","timestamp":1650240000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Neo-Fregean logicists claim that Hume\u2019s Principle (<jats:roman>HP<\/jats:roman>) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that <jats:roman>HP<\/jats:roman> is not deductively conservative over pure axiomatic second-order logic. This seems to preclude <jats:roman>HP<\/jats:roman> from being true by fiat. In this paper, we study Richard Kimberly Heck\u2019s Two-Sorted Frege Arithmetic (<jats:roman>2FA<\/jats:roman>), a variation on <jats:roman>HP<\/jats:roman> which has been thought to be deductively conservative over second-order logic. We show that it isn\u2019t. In fact, <jats:roman>2FA<\/jats:roman> is not conservative over <jats:italic>n<\/jats:italic>-th order logic, for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000156_inline1.png\"\/><jats:tex-math>\n$n \\geq 2$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. It follows that in the usual one-sorted setting, <jats:roman>HP<\/jats:roman> is not deductively Field-conservative over second- or higher-order logic.<\/jats:p>","DOI":"10.1017\/s1755020322000156","type":"journal-article","created":{"date-parts":[[2022,4,18]],"date-time":"2022-04-18T08:16:48Z","timestamp":1650269808000},"page":"1199-1232","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["TWO-SORTED FREGE ARITHMETIC IS NOT CONSERVATIVE"],"prefix":"10.1017","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8150-9658","authenticated-orcid":false,"given":"STEPHEN","family":"MACKERETH","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1275-315X","authenticated-orcid":false,"given":"JEREMY","family":"AVIGAD","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2022,4,18]]},"reference":[{"key":"S1755020322000156_r28","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581007"},{"key":"S1755020322000156_r10","volume-title":"Die Grundlagen der Arithmetik: Eine logisch mathematische Untersuchung \u00fcber den Begriff der Zahl","author":"Frege","year":"1884"},{"key":"S1755020322000156_r12","volume-title":"Grundgesetze der Arithmetik","author":"Frege","year":"1903"},{"key":"S1755020322000156_r13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22156-3"},{"key":"S1755020322000156_r26","volume-title":"Foundations without Foundationalism: A Case for Second-Order Logic","author":"Shapiro","year":"1991"},{"key":"S1755020322000156_r4","doi-asserted-by":"publisher","DOI":"10.1017\/9781108178600"},{"key":"S1755020322000156_r8","volume-title":"A Mathematical Introduction to Logic","author":"Enderton","year":"2001"},{"key":"S1755020322000156_r20","doi-asserted-by":"crossref","DOI":"10.1017\/9781316716854","volume-title":"Aspects of Incompleteness","volume":"10","author":"Lindstr\u00f6m","year":"2017"},{"key":"S1755020322000156_r18","volume-title":"Set Theory","author":"Jech","year":"2002"},{"key":"S1755020322000156_r25","doi-asserted-by":"publisher","DOI":"10.1080\/01445348708837116"},{"key":"S1755020322000156_r11","volume-title":"Grundgesetze der Arithmetik","author":"Frege","year":"1893"},{"key":"S1755020322000156_r27","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/7.3.293"},{"key":"S1755020322000156_r14","doi-asserted-by":"publisher","DOI":"10.1093\/0198236395.001.0001"},{"key":"S1755020322000156_r22","volume-title":"Abstractionism: Essays in Philosophy of Mathematics","author":"Linnebo","year":"2016"},{"key":"S1755020322000156_r5","doi-asserted-by":"publisher","DOI":"10.1093\/bjps\/axr047"},{"key":"S1755020322000156_r16","volume-title":"Frege\u2019s Theorem","author":"Heck","year":"2011"},{"key":"S1755020322000156_r23","doi-asserted-by":"publisher","DOI":"10.1093\/bjps\/54.1.103"},{"key":"S1755020322000156_r30","doi-asserted-by":"publisher","DOI":"10.1093\/bjps\/axu035"},{"key":"S1755020322000156_r7","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020318000230"},{"key":"S1755020322000156_r33","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1082637613"},{"key":"S1755020322000156_r35","first-page":"8","volume-title":"Atti del IV Congresso Internazionale Dei Matematici","author":"Zermelo","year":"1909b"},{"key":"S1755020322000156_r3","doi-asserted-by":"publisher","DOI":"10.1515\/9780691187068"},{"key":"S1755020322000156_r24","doi-asserted-by":"publisher","DOI":"10.1007\/s11229-007-9260-z"},{"key":"S1755020322000156_r1","volume-title":"Einf\u00fchrung in die mathematische Logik, Teil III: Pr\u00e4dikatenlogik h\u00f6herer Stufe","author":"Asser","year":"1981"},{"key":"S1755020322000156_r9","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780199246182.001.0001","volume-title":"The Limits of Abstraction","author":"Fine","year":"2002"},{"key":"S1755020322000156_r2","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1093\/oso\/9780198236542.003.0017","volume-title":"The Philosophy of Mathematics Today","author":"Boolos","year":"1998"},{"key":"S1755020322000156_r17","volume-title":"Reading Frege\u2019s Grundgesetze","author":"Heck","year":"2012"},{"key":"S1755020322000156_r31","volume-title":"Proof Theory","author":"Takeuti","year":"1987"},{"key":"S1755020322000156_r32","volume-title":"The Stanford Encyclopedia of Philosophy","author":"V\u00e4\u00e4n\u00e4nen","year":"2021"},{"key":"S1755020322000156_r29","first-page":"425","article-title":"Zu H. 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Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}