{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T22:04:25Z","timestamp":1747173865950,"version":"3.40.5"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2022,2,7]],"date-time":"2022-02-07T00:00:00Z","timestamp":1644192000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2024,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Neo-Fregeanism aims to provide a possible route to knowledge of arithmetic via Hume\u2019s principle, but this is of only limited significance if it cannot account for how the vast majority of arithmetic knowledge, accrued by ordinary people, is obtained. I argue that Hume\u2019s principle does not capture what is ordinarily meant by numerical identity, but that we can do much better by buttressing plural logic with plural versions of the ancestral operator, obtaining natural and plausible characterizations of various key arithmetic concepts, including finiteness, equinumerosity and addition and multiplication of cardinality\u2014revealing these to be logical concepts, and obtaining much of ordinary arithmetic knowledge as logical knowledge. Supplementing this with an abstraction principle and a simple axiom of infinity (known either empirically or modally) we obtain a full interpretation of arithmetic.<\/jats:p>","DOI":"10.1017\/s1755020322000041","type":"journal-article","created":{"date-parts":[[2022,2,7]],"date-time":"2022-02-07T07:02:20Z","timestamp":1644217340000},"page":"305-342","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["PLURAL ANCESTRAL LOGIC AS THE LOGIC OF ARITHMETIC"],"prefix":"10.1017","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8439-8068","authenticated-orcid":false,"given":"OLIVER","family":"TATTON-BROWN","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2022,2,7]]},"reference":[{"key":"S1755020322000041_r13","doi-asserted-by":"publisher","DOI":"10.2307\/2268974"},{"key":"S1755020322000041_r18","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780195141924.001.0001"},{"key":"S1755020322000041_r23","first-page":"162","volume-title":"Abstractionism: Essays in Philosophy of Mathematics","author":"Wright","year":"2016"},{"key":"S1755020322000041_r2","doi-asserted-by":"publisher","DOI":"10.2307\/2185003"},{"volume-title":"Frege\u2019s Theorem","year":"2011","author":"Heck","key":"S1755020322000041_r8"},{"key":"S1755020322000041_r1","doi-asserted-by":"publisher","DOI":"10.2307\/2026308"},{"volume-title":"Frege\u2019s Conception of Numbers as Objects","year":"1983","author":"Wright","key":"S1755020322000041_r22"},{"key":"S1755020322000041_r14","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198744382.001.0001"},{"key":"S1755020322000041_r20","doi-asserted-by":"publisher","DOI":"10.1002\/tht3.399"},{"volume-title":"A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics","year":"1997","author":"Burgess","key":"S1755020322000041_r3"},{"key":"S1755020322000041_r5","first-page":"333","volume-title":"Abstractionism: Essays in Philosophy of Mathematics","author":"Hale","year":"2016"},{"key":"S1755020322000041_r7","doi-asserted-by":"publisher","DOI":"10.1007\/s10763-015-9707-5"},{"volume-title":"Mathematical Thought and Its Objects","year":"2009","author":"Parsons","key":"S1755020322000041_r15"},{"volume-title":"Foundations Without Foundationalism: A Case for Second-Order Logic","year":"1991","author":"Shapiro","key":"S1755020322000041_r16"},{"key":"S1755020322000041_r21","doi-asserted-by":"publisher","DOI":"10.1111\/1467-8349.00111"},{"key":"S1755020322000041_r17","doi-asserted-by":"publisher","DOI":"10.1093\/analys\/68.1.1"},{"key":"S1755020322000041_r12","doi-asserted-by":"publisher","DOI":"10.2307\/2267976"},{"key":"S1755020322000041_r19","doi-asserted-by":"publisher","DOI":"10.1080\/01445348608837096"},{"volume-title":"Pythagoras and the Pythagoreans","year":"2001","author":"Kahn","key":"S1755020322000041_r9"},{"key":"S1755020322000041_r24","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-005-9015-6"},{"key":"S1755020322000041_r10","doi-asserted-by":"publisher","DOI":"10.1080\/07468342.1989.11973245"},{"key":"S1755020322000041_r4","doi-asserted-by":"publisher","DOI":"10.1007\/s11229-015-0784-3"},{"key":"S1755020322000041_r11","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780199641314.001.0001"},{"volume-title":"The Reason\u2019s Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics","year":"2004","author":"Hale","key":"S1755020322000041_r6"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020322000041","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T13:20:13Z","timestamp":1716816013000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020322000041\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,7]]},"references-count":24,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,6]]}},"alternative-id":["S1755020322000041"],"URL":"https:\/\/doi.org\/10.1017\/s1755020322000041","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"type":"print","value":"1755-0203"},{"type":"electronic","value":"1755-0211"}],"subject":[],"published":{"date-parts":[[2022,2,7]]},"assertion":[{"value":"\u00a9 The Author(s), 2022. Published by Cambridge University Press on behalf of Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}