{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T04:52:34Z","timestamp":1784695954566,"version":"3.55.0"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2024,5,31]],"date-time":"2024-05-31T00:00:00Z","timestamp":1717113600000},"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":[[2024,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper proposes and partially defends a novel philosophy of arithmetic\u2014<jats:sc>finitary upper logicism<\/jats:sc>. According to it, the natural numbers are finite cardinalities\u2014conceived of as properties of properties\u2014and arithmetic is nothing but higher-order modal logic. <jats:sc>Finitary upper logicism<\/jats:sc> is furthermore essentially committed to the logicality of <jats:sc>finitary plenitude<\/jats:sc>, the principle according to which every finite cardinality could have been instantiated. Among other things, it is proved in the paper that second-order Peano arithmetic is interpretable, on the basis of the finite cardinalities\u2019 conception of the natural numbers, in a weak modal type theory consisting of the modal logic <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020324000145_inline1.png\"\/><jats:tex-math>\n$\\mathsf {K}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, negative free quantified logic, a contingentist-friendly comprehension principle, and <jats:sc>finitary plenitude<\/jats:sc>. By replacing <jats:sc>finitary plenitude<\/jats:sc> for the <jats:sc>axiom of infinity<\/jats:sc> this result constitutes a significant improvement on Russell and Whitehead\u2019s interpretation of second-order Peano arithmetic, itself based on the finite cardinalities\u2019 conception of the natural numbers.<\/jats:p>","DOI":"10.1017\/s1755020324000145","type":"journal-article","created":{"date-parts":[[2024,5,31]],"date-time":"2024-05-31T09:35:59Z","timestamp":1717148159000},"page":"1172-1247","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["FINITARY UPPER LOGICISM"],"prefix":"10.1017","volume":"17","author":[{"given":"BRUNO","family":"JACINTO","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2024,5,31]]},"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020324000145","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T10:53:34Z","timestamp":1735815214000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020324000145\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,31]]},"references-count":0,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["S1755020324000145"],"URL":"https:\/\/doi.org\/10.1017\/s1755020324000145","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,31]]},"assertion":[{"value":"\u00a9 The Author(s), 2024. 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"}}]}}