{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T23:29:40Z","timestamp":1770679780293,"version":"3.49.0"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2021,7,15]],"date-time":"2021-07-15T00:00:00Z","timestamp":1626307200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,7,15]],"date-time":"2021-07-15T00:00:00Z","timestamp":1626307200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Philos Logic"],"published-print":{"date-parts":[[2021,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The notion of grounding is usually conceived as an objective and explanatory relation. It connects two relata if one\u2014the ground\u2014determines or explains the other\u2014the consequence. In the contemporary literature on grounding, much effort has been devoted to logically characterize the formal aspects of grounding, but a major hard problem remains: defining suitable grounding principles for universal and existential formulae. Indeed, several grounding principles for quantified formulae have been proposed, but all of them are exposed to paradoxes in some very natural contexts of application. We introduce in this paper a first-order formal system that captures the notion of grounding and avoids the paradoxes in a novel and non-trivial way. The system we present formally develops Bolzano\u2019s ideas on grounding by employing Hilbert\u2019s<jats:italic>\u03b5<\/jats:italic>-terms and an adapted version of Fine\u2019s theory of arbitrary objects.<\/jats:p>","DOI":"10.1007\/s10992-021-09604-w","type":"journal-article","created":{"date-parts":[[2021,7,15]],"date-time":"2021-07-15T10:04:25Z","timestamp":1626343465000},"page":"1417-1448","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Grounding, Quantifiers, and Paradoxes"],"prefix":"10.1007","volume":"50","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7415-5839","authenticated-orcid":false,"given":"Francesco A.","family":"Genco","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesca","family":"Poggiolesi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1932-5484","authenticated-orcid":false,"given":"Lorenzo","family":"Rossi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,7,15]]},"reference":[{"key":"9604_CR1","unstructured":"Avigad, J., & Zach, R. (2020). The epsilon calculus. In E. N. Zalta (Ed.) The Stanford Encyclopedia of Philosophy (Fall 2020 Edition). https:\/\/plato.stanford.edu\/archives\/fall2020\/entries\/epsilon-calculus\/"},{"key":"9604_CR2","unstructured":"Bell, J.L. (2016). Infinitary logic. In E. N. Zalta (Ed.) The Stanford Encyclopedia of Philosophy (Winter 2016 Edition). https:\/\/plato.stanford.edu\/archives\/win2016\/entries\/logic-infinitary\/"},{"key":"9604_CR3","first-page":"281","volume":"211","author":"A Betti","year":"2010","unstructured":"Betti, A. (2010). Explanation in metaphysics and Bolzano\u2019s theory of ground and consequence. Logique & Analyse, 211, 281\u2013316.","journal-title":"Logique & Analyse"},{"key":"9604_CR4","volume-title":"Theory of science","author":"B Bolzano","year":"2014","unstructured":"Bolzano, B. (2014). Theory of science. Oxford: Oxford University Press. Translated by Rolf George and Paul Rusnok."},{"key":"9604_CR5","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198790396.001.0001","volume-title":"Philosophy and model theory","author":"T Button","year":"2018","unstructured":"Button, T., & Walsh, S. (2018). Philosophy and model theory. Oxford: Oxford University Press."},{"issue":"211","key":"9604_CR6","first-page":"251","volume":"53","author":"Correia F.","year":"2010","unstructured":"Correia F. (2010). Grounding and truth-functions. Logique et Analyse, 53(211), 251\u2013279.","journal-title":"Logique et Analyse"},{"issue":"1","key":"9604_CR7","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1017\/S1755020313000300","volume":"7","author":"F Correia","year":"2014","unstructured":"Correia, F. (2014). Logical grounds. The Review of Symbolic Logic, 7(1), 31\u201359.","journal-title":"The Review of Symbolic Logic"},{"key":"9604_CR8","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139149136","volume-title":"Methaphysical grounding: understanding the structure of reality","author":"F Correia","year":"2012","unstructured":"Correia, F., & Schnieder, B. (2012). Methaphysical grounding: understanding the structure of reality. Cambridge: Cambridge University Press."},{"key":"9604_CR9","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1093\/aristoteliansupp\/57.1.55","volume":"57","author":"K Fine","year":"1983","unstructured":"Fine, K. (1983). A defence of arbitrary objects. Proceedings of the Aristotelian Society, Supplementary Volumes, 57, 55\u201389.","journal-title":"Proceedings of the Aristotelian Society, Supplementary Volumes"},{"issue":"1","key":"9604_CR10","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/BF00542649","volume":"14","author":"K Fine","year":"1985","unstructured":"Fine, K. (1985). Natural deduction and arbitrary objects. Journal of Philosophical Logic, 14(1), 57\u2013107.","journal-title":"Journal of Philosophical Logic"},{"key":"9604_CR11","volume-title":"Reasoning with arbitrary objects","author":"K Fine","year":"1985","unstructured":"Fine, K. (1985). Reasoning with arbitrary objects. Oxford: Blackwell."},{"issue":"1","key":"9604_CR12","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1215\/00294527-2010-007","volume":"51","author":"K Fine","year":"2010","unstructured":"Fine, K. (2010). Some puzzles of ground. Notre Dame Journal of Formal Logic, 51(1), 97\u2013118.","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"9604_CR13","first-page":"37","volume-title":"Guide to ground","author":"K Fine","year":"2012","unstructured":"Fine, K. (2012). Guide to ground, (pp. 37\u201380). Cambridge: Cambridge University Press."},{"issue":"1","key":"9604_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S1755020311000086","volume":"5","author":"K Fine","year":"2012","unstructured":"Fine, K. (2012). The pure logic of ground. The Review of Symbolic Logic, 5(1), 1\u201325.","journal-title":"The Review of Symbolic Logic"},{"key":"9604_CR15","doi-asserted-by":"crossref","unstructured":"Hilbert, D. (1922). Neubegr\u00fcndung der Mathematik. Erste Mitteilung. In Abhandlungen aus dem Seminar der Hamburgischen Universit\u00e4t, (Vol. 1 pp. 157\u201377).","DOI":"10.1007\/BF02940589"},{"key":"9604_CR16","doi-asserted-by":"publisher","DOI":"10.1017\/9781139600293","volume-title":"The metaphysics and mathematics of arbitrary objects","author":"L Horsten","year":"2019","unstructured":"Horsten, L. (2019). The metaphysics and mathematics of arbitrary objects. Cambridge: Cambridge University Press."},{"issue":"2","key":"9604_CR17","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/s10992-017-9444-z","volume":"47","author":"J Korbmacher","year":"2018","unstructured":"Korbmacher, J. (2018). Axiomatic theories of partial ground II. Journal of Philosophical Logic, 47(2), 193\u2013226.","journal-title":"Journal of Philosophical Logic"},{"issue":"2","key":"9604_CR18","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1002\/tht3.77","volume":"2","author":"S Kr\u00e4mer","year":"2013","unstructured":"Kr\u00e4mer, S. (2013). A simpler puzzle of ground. Thought: A Journal of Philosophy, 2(2), 85\u201389.","journal-title":"Thought: A Journal of Philosophy"},{"key":"9604_CR19","volume-title":"Leisenring. Mathematical logic and Hilbert\u2019s epsilon-symbol","author":"C Albert","year":"1969","unstructured":"Albert, C. (1969). Leisenring. Mathematical logic and Hilbert\u2019s epsilon-symbol. London: Macdonald."},{"key":"9604_CR20","volume-title":"The possibility of metaphysics: Substance, identity, and time","author":"JE Lowe","year":"1998","unstructured":"Lowe, J.E. (1998). The possibility of metaphysics: Substance, identity, and time. Oxford: Clarendon Press."},{"key":"9604_CR21","doi-asserted-by":"crossref","unstructured":"McSweeney, M.M. (2020). Grounding logically complex facts. In M. Raven (Ed.) Routledge handbook of metaphysical grounding. Forthcoming. Routledge.","DOI":"10.4324\/9781351258845-43"},{"issue":"2","key":"9604_CR22","doi-asserted-by":"publisher","first-page":"156","DOI":"10.1017\/apa.2019.40","volume":"6","author":"M McSweeney","year":"2020","unstructured":"McSweeney, M. (2020). Debunking logical ground: Distinguishing metaphysics from semantics. Journal of the American Philosophical Association, 6 (2), 156\u2013170.","journal-title":"Journal of the American Philosophical Association"},{"key":"9604_CR23","unstructured":"Moschovakis, Y.N. (2008). Elementary induction on abstract structures. New York and Amsterdam: North-Holland and Elsevier (1974), unabridged republication by Dover Books."},{"issue":"238","key":"9604_CR24","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1111\/j.1467-9213.2009.642.x","volume":"60","author":"A Paseau","year":"2010","unstructured":"Paseau, A. (2010). Defining ultimate ontological basis and the fundamental layer. The Philosophical Quarterly, 60(238), 169\u2013175.","journal-title":"The Philosophical Quarterly"},{"key":"9604_CR25","doi-asserted-by":"publisher","first-page":"3147","DOI":"10.1007\/s11229-015-0923-x","volume":"193","author":"F Poggiolesi","year":"2016","unstructured":"Poggiolesi, F. (2016). On defining the notion of complete and immediate formal grounding. Synthese, 193, 3147\u20133167.","journal-title":"Synthese"},{"key":"9604_CR26","doi-asserted-by":"publisher","first-page":"1231","DOI":"10.1007\/s11229-016-1265-z","volume":"195","author":"F Poggiolesi","year":"2018","unstructured":"Poggiolesi, F. (2018). On constructing a logic for the notion of complete and immediate formal grounding. Synthese, 195, 1231\u20131254.","journal-title":"Synthese"},{"key":"9604_CR27","doi-asserted-by":"crossref","unstructured":"Poggiolesi, F. (2020). Logics of grounding. In M.J. Raven (Ed.) Routledge handbook of metaphysical grounding (pp. 213\u2013227). Routledge.","DOI":"10.4324\/9781351258845-21"},{"key":"9604_CR28","doi-asserted-by":"crossref","unstructured":"Rosen, G. (2010). Metaphysical dependence: Grounding and reduction explanation in mathematics. In B. Hale A. Hoffmann (Eds.) Modality: Metaphysics, Logic, and Epistemology (pp. 109\u2013136). Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199565818.003.0007"},{"issue":"8","key":"9604_CR29","first-page":"1985","volume":"175","author":"S Roski","year":"2018","unstructured":"Roski, S. (2018). Grounding and the explanatory role of generalizations. Journal of the History of Philosophy, 175(8), 1985\u20132003.","journal-title":"Journal of the History of Philosophy"},{"issue":"3","key":"9604_CR30","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1017\/S1755020313000154","volume":"6","author":"A Rumberg","year":"2013","unstructured":"Rumberg, A. (2013). Bolzano\u2019s concept of grounding (Abfolge) against the background of normal proofs. Review of Symbolic Logic, 6(3), 424\u2013459.","journal-title":"Review of Symbolic Logic"},{"issue":"1","key":"9604_CR31","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1215\/00318108-2009-025","volume":"119","author":"J Schaffer","year":"2010","unstructured":"Schaffer, J. (2010). Monism: The priority of the whole. The Philosophical Review, 119(1), 31\u201376.","journal-title":"The Philosophical Review"},{"issue":"3","key":"9604_CR32","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1017\/S1755020311000104","volume":"4","author":"B Schnieder","year":"2011","unstructured":"Schnieder, B. (2011). A logic for because. Review of Symbolic Logic, 4(3), 445\u2013465.","journal-title":"Review of Symbolic Logic"},{"key":"9604_CR33","doi-asserted-by":"publisher","first-page":"747","DOI":"10.1007\/s11098-018-1204-6","volume":"177","author":"T Sider","year":"2020","unstructured":"Sider, T. (2020). Ground grounded. Philos Stud, 177, 747\u2013767.","journal-title":"Philos Stud"},{"key":"9604_CR34","first-page":"175","volume":"41","author":"BH Slater","year":"1991","unstructured":"Slater, B.H. (1991). The epsilon calculus and its applications. Grazer Philosophische Studien, 41, 175\u2013205.","journal-title":"Grazer Philosophische Studien"},{"key":"9604_CR35","volume-title":"Basic proof theory","author":"AS Troelstra","year":"1996","unstructured":"Troelstra, A.S., & Schwichtenberg, H. (1996). Basic proof theory. Cambridge: Cambridge University Press."},{"issue":"4","key":"9604_CR36","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1007\/s10992-017-9442-1","volume":"47","author":"J Woods","year":"2018","unstructured":"Woods, J. (2018). Emptying a paradox of ground. Journal of Philosophical Logic, 47(4), 631\u2013648.","journal-title":"Journal of Philosophical Logic"},{"key":"9604_CR37","doi-asserted-by":"crossref","unstructured":"Zach, R. (2017). Semantics and proof theory of the epsilon calculus. In Proceedings of the 7th Indian conference on logic and its applications, (Vol. 10119 pp. 27\u201347). Springer.","DOI":"10.1007\/978-3-662-54069-5_4"}],"container-title":["Journal of Philosophical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10992-021-09604-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10992-021-09604-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10992-021-09604-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T10:30:58Z","timestamp":1672828258000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10992-021-09604-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,15]]},"references-count":37,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["9604"],"URL":"https:\/\/doi.org\/10.1007\/s10992-021-09604-w","relation":{},"ISSN":["0022-3611","1573-0433"],"issn-type":[{"value":"0022-3611","type":"print"},{"value":"1573-0433","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,15]]},"assertion":[{"value":"10 April 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 July 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interests"}}]}}