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Math. Logic"],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In Bagaria (J Symb Log 81(2), 584\u2013604, 2016), Bagaria and V\u00e4\u00e4n\u00e4nen developed a framework for studying the large cardinal strength of <jats:italic>downwards<\/jats:italic> L\u00f6wenheim-Skolem theorems and related set theoretic reflection properties. The main tool was the notion of <jats:italic>symbiosis<\/jats:italic>, originally introduced by the third author in V\u00e4\u00e4n\u00e4nen (Applications of set theory to generalized quantifiers. PhD thesis, University of Manchester, 1967); V\u00e4\u00e4n\u00e4nen (in Logic Colloquium \u201978 (Mons, 1978), volume\u00a097 of Stud. Logic Foundations Math., pages 391\u2013421. North-Holland, Amsterdam 1979) <jats:italic>Symbiosis<\/jats:italic> provides a way of relating model theoretic properties of strong logics to definability in set theory. In this paper we continue the systematic investigation of <jats:italic>symbiosis<\/jats:italic> and apply it to <jats:italic>upwards<\/jats:italic> L\u00f6wenheim-Skolem theorems and reflection principles. To achieve this, we need to adapt the notion of <jats:italic>symbiosis<\/jats:italic> to a new form, called <jats:italic>bounded symbiosis<\/jats:italic>. As one easy application, we obtain upper and lower bounds for the large cardinal strength of upwards L\u00f6wenheim\u2013Skolem-type principles for second order logic.<\/jats:p>","DOI":"10.1007\/s00153-024-00955-0","type":"journal-article","created":{"date-parts":[[2024,12,23]],"date-time":"2024-12-23T16:22:42Z","timestamp":1734970962000},"page":"579-603","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bounded symbiosis and upwards reflection"],"prefix":"10.1007","volume":"64","author":[{"given":"Lorenzo","family":"Galeotti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1763-876X","authenticated-orcid":false,"given":"Yurii","family":"Khomskii","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jouko","family":"V\u00e4\u00e4n\u00e4nen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,12,23]]},"reference":[{"issue":"3\u20134","key":"955_CR1","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1007\/s00153-011-0261-8","volume":"51","author":"J Bagaria","year":"2012","unstructured":"Bagaria, J.: $$C^{(n)}$$-cardinals. 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