{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T09:34:59Z","timestamp":1763544899507,"version":"3.37.3"},"reference-count":23,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2018,11,26]],"date-time":"2018-11-26T00:00:00Z","timestamp":1543190400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P 27527","M 2461"],"award-info":[{"award-number":["P 27527","M 2461"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,1,23]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper is part of a project that is based on the notion of a dialectical system, introduced by Magari as a way of capturing trial and error mathematics. In Amidei et al. (2016, Rev. Symb. Logic, 9, 1\u201326) and Amidei et al. (2016, Rev. Symb. Logic, 9, 299\u2013324), we investigated the expressive and computational power of dialectical systems, and we compared them to a new class of systems, that of quasi-dialectical systems, that enrich Magari\u2019s systems with a natural mechanism of revision. In the present paper we consider a third class of systems, that of $p$-dialectical systems, that naturally combine features coming from the two other cases. We prove several results about $p$-dialectical systems and the sets that they represent. Then we focus on the completions of first-order theories. In doing so, we consider systems with connectives, i.e. systems that encode the rules of classical logic. We show that any consistent system with connectives represents the completion of a given theory. We prove that dialectical and $q$-dialectical systems coincide with respect to the completions that they can represent. Yet, $p$-dialectical systems are more powerful; we exhibit a $p$-dialectical system representing a completion of Peano Arithmetic that is neither dialectical nor $q$-dialectical.<\/jats:p>","DOI":"10.1093\/logcom\/exy033","type":"journal-article","created":{"date-parts":[[2018,11,22]],"date-time":"2018-11-22T12:12:46Z","timestamp":1542888766000},"page":"157-184","source":"Crossref","is-referenced-by-count":2,"title":["Trial and error mathematics: Dialectical systems and completions of theories"],"prefix":"10.1093","volume":"29","author":[{"given":"Jacopo","family":"Amidei","sequence":"first","affiliation":[{"name":"School of Computing & Communications, The Open University, Milton Keynes, UK"}]},{"given":"Uri","family":"Andrews","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA"}]},{"given":"Duccio","family":"Pianigiani","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria Informatica e Scienze Matematiche, Universit\u00e0 Degli Studi di Siena, I-53100 Siena, Italy"}]},{"given":"Luca","family":"San Mauro","sequence":"first","affiliation":[{"name":"Institute for Discrete Mathematics and Geometry, Vienna University of Technology, 1040 Vienna, Austria"}]},{"given":"Andrea","family":"Sorbi","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria Informatica e Scienze Matematiche, Universit\u00e0 Degli Studi di Siena, I-53100 Siena, Italy"}]}],"member":"286","published-online":{"date-parts":[[2018,11,26]]},"reference":[{"key":"2020042900055728600_ref1","doi-asserted-by":"crossref","first-page":"510","DOI":"10.2307\/2274239","article-title":"On the logic of theory change: partial meet contraction and revision functions","volume":"50","author":"Alchourr\u00f3n","year":"1985","journal-title":"Journal of Symbolic Logic"},{"key":"2020042900055728600_ref2","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1017\/S1755020315000404","article-title":"Trial and error mathematics I: dialectical and quasi-dialectical systems","volume":"9","author":"Amidei","year":"2016","journal-title":"Review of Symbolic Logic"},{"key":"2020042900055728600_ref3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S1755020316000253","article-title":"Trial and error mathematics II: dialectical sets and quasi-dialectical sets, their degrees, and their distribution within the class of limit sets","volume":"9","author":"Amidei","year":"2016","journal-title":"Review of Symbolic Logic"},{"key":"2020042900055728600_ref4","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/B978-044451541-4\/50006-3","article-title":"Logic in philosophy","volume-title":"Philosophy of Logic","author":"van Benthem","year":"2007"},{"key":"2020042900055728600_ref5","first-page":"51","article-title":"Aspetti ricorsivi degli insiemi dialettici","volume":"9","author":"Bernardi","year":"1974","journal-title":"Bollettino della Unione Matematica Italiana. 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