{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T17:38:00Z","timestamp":1782927480514,"version":"3.54.5"},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2018,6,13]],"date-time":"2018-06-13T00:00:00Z","timestamp":1528848000000},"content-version":"unspecified","delay-in-days":12,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2018,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Mathematicians prove theorems in a semi-formal setting, providing what we\u2019ll call <jats:italic>informal proofs<\/jats:italic>. There are various philosophical reasons not to reduce informal provability to formal provability within some appropriate axiomatic theory (Leitgeb, 2009; Marfori, 2010; Tanswell, 2015), but the main worry is that we seem committed to all instances of the so-called reflection schema: <jats:italic>B<\/jats:italic>(<jats:italic>\u03c6<\/jats:italic>) \u2192 <jats:italic>\u03c6<\/jats:italic> (where <jats:italic>B<\/jats:italic> stands for the informal provability predicate). Yet, adding all its instances to any theory for which L\u00f6b\u2019s theorem for <jats:italic>B<\/jats:italic> holds leads to inconsistency.<\/jats:p><jats:p>Currently existing approaches (Shapiro, 1985; Horsten, 1996, 1998) to formalizing the properties of informal provability avoid contradiction at a rather high price. They either drop one of the Hilbert-Bernays conditions for the provability predicate, or use a provability operator that cannot consistently be treated as a predicate.<\/jats:p><jats:p>Inspired by (Kripke, 1975), we investigate the strategy which changes the underlying logic and treats informal provability as a partial notion. We use non-deterministic matrices to develop a three-valued logic of informal provability, which avoids some of the above mentioned problems.<\/jats:p>","DOI":"10.1017\/s1755020317000363","type":"journal-article","created":{"date-parts":[[2018,6,13]],"date-time":"2018-06-13T06:31:43Z","timestamp":1528871503000},"page":"207-223","source":"Crossref","is-referenced-by-count":13,"title":["MANY-VALUED LOGIC OF INFORMAL PROVABILITY: A NON-DETERMINISTIC STRATEGY"],"prefix":"10.1017","volume":"11","author":[{"given":"PAWEL","family":"PAWLOWSKI","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"RAFAL","family":"URBANIAK","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2018,6,13]]},"reference":[{"key":"S1755020317000363_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-3687-4"},{"key":"S1755020317000363_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-010-9280-4"},{"key":"S1755020317000363_ref9","doi-asserted-by":"publisher","DOI":"10.2307\/2024634"},{"key":"S1755020317000363_ref5","doi-asserted-by":"publisher","DOI":"10.1080\/11663081.2015.1011543"},{"key":"S1755020317000363_ref7","doi-asserted-by":"publisher","DOI":"10.1023\/A:1005016405987"},{"key":"S1755020317000363_ref10","doi-asserted-by":"publisher","DOI":"10.1057\/9780230245198_13"},{"key":"S1755020317000363_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70138-1"},{"key":"S1755020317000363_ref8","first-page":"203","volume-title":"Principles of Truth","author":"Horsten","year":"2002"},{"key":"S1755020317000363_ref14","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/nkv008"},{"key":"S1755020317000363_ref4","volume-title":"The Logic of Provability","author":"Boolos","year":"1993"},{"key":"S1755020317000363_ref6","doi-asserted-by":"publisher","DOI":"10.2307\/2275785"},{"key":"S1755020317000363_ref1","first-page":"227","volume-title":"Handbook of Philosophical Logic","volume":"Vol. 16","author":"Gabbay","year":"2011"},{"key":"S1755020317000363_ref2","first-page":"227","article-title":"A rich paraconsistent extension of full positive logic","volume":"47","author":"Batens","year":"2004","journal-title":"Logique Et Analyse"},{"key":"S1755020317000363_ref3","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/7.3.322"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020317000363","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,14]],"date-time":"2019-04-14T16:17:20Z","timestamp":1555258640000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020317000363\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6]]},"references-count":14,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,6]]}},"alternative-id":["S1755020317000363"],"URL":"https:\/\/doi.org\/10.1017\/s1755020317000363","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6]]}}}