{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T11:47:58Z","timestamp":1759146478780},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2018,12,26]],"date-time":"2018-12-26T00:00:00Z","timestamp":1545782400000},"content-version":"unspecified","delay-in-days":0,"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":[[2019,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper is a follow-up to [4], in which a mistake in [6] (which spread also to [9]) was corrected. We give a strenghtening of the main result on the semantical nonconservativity of the theory of PT<jats:sup>\u2212<\/jats:sup>with internal induction for total formulae<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1755020318000400_inline1\"\/><jats:tex-math>${(\\rm{P}}{{\\rm{T}}^ - } + {\\rm{INT}}\\left( {{\\rm{tot}}} \\right)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, denoted by PT<jats:sup>\u2212<\/jats:sup>in [9]). We show that if to PT<jats:sup>\u2212<\/jats:sup>the axiom of internal induction for<jats:italic>all<\/jats:italic>arithmetical formulae is added (giving<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1755020318000400_inline2\"\/><jats:tex-math>${\\rm{P}}{{\\rm{T}}^ - } + {\\rm{INT}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>), then this theory is semantically stronger than<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1755020318000400_inline3\"\/><jats:tex-math>${\\rm{P}}{{\\rm{T}}^ - } + {\\rm{INT}}\\left( {{\\rm{tot}}} \\right)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In particular the latter is not relatively truth definable (in the sense of [11]) in the former. Last but not least, we provide an axiomatic theory of truth which meets the requirements put forward by Fischer and Horsten in [9]. The truth theory we define is based on Weak Kleene Logic instead of the Strong one.<\/jats:p>","DOI":"10.1017\/s1755020318000400","type":"journal-article","created":{"date-parts":[[2018,12,26]],"date-time":"2018-12-26T10:27:53Z","timestamp":1545820073000},"page":"144-172","source":"Crossref","is-referenced-by-count":5,"title":["MODELS OF POSITIVE TRUTH"],"prefix":"10.1017","volume":"12","author":[{"given":"MATEUSZ","family":"\u0141E\u0141YK","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"BARTOSZ","family":"WCIS\u0141O","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2018,12,26]]},"reference":[{"key":"S1755020318000400_ref26","doi-asserted-by":"publisher","DOI":"10.2307\/2273620"},{"key":"S1755020318000400_ref24","first-page":"493","article-title":"Proof and truth: Through thick and thin","volume":"95","author":"Shapiro","year":"1998","journal-title":"Jornal of Philosophy"},{"key":"S1755020318000400_ref23","unstructured":"[23] Schindler R. ACA0 and ${\\rm{\\Pi }}_1^1 - {\\rm{C}}{{\\rm{A}}_0}$ and the semantics of arithmetic and BG and , ${\\rm{BG}} + {\\rm{\\Sigma }}_1^1 - {\\rm{Ind}}$ , and the semantics of set theory. Unpublished manuscript. 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