In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics co-logic and /?-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Q-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing. This paper is concerned with strong logics of first and second order. At the most abstract level, a strong logic of first-order has the following general form: Let L be a first-order language and let (x) be a formula that defines a class of L-structures. Then, for a recursively enumerable set T of sentences of L, and for a sentence ip of L set
{"title":"Strong logics of first and second order","authors":"P. Koellner","doi":"10.2178/bsl/1264433796","DOIUrl":"https://doi.org/10.2178/bsl/1264433796","url":null,"abstract":"In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics co-logic and /?-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Q-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing. This paper is concerned with strong logics of first and second order. At the most abstract level, a strong logic of first-order has the following general form: Let L be a first-order language and let (x) be a formula that defines a class of L-structures. Then, for a recursively enumerable set T of sentences of L, and for a sentence ip of L set","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"37 1 1","pages":"1-36"},"PeriodicalIF":0.6,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72953192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present disclosure involves a method of identifying animals involving the use of a universal identification scheme capable of identifying individual animals anywhere in the world such that data may collected for the animals over their entire life cycle.
{"title":"Fifty years of the spectrum problem: survey and new results","authors":"Arnaud Durand, N. Jones, J. Makowsky, Malika More","doi":"10.2178/bsl.1804020","DOIUrl":"https://doi.org/10.2178/bsl.1804020","url":null,"abstract":"The present disclosure involves a method of identifying animals involving the use of a universal identification scheme capable of identifying individual animals anywhere in the world such that data may collected for the animals over their entire life cycle.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"61 1","pages":"505-553"},"PeriodicalIF":0.6,"publicationDate":"2009-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83969130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"15th Workshop on Logic, Language, Information and Computation (WoLLIC 2008)","authors":"W. Hodges, Fairouz Kamareddine, R. D. Queiroz","doi":"10.2178/BSL/1231081466","DOIUrl":"https://doi.org/10.2178/BSL/1231081466","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"550 - 551"},"PeriodicalIF":0.6,"publicationDate":"2008-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2178/BSL/1231081466","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68346824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Abadi, Y. Abe, Andreas Abel, F. Abeles, Andrew Aberdein, J. D. Abernethy, B. Adams, Klaus Aehlig, F. Aeschbach, Henry Africk
This membership list may not be used for commercial purposes, for bulk mailing, or to prepare mailing lists, without written permission from the Association. For information regarding use of this list, contact the Secretary-Treasurer of the Association. Corrections to this list should be sent to Association Business Office: A.S.L., Box 742, Vassar College, 124 Raymond Ave., Poughkeepsie, N.Y. 12604 U.S.A.; email: asl@vassar.edu.
{"title":"Individual Members 2008","authors":"M. Abadi, Y. Abe, Andreas Abel, F. Abeles, Andrew Aberdein, J. D. Abernethy, B. Adams, Klaus Aehlig, F. Aeschbach, Henry Africk","doi":"10.2178/bsl/1231081469","DOIUrl":"https://doi.org/10.2178/bsl/1231081469","url":null,"abstract":"This membership list may not be used for commercial purposes, for bulk mailing, or to prepare mailing lists, without written permission from the Association. For information regarding use of this list, contact the Secretary-Treasurer of the Association. Corrections to this list should be sent to Association Business Office: A.S.L., Box 742, Vassar College, 124 Raymond Ave., Poughkeepsie, N.Y. 12604 U.S.A.; email: asl@vassar.edu.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"559 - 614"},"PeriodicalIF":0.6,"publicationDate":"2008-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68346725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Geraldine Brady. From Peirce to Skolem. A neglected chapter in the history of logic . Elsevier, Amsterdam, 2000, xi + 468 pp.","authors":"John Corcoran","doi":"10.2178/BSL/1231081463","DOIUrl":"https://doi.org/10.2178/BSL/1231081463","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"541 - 544"},"PeriodicalIF":0.6,"publicationDate":"2008-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2178/BSL/1231081463","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68346759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Second Indian Winter School on Logic","authors":"M. Banerjee, Anil Seth","doi":"10.2178/BSL/1231081464","DOIUrl":"https://doi.org/10.2178/BSL/1231081464","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"546 - 547"},"PeriodicalIF":0.6,"publicationDate":"2008-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2178/BSL/1231081464","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68346768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Errata to “Cohen and Set Theory” [1]","authors":"A. Kanamori","doi":"10.2178/bsl/1231081467","DOIUrl":"https://doi.org/10.2178/bsl/1231081467","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"552 - 552"},"PeriodicalIF":0.6,"publicationDate":"2008-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2178/bsl/1231081467","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68346961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we give probably an exhaustive analysis of the geometry of solids which was sketched by Tarski in his short paper [20, 21]. We show that in order to prove theorems stated in [20, 21] one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e., with solids. We show that once having adopted such a solution Tarski's Postulate 4 can be omitted, together with its versions 4′ and 4″. We also prove that the equivalence of postulates 4, 4′ and 4″ is not provable in any theory whose domain contains objects other than solids. Moreover, we show that the concentricity relation as defined by Tarski must be transitive in the largest class of structures satisfying Tarski's axioms. We build a model (in three-dimensional Euclidean space) of the theory of so called T*-structures and present the proof of the fact that this is the only (up to isomorphism) model of this theory. Moreover, we propose different categorical axiomatizations of the geometry of solids. In the final part of the paper we answer the question concerning the logical status (within the theory of T*-structures) of the definition of the concentricity relation given by Tarski.
{"title":"Full Development of Tarski's Geometry of Solids","authors":"Rafał Gruszczyński, A. Pietruszczak","doi":"10.2178/bsl/1231081462","DOIUrl":"https://doi.org/10.2178/bsl/1231081462","url":null,"abstract":"Abstract In this paper we give probably an exhaustive analysis of the geometry of solids which was sketched by Tarski in his short paper [20, 21]. We show that in order to prove theorems stated in [20, 21] one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e., with solids. We show that once having adopted such a solution Tarski's Postulate 4 can be omitted, together with its versions 4′ and 4″. We also prove that the equivalence of postulates 4, 4′ and 4″ is not provable in any theory whose domain contains objects other than solids. Moreover, we show that the concentricity relation as defined by Tarski must be transitive in the largest class of structures satisfying Tarski's axioms. We build a model (in three-dimensional Euclidean space) of the theory of so called T*-structures and present the proof of the fact that this is the only (up to isomorphism) model of this theory. Moreover, we propose different categorical axiomatizations of the geometry of solids. In the final part of the paper we answer the question concerning the logical status (within the theory of T*-structures) of the definition of the concentricity relation given by Tarski.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"481 - 540"},"PeriodicalIF":0.6,"publicationDate":"2008-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2178/bsl/1231081462","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68346751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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