Pub Date : 2022-11-01DOI: 10.1215/00294527-2022-0029
L. Humberstone, Steven T. Kuhn
A notable early result of David Makinson establishes that every monotone modal logic can be extended to L I , L V or L F , and every antitone logic, to L N , L V or L F , where L I , L N , L V and L F are logics axiomatized, respectively, by the schemas (cid:50) α ↔ α , (cid:50) α ↔ ¬ α , (cid:50) α ↔ ⊤ and (cid:50) α ↔ ⊥ . We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: L V , L F and the minimum amphitone logic AM axiomatized by the schema (cid:50) α → (cid:50) β . These logics, along with L I , L N and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the non-modal connectives. We close by discussing these divergences and conditions under which our results do carry over.
{"title":"Modal Logics That Are Both Monotone and Antitone: Makinson’s Extension Results and Affinities between Logics","authors":"L. Humberstone, Steven T. Kuhn","doi":"10.1215/00294527-2022-0029","DOIUrl":"https://doi.org/10.1215/00294527-2022-0029","url":null,"abstract":"A notable early result of David Makinson establishes that every monotone modal logic can be extended to L I , L V or L F , and every antitone logic, to L N , L V or L F , where L I , L N , L V and L F are logics axiomatized, respectively, by the schemas (cid:50) α ↔ α , (cid:50) α ↔ ¬ α , (cid:50) α ↔ ⊤ and (cid:50) α ↔ ⊥ . We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: L V , L F and the minimum amphitone logic AM axiomatized by the schema (cid:50) α → (cid:50) β . These logics, along with L I , L N and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the non-modal connectives. We close by discussing these divergences and conditions under which our results do carry over.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46460894","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}
Pub Date : 2022-11-01DOI: 10.1215/00294527-2022-0032
Yale Weiss
{"title":"Did Aristotle Endorse Aristotle’s Thesis? A Case Study in Aristotle’s Metalogic","authors":"Yale Weiss","doi":"10.1215/00294527-2022-0032","DOIUrl":"https://doi.org/10.1215/00294527-2022-0032","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43198115","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}
Pub Date : 2022-08-01DOI: 10.1215/00294527-2022-0017
Fausto Barbero, Fan Yang
{"title":"Characterizing Counterfactuals and Dependencies over (Generalized) Causal Teams","authors":"Fausto Barbero, Fan Yang","doi":"10.1215/00294527-2022-0017","DOIUrl":"https://doi.org/10.1215/00294527-2022-0017","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42801668","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}
Pub Date : 2022-08-01DOI: 10.1215/00294527-2022-0020
John Grant
{"title":"Measuring Inconsistency in Some Logics with Tense Operators","authors":"John Grant","doi":"10.1215/00294527-2022-0020","DOIUrl":"https://doi.org/10.1215/00294527-2022-0020","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48716008","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}
Pub Date : 2022-05-01DOI: 10.1215/00294527-2022-0014
Jim de Groot, D. Pattinson
{"title":"Monotone Subintuitionistic Logic: Duality and Transfer Results","authors":"Jim de Groot, D. Pattinson","doi":"10.1215/00294527-2022-0014","DOIUrl":"https://doi.org/10.1215/00294527-2022-0014","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42341675","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}
Pub Date : 2022-05-01DOI: 10.1215/00294527-2022-0009
Hitoshi Omori, Michael De
{"title":"Shrieking, Shrugging, and the Australian Plan","authors":"Hitoshi Omori, Michael De","doi":"10.1215/00294527-2022-0009","DOIUrl":"https://doi.org/10.1215/00294527-2022-0009","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46455639","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}
Pub Date : 2022-03-16DOI: 10.1215/00294527-2022-0027
D. Hirschfeldt, Sarah C. Reitzes
This paper is part of a line of research on the computability-theoretic and reverse-mathematical strength of versions of Hindman’s Theorem [6] that began with the work of Blass, Hirst, and Simpson [1], and has seen considerable interest recently. We assume basic familiarity with computability theory and reverse mathematics, at the level of the background material in [8], for instance. On the reverse mathematics side, the two major systems with which we will be concerned are RCA0, the usual weak base system for reverse mathematics, which corresponds roughly to computable mathematics; and ACA0, which corresponds roughly to arithmetic mathematics. For principles P of the form (∀X) [Φ(X) → (∃Y ) Ψ(X, Y )], we call any X such that Φ(X) holds an instance of P , and any Y such that Ψ(X, Y ) holds a solution to X . We begin by introducing some related combinatorial principles. For a set S, let [S] be the set of n-element subsets of S. Ramsey’s Theorem (RT) is the statement that for every n and every coloring of [N] with finitely many colors, there is an infinite set H that is homogeneous for c, which means that all elements of [H ] have the same color. There has been a great deal of work on computability-theoretic and reverse-mathematical aspects of versions of Ramsey’s Theorem, such as RTnk , which is RT restricted to colorings of [N] n with k many colors. (See e.g. [8].)
{"title":"Thin Set Versions of Hindman’s Theorem","authors":"D. Hirschfeldt, Sarah C. Reitzes","doi":"10.1215/00294527-2022-0027","DOIUrl":"https://doi.org/10.1215/00294527-2022-0027","url":null,"abstract":"This paper is part of a line of research on the computability-theoretic and reverse-mathematical strength of versions of Hindman’s Theorem [6] that began with the work of Blass, Hirst, and Simpson [1], and has seen considerable interest recently. We assume basic familiarity with computability theory and reverse mathematics, at the level of the background material in [8], for instance. On the reverse mathematics side, the two major systems with which we will be concerned are RCA0, the usual weak base system for reverse mathematics, which corresponds roughly to computable mathematics; and ACA0, which corresponds roughly to arithmetic mathematics. For principles P of the form (∀X) [Φ(X) → (∃Y ) Ψ(X, Y )], we call any X such that Φ(X) holds an instance of P , and any Y such that Ψ(X, Y ) holds a solution to X . We begin by introducing some related combinatorial principles. For a set S, let [S] be the set of n-element subsets of S. Ramsey’s Theorem (RT) is the statement that for every n and every coloring of [N] with finitely many colors, there is an infinite set H that is homogeneous for c, which means that all elements of [H ] have the same color. There has been a great deal of work on computability-theoretic and reverse-mathematical aspects of versions of Ramsey’s Theorem, such as RTnk , which is RT restricted to colorings of [N] n with k many colors. (See e.g. [8].)","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46765150","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}
Pub Date : 2022-03-04DOI: 10.1215/00294527-10701451
V. Fischer, Marlene Koelbing, Wolfgang Wohofsky
Using a game characterization of distributivity, we show that base matrices for $mathcal{P}(omega)/text{fin}$ of regular height larger than $mathfrak{h}$ necessarily have maximal branches which are not cofinal.
{"title":"Games on Base Matrices","authors":"V. Fischer, Marlene Koelbing, Wolfgang Wohofsky","doi":"10.1215/00294527-10701451","DOIUrl":"https://doi.org/10.1215/00294527-10701451","url":null,"abstract":"Using a game characterization of distributivity, we show that base matrices for $mathcal{P}(omega)/text{fin}$ of regular height larger than $mathfrak{h}$ necessarily have maximal branches which are not cofinal.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48033427","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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