Pub Date : 2023-08-01DOI: 10.1215/00294527-2023-0008
Jeremiah Joven Joaquin
This paper presents a weak Kleene approach to conditionals that preserves some salient formal features of conditionals, particularly their interdefinability with Boolean logical connectives. I argue that such an approach fares better than other proposed weak logics of conditionals in this regard. In particular, it fares better than the logics proposed by Cooper, Cantwell, Farrell, De Finetti, Égré, Rossi, and Sprenger.
{"title":"Weak Kleene and Other Weak Logics of Conditionals","authors":"Jeremiah Joven Joaquin","doi":"10.1215/00294527-2023-0008","DOIUrl":"https://doi.org/10.1215/00294527-2023-0008","url":null,"abstract":"This paper presents a weak Kleene approach to conditionals that preserves some salient formal features of conditionals, particularly their interdefinability with Boolean logical connectives. I argue that such an approach fares better than other proposed weak logics of conditionals in this regard. In particular, it fares better than the logics proposed by Cooper, Cantwell, Farrell, De Finetti, Égré, Rossi, and Sprenger.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135005604","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 : 2023-08-01DOI: 10.1215/00294527-2023-0010
Jaruwat Rodbanjong, Athipat Thamrongthanyalak
Here we work in an arbitrary o-minimal expansion of a divisible ordered abelian group. We say that a definable ring is definably semiprime if squares of nontrivial two-sided ideals definable in the expansion are nontrivial. We prove a definable version of Wedderburn–Artin theorem and give a characterization of definably semiprime rings.
{"title":"Definable Version of Wedderburn–Artin Theorem in O-Minimal Structures","authors":"Jaruwat Rodbanjong, Athipat Thamrongthanyalak","doi":"10.1215/00294527-2023-0010","DOIUrl":"https://doi.org/10.1215/00294527-2023-0010","url":null,"abstract":"Here we work in an arbitrary o-minimal expansion of a divisible ordered abelian group. We say that a definable ring is definably semiprime if squares of nontrivial two-sided ideals definable in the expansion are nontrivial. We prove a definable version of Wedderburn–Artin theorem and give a characterization of definably semiprime rings.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135055593","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 : 2023-08-01DOI: 10.1215/00294527-2023-0012
Brice Halimi
The purpose of this paper is to generalize Kripke semantics for propositional modal logic by geometrizing it, that is, by considering the space underlying the collection of all possible worlds as an important semantic feature in its own right, so as to take the idea of accessibility seriously. The resulting new modal semantics is worked out in a setting coming from Riemannian geometry, where Kripke semantics is shown to correspond to a particular case, namely, the discrete one. Several correspondence results, established between variants of well-known modal systems and corresponding geometric properties, illustrate the import of this new framework.
{"title":"Geometric Modal Logic","authors":"Brice Halimi","doi":"10.1215/00294527-2023-0012","DOIUrl":"https://doi.org/10.1215/00294527-2023-0012","url":null,"abstract":"The purpose of this paper is to generalize Kripke semantics for propositional modal logic by geometrizing it, that is, by considering the space underlying the collection of all possible worlds as an important semantic feature in its own right, so as to take the idea of accessibility seriously. The resulting new modal semantics is worked out in a setting coming from Riemannian geometry, where Kripke semantics is shown to correspond to a particular case, namely, the discrete one. Several correspondence results, established between variants of well-known modal systems and corresponding geometric properties, illustrate the import of this new framework.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135053037","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 : 2023-08-01DOI: 10.1215/00294527-2023-0006
Edwin Mares
This paper begins with a discussion of C. I. Lewis’s theory of meaning in his book, An Analysis of Knowledge and Valuation (1946) and his pragmatic theory of analyticity and necessity. I bring this theories together with some remarks that he makes in an appendix to the second edition of Symbolic Logic to construct an algebraic semantics for his logics S2 and S3. These logics and their semantics are compared and evaluated with regard to how well they implement Lewis’s theories of meaning and analyticity.
{"title":"C. I. Lewis’s Intensional Semantics","authors":"Edwin Mares","doi":"10.1215/00294527-2023-0006","DOIUrl":"https://doi.org/10.1215/00294527-2023-0006","url":null,"abstract":"This paper begins with a discussion of C. I. Lewis’s theory of meaning in his book, An Analysis of Knowledge and Valuation (1946) and his pragmatic theory of analyticity and necessity. I bring this theories together with some remarks that he makes in an appendix to the second edition of Symbolic Logic to construct an algebraic semantics for his logics S2 and S3. These logics and their semantics are compared and evaluated with regard to how well they implement Lewis’s theories of meaning and analyticity.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135055594","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 : 2023-08-01DOI: 10.1215/00294527-2023-0005
Kyle Gannon, Jinhe Ye
Motivated by the theory of domination for types, we introduce a notion of domination for Keisler measures called extension domination. We argue that this variant of domination behaves similarly to its typesetting counterpart. We prove that extension domination extends domination for types and that it forms a preorder on the space of global Keisler measures. We then explore some basic properties related to this notion (e.g., approximations by formulas, closure under localizations, convex combinations). We also prove a few preservation theorems and provide some explicit examples.
{"title":"An Invitation to Extension Domination","authors":"Kyle Gannon, Jinhe Ye","doi":"10.1215/00294527-2023-0005","DOIUrl":"https://doi.org/10.1215/00294527-2023-0005","url":null,"abstract":"Motivated by the theory of domination for types, we introduce a notion of domination for Keisler measures called extension domination. We argue that this variant of domination behaves similarly to its typesetting counterpart. We prove that extension domination extends domination for types and that it forms a preorder on the space of global Keisler measures. We then explore some basic properties related to this notion (e.g., approximations by formulas, closure under localizations, convex combinations). We also prove a few preservation theorems and provide some explicit examples.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135002828","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 : 2023-08-01DOI: 10.1215/00294527-2023-0007
Marco Grossi
In the substitutional framework, validity is truth under all substitutions of the nonlogical vocabulary. I develop a theory where □ is interpreted as substitutional validity. I show how to prove soundness and completeness for common modal calculi using this definition.
{"title":"Substitutional Validity for Modal Logic","authors":"Marco Grossi","doi":"10.1215/00294527-2023-0007","DOIUrl":"https://doi.org/10.1215/00294527-2023-0007","url":null,"abstract":"In the substitutional framework, validity is truth under all substitutions of the nonlogical vocabulary. I develop a theory where □ is interpreted as substitutional validity. I show how to prove soundness and completeness for common modal calculi using this definition.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135003739","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 : 2023-08-01DOI: 10.1215/00294527-2023-0011
Naosuke Matsuda
A set F of Boolean functions is said to be functionally complete if every Boolean function is definable by combining functions in F. Post clarified when a set of Boolean functions is functionally complete (with respect to classical semantics). In this paper, by extending Post’s theorem, we clarify when a set of Boolean functions is functionally complete with respect to Kripke semantics.
{"title":"Definability of Boolean Functions in Kripke Semantics","authors":"Naosuke Matsuda","doi":"10.1215/00294527-2023-0011","DOIUrl":"https://doi.org/10.1215/00294527-2023-0011","url":null,"abstract":"A set F of Boolean functions is said to be functionally complete if every Boolean function is definable by combining functions in F. Post clarified when a set of Boolean functions is functionally complete (with respect to classical semantics). In this paper, by extending Post’s theorem, we clarify when a set of Boolean functions is functionally complete with respect to Kripke semantics.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135051656","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 : 2023-08-01DOI: 10.1215/00294527-2023-0009
Borisa Kuzeljevic, Dilip Raghavan, Jonathan L. Verner
We show that $mathrm{MA}_{kappa}$ implies that each collection of ${P}_{mathfrak c}$-points of size at most $kappa$ which has a $P_{mathfrak c}$-point as an $RK$ upper bound also has a ${P}_{mathfrak c}$-point as an $RK$ lower bound.
{"title":"Lower Bounds of Sets of P-points","authors":"Borisa Kuzeljevic, Dilip Raghavan, Jonathan L. Verner","doi":"10.1215/00294527-2023-0009","DOIUrl":"https://doi.org/10.1215/00294527-2023-0009","url":null,"abstract":"We show that $mathrm{MA}_{kappa}$ implies that each collection of ${P}_{mathfrak c}$-points of size at most $kappa$ which has a $P_{mathfrak c}$-point as an $RK$ upper bound also has a ${P}_{mathfrak c}$-point as an $RK$ lower bound.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135005605","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 : 2023-05-01DOI: 10.1215/00294527-10670082
Krzysztof A. Krawczyk
{"title":"Deduction Theorem in Congruential Modal Logics","authors":"Krzysztof A. Krawczyk","doi":"10.1215/00294527-10670082","DOIUrl":"https://doi.org/10.1215/00294527-10670082","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43973220","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 : 2023-05-01DOI: 10.1215/00294527-10670103
U. Petersen
{"title":"Dialetheias and Numbers Distinct from Themselves","authors":"U. Petersen","doi":"10.1215/00294527-10670103","DOIUrl":"https://doi.org/10.1215/00294527-10670103","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45434702","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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