Pub Date : 2020-11-01DOI: 10.1215/00294527-2020-0025
Adam Přenosil
The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth (truth and nonfalsity). Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued logics, for example, the logic of truth and exact truth or the logic of truth and material equivalence. These combined systems are consequence relations which allow us to express implications involving more than one of these features of propositions.
{"title":"Four-Valued Logics of Truth, Nonfalsity, Exact Truth, and Material Equivalence","authors":"Adam Přenosil","doi":"10.1215/00294527-2020-0025","DOIUrl":"https://doi.org/10.1215/00294527-2020-0025","url":null,"abstract":"The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth (truth and nonfalsity). Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued logics, for example, the logic of truth and exact truth or the logic of truth and material equivalence. These combined systems are consequence relations which allow us to express implications involving more than one of these features of propositions.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"61 1","pages":"601-621"},"PeriodicalIF":0.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49311065","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 : 2020-10-20DOI: 10.1215/00294527-2022-0002
Annalisa Conversano, M. Mamino
We prove that a first-order structure defines (resp. interprets) every connected Lie group if and only if it defines (resp. interprets) the real field expanded with a predicate for the integers.
{"title":"One Lie Group to Define Them All","authors":"Annalisa Conversano, M. Mamino","doi":"10.1215/00294527-2022-0002","DOIUrl":"https://doi.org/10.1215/00294527-2022-0002","url":null,"abstract":"We prove that a first-order structure defines (resp. interprets) every connected Lie group if and only if it defines (resp. interprets) the real field expanded with a predicate for the integers.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45028956","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 : 2020-09-01DOI: 10.1215/00294527-2020-0019
A. Pruss
{"title":"Erratum for \"Conditionals and Conditional Probabilities without Triviality\"","authors":"A. Pruss","doi":"10.1215/00294527-2020-0019","DOIUrl":"https://doi.org/10.1215/00294527-2020-0019","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"47 1","pages":"501"},"PeriodicalIF":0.7,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87595831","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 : 2020-09-01DOI: 10.1215/00294527-2020-0016
Jon Erling Litland
{"title":"Prospects for a Theory of Decycling","authors":"Jon Erling Litland","doi":"10.1215/00294527-2020-0016","DOIUrl":"https://doi.org/10.1215/00294527-2020-0016","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"84 1","pages":"467-499"},"PeriodicalIF":0.7,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83806800","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 : 2020-09-01DOI: 10.1215/00294527-2020-0012
David Chodounský, J. Zapletal
{"title":"Ideals and Their Generic Ultrafilters","authors":"David Chodounský, J. Zapletal","doi":"10.1215/00294527-2020-0012","DOIUrl":"https://doi.org/10.1215/00294527-2020-0012","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"208 1","pages":"403-408"},"PeriodicalIF":0.7,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76429199","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 : 2020-08-25DOI: 10.1305/ndjfl/1093637871
G. Boolos
www.tlsbooks.com Alphabetical Order Read the words in each row. Write the number 1 by the word that is first in alphabetical order. Write the number 2 by the word that is second in alphabetical order. Write the number 3 by the word that is last in alphabetical order. The first one is done for you. ____ cute ____ turtle ____ little ____ snake ____ gopher ____ cobra ____ cactus ____ plant ____ grow ____ rabbits ____ fast ____ hop ____ cookies ____ milk ____ full ____ camping ____ flashlight ____ tent ____ truck ____ jeep ____ car ____ quickly ____ gerbils ____ run ____ I ____ support ____ zoos ____ needs ____ earth ____ respect ____ sting ____ scorpions ____ sometimes ____ useful ____ are ____ almanacs ____ chase ____ cats ____ mice ____ Asia ____ Europe ____ Australia ____ develop ____ grow ____ learn ____ math ____ numbers ____ googol ____ drum ____ flute ____ cello ____ plane ____ ship ____ train 1 3 2
{"title":"Alphabetical order","authors":"G. Boolos","doi":"10.1305/ndjfl/1093637871","DOIUrl":"https://doi.org/10.1305/ndjfl/1093637871","url":null,"abstract":"www.tlsbooks.com Alphabetical Order Read the words in each row. Write the number 1 by the word that is first in alphabetical order. Write the number 2 by the word that is second in alphabetical order. Write the number 3 by the word that is last in alphabetical order. The first one is done for you. ____ cute ____ turtle ____ little ____ snake ____ gopher ____ cobra ____ cactus ____ plant ____ grow ____ rabbits ____ fast ____ hop ____ cookies ____ milk ____ full ____ camping ____ flashlight ____ tent ____ truck ____ jeep ____ car ____ quickly ____ gerbils ____ run ____ I ____ support ____ zoos ____ needs ____ earth ____ respect ____ sting ____ scorpions ____ sometimes ____ useful ____ are ____ almanacs ____ chase ____ cats ____ mice ____ Asia ____ Europe ____ Australia ____ develop ____ grow ____ learn ____ math ____ numbers ____ googol ____ drum ____ flute ____ cello ____ plane ____ ship ____ train 1 3 2","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"36 6","pages":"214-215"},"PeriodicalIF":0.7,"publicationDate":"2020-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1305/ndjfl/1093637871","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72371441","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 : 2020-06-18DOI: 10.1215/00294527-1715653
Thomas F. Icard, J. Joosten
The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $Gamma$. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set $Gamma$, where each sentence in $Gamma$ has a well understood (meta)-mathematical content in T, the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between PRA and I$Sigma_1$. We then study interpretability logics, obtaining some upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that IL(PRA) $subset$ ILM.
{"title":"Provability and Interpretability Logics with Restricted Realizations","authors":"Thomas F. Icard, J. Joosten","doi":"10.1215/00294527-1715653","DOIUrl":"https://doi.org/10.1215/00294527-1715653","url":null,"abstract":"The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $Gamma$. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set $Gamma$, where each sentence in $Gamma$ has a well understood (meta)-mathematical content in T, the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between PRA and I$Sigma_1$. We then study interpretability logics, obtaining some upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that IL(PRA) $subset$ ILM.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"55 1","pages":"133-154"},"PeriodicalIF":0.7,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78394337","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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