Pub Date : 2023-11-28DOI: 10.1017/s1755020323000357
DANIELE MUNDICI
Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class $Omega $ of uniform ultrafilters generates a $Delta $-closed logic ${mathcal {L}}_Omega $. ${mathcal {L}}_Omega $ is $omega $-relatively compact iff some $Din Omega $ fails to be $omega _1$-complete iff ${mathcal {L}}_Omega $ does not contain the quantifier “there are uncountably many.” If
{"title":"LOGICS FROM ULTRAFILTERS","authors":"DANIELE MUNDICI","doi":"10.1017/s1755020323000357","DOIUrl":"https://doi.org/10.1017/s1755020323000357","url":null,"abstract":"<p>Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221125331656-0178:S1755020323000357:S1755020323000357_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$Omega $</span></span></img></span></span> of uniform ultrafilters generates a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221125331656-0178:S1755020323000357:S1755020323000357_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$Delta $</span></span></img></span></span>-closed logic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221125331656-0178:S1755020323000357:S1755020323000357_inline3.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal {L}}_Omega $</span></span></img></span></span>. <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221125331656-0178:S1755020323000357:S1755020323000357_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal {L}}_Omega $</span></span></img></span></span> is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221125331656-0178:S1755020323000357:S1755020323000357_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$omega $</span></span></img></span></span>-relatively compact iff some <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221125331656-0178:S1755020323000357:S1755020323000357_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$Din Omega $</span></span></img></span></span> fails to be <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221125331656-0178:S1755020323000357:S1755020323000357_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$omega _1$</span></span></img></span></span>-complete iff <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221125331656-0178:S1755020323000357:S1755020323000357_inline8.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal {L}}_Omega $</span></span></img></span></span> does not contain the quantifier “there are uncountably many.” If <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221125331656-0178:S1755020323000357:S1755020323000","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139020521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-27DOI: 10.1017/s1755020322000193
ALEXANDER W. KOCUREK
Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical interpretation of the connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators.
{"title":"THE LOGIC OF HYPERLOGIC. PART A: FOUNDATIONS","authors":"ALEXANDER W. KOCUREK","doi":"10.1017/s1755020322000193","DOIUrl":"https://doi.org/10.1017/s1755020322000193","url":null,"abstract":"<p>Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical interpretation of the connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators.</p>","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"105 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138544258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-25DOI: 10.1017/s1755020322000181
TOBY MEADOWS
In providing a good foundation for mathematics, set theorists often aim to develop the strongest theories possible and avoid those theories that place undue restrictions on the capacity to possess strength. For example, adding a measurable cardinal to $ZFC$ is thought to give a stronger theory than adding $V=L$ and the latter is thought to be more restrictive than the former. The two main proponents of this style of account are Penelope Maddy and John Steel. In this paper, I’ll offer a third account that is intended to provide a simple analysis of restrictiveness based on the algebraic concept of retraction in the category of theories. I will also deliver some results and arguments that suggest some plausible alternative approaches to analyzing restrictiveness do not live up to their intuitive motivation.
{"title":"WHAT IS A RESTRICTIVE THEORY?","authors":"TOBY MEADOWS","doi":"10.1017/s1755020322000181","DOIUrl":"https://doi.org/10.1017/s1755020322000181","url":null,"abstract":"<p>In providing a good foundation for mathematics, set theorists often aim to develop the strongest theories possible and avoid those theories that place undue restrictions on the capacity to possess strength. For example, adding a measurable cardinal to <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230109184254371-0886:S1755020322000181:S1755020322000181_inline1.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$ZFC$\u0000</span></span>\u0000</span>\u0000</span> is thought to give a stronger theory than adding <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230109184254371-0886:S1755020322000181:S1755020322000181_inline2.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$V=L$\u0000</span></span>\u0000</span>\u0000</span> and the latter is thought to be more restrictive than the former. The two main proponents of this style of account are Penelope Maddy and John Steel. In this paper, I’ll offer a third account that is intended to provide a simple analysis of restrictiveness based on the algebraic concept of retraction in the category of theories. I will also deliver some results and arguments that suggest some plausible alternative approaches to analyzing restrictiveness do not live up to their intuitive motivation.</p>","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"9 1","pages":"1-42"},"PeriodicalIF":0.0,"publicationDate":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138544317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$Omega $ of uniform ultrafilters generates a 
$Delta $-closed logic 
${mathcal {L}}_Omega $. 
${mathcal {L}}_Omega $ is 
$omega $-relatively compact iff some 
$Din Omega $ fails to be 
$omega _1$-complete iff 
${mathcal {L}}_Omega $ does not contain the quantifier “there are uncountably many.” If 
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