Pub Date : 2020-06-14DOI: 10.1215/00294527-2021-0032
Carlo Nicolai
The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in derivations. Next, we notice that cut remains eliminable when suitable arithmetical axioms are added to the system. Finally, we establish a direct link between cut-free derivability in infinitary formulations of the systems considered and fixed-point semantics. Noticeably, unlike what happens with other background logics, such links are established without imposing any restriction to the premisses of the truth rules.
{"title":"Cut Elimination for Systems of Transparent Truth with Restricted Initial Sequents","authors":"Carlo Nicolai","doi":"10.1215/00294527-2021-0032","DOIUrl":"https://doi.org/10.1215/00294527-2021-0032","url":null,"abstract":"The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in derivations. Next, we notice that cut remains eliminable when suitable arithmetical axioms are added to the system. Finally, we establish a direct link between cut-free derivability in infinitary formulations of the systems considered and fixed-point semantics. Noticeably, unlike what happens with other background logics, such links are established without imposing any restriction to the premisses of the truth rules.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"30 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86272005","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 : 2020-05-01DOI: 10.1215/00294527-2020-0008
A. Berenstein, R. Zamora
{"title":"Isometry Groups of Borel Randomizations","authors":"A. Berenstein, R. Zamora","doi":"10.1215/00294527-2020-0008","DOIUrl":"https://doi.org/10.1215/00294527-2020-0008","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"9 1","pages":"297-316"},"PeriodicalIF":0.7,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82243975","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 : 2020-05-01DOI: 10.1215/00294527-2020-0009
V. Halbach
Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: An argument is defined to be logically valid iff there is no substitution instance with true premisses and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: Logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or not. It is shown that the resulting notions of logical consequence are extensionally equivalent to versions of first-order provability and thus model-theoretic consequence. Every modeltheoretic interpretation has a substitutional counterpart, but not vice versa. In particular, in contrast to the model-theoretic account, there is a trivial intended interpretation on the substitutional account, namely the homophonic interpretation that does not substitute anything. Applications to free logic, second-order logic, and theories and languages other than set theory are sketched. 1 The Substitutional Analysis of Logical Consequence In what could be called semantic theories of logical consequence – as opposed to proof-theoretic analyses –, logical consequence is defined as truth preservation under all interpretations. In model-theoretic semantics, interpretations are conceived as formal set-theoretic models. However, this is only a very recent understanding of interpretation. Traditionally, in order to refute the formal validity of an argument, logicians showed that there is a substitution instance with true premisses but a false conclusion. Such an interpretation is a substitutional counterexample to the argument in question. In the present paper I attempt to revive the substitutional understanding of interpretation and make the informal substitutional account precise in a mathematical setting for first-order predicate logic. The substitutional account of logical consequence advanced in this paper contrasts with earlier substitutional definitions of logical truth and consequence by Quine [20] 2010 Mathematics Subject Classification: Primary 03A05, 03B10
{"title":"Formal Notes on the Substitutional Analysis of Logical Consequence","authors":"V. Halbach","doi":"10.1215/00294527-2020-0009","DOIUrl":"https://doi.org/10.1215/00294527-2020-0009","url":null,"abstract":"Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: An argument is defined to be logically valid iff there is no substitution instance with true premisses and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: Logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or not. It is shown that the resulting notions of logical consequence are extensionally equivalent to versions of first-order provability and thus model-theoretic consequence. Every modeltheoretic interpretation has a substitutional counterpart, but not vice versa. In particular, in contrast to the model-theoretic account, there is a trivial intended interpretation on the substitutional account, namely the homophonic interpretation that does not substitute anything. Applications to free logic, second-order logic, and theories and languages other than set theory are sketched. 1 The Substitutional Analysis of Logical Consequence In what could be called semantic theories of logical consequence – as opposed to proof-theoretic analyses –, logical consequence is defined as truth preservation under all interpretations. In model-theoretic semantics, interpretations are conceived as formal set-theoretic models. However, this is only a very recent understanding of interpretation. Traditionally, in order to refute the formal validity of an argument, logicians showed that there is a substitution instance with true premisses but a false conclusion. Such an interpretation is a substitutional counterexample to the argument in question. In the present paper I attempt to revive the substitutional understanding of interpretation and make the informal substitutional account precise in a mathematical setting for first-order predicate logic. The substitutional account of logical consequence advanced in this paper contrasts with earlier substitutional definitions of logical truth and consequence by Quine [20] 2010 Mathematics Subject Classification: Primary 03A05, 03B10","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"3 1","pages":"317-339"},"PeriodicalIF":0.7,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90730615","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 : 2020-05-01DOI: 10.1215/00294527-2020-0007
J. Beall, Caleb Camrud
There is a natural ‘combinatorial argument’ for the philosophical view that if the standard (socalled classical) account of logical consequence (henceforth, logic) is right about logic’s fundamental truth values (viz., The True and The False), then FDE, a well-known subclassical logic [1, 2, 6, 8], is a more natural account of the space of ‘logical possibility’. Without officially endorsing that argument, our aim in this note is to defend it from an otherwise powerful objection. Our defense rests on an explicit generalization of a result by Priest [11]. In particular, by way of answering the target objection, we explicitly show that after combining the standard (classical) values in {>,⊥} to get a space of four values, as FDE demands, the given process of combining values ‘all the way up’ to α many values, for any ordinal α, results in the same account of logical consequence (viz., FDE).
{"title":"A Note on FDE \"All the Way Up\"","authors":"J. Beall, Caleb Camrud","doi":"10.1215/00294527-2020-0007","DOIUrl":"https://doi.org/10.1215/00294527-2020-0007","url":null,"abstract":"There is a natural ‘combinatorial argument’ for the philosophical view that if the standard (socalled classical) account of logical consequence (henceforth, logic) is right about logic’s fundamental truth values (viz., The True and The False), then FDE, a well-known subclassical logic [1, 2, 6, 8], is a more natural account of the space of ‘logical possibility’. Without officially endorsing that argument, our aim in this note is to defend it from an otherwise powerful objection. Our defense rests on an explicit generalization of a result by Priest [11]. In particular, by way of answering the target objection, we explicitly show that after combining the standard (classical) values in {>,⊥} to get a space of four values, as FDE demands, the given process of combining values ‘all the way up’ to α many values, for any ordinal α, results in the same account of logical consequence (viz., FDE).","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"13 1","pages":"283-296"},"PeriodicalIF":0.7,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90040034","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 : 2020-05-01DOI: 10.1215/00294527-2020-0002
G. Shen
{"title":"A Note on Strongly Almost Disjoint Families","authors":"G. Shen","doi":"10.1215/00294527-2020-0002","DOIUrl":"https://doi.org/10.1215/00294527-2020-0002","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"80 1","pages":"227-231"},"PeriodicalIF":0.7,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78986858","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 : 2020-05-01DOI: 10.1215/00294527-2020-0005
K. Ng, Hongyuan Yu
We study the relationship between effective domination properties and the bounded jump. We answer two open questions about the bounded jump: (1) We prove that the analogue of Sacks jump inversion fails for the bounded jump and the wtt-reducibility. (2) We prove that no c.e. bounded high set can be low by showing that they all have to be Turing complete. We characterize the class of c.e. bounded high sets as being those sets computing the Halting problem via a reduction with use bounded by an ω-c.e. function. We define several notions of a c.e. set being effectively dominant, and show that together with the bounded high sets they form a proper hierarchy.
{"title":"Effective Domination and the Bounded Jump","authors":"K. Ng, Hongyuan Yu","doi":"10.1215/00294527-2020-0005","DOIUrl":"https://doi.org/10.1215/00294527-2020-0005","url":null,"abstract":"We study the relationship between effective domination properties and the bounded jump. We answer two open questions about the bounded jump: (1) We prove that the analogue of Sacks jump inversion fails for the bounded jump and the wtt-reducibility. (2) We prove that no c.e. bounded high set can be low by showing that they all have to be Turing complete. We characterize the class of c.e. bounded high sets as being those sets computing the Halting problem via a reduction with use bounded by an ω-c.e. function. We define several notions of a c.e. set being effectively dominant, and show that together with the bounded high sets they form a proper hierarchy.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"18 8 1","pages":"203-225"},"PeriodicalIF":0.7,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86873029","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 : 2020-02-28DOI: 10.1215/00294527-2021-0013
Pantelis E. Eleftheriou, O. Sánchez, N. Regnault
Let $(R, delta)$ be a closed ordered differential field, and $C$ its field of constants. In this note, we prove that for sets definable in the pair $(R, C)$, the $delta$-dimension and the large dimension coincide. As an application, we characterize the definable sets that are internal to $C$, as those sets that are definable in $(R, C)$ and have $delta$-dimension $0$. We further show that having $delta$-dimension $0$ does not generally imply co-analyzability in $C$.
{"title":"On Coincidence of Dimensions in Closed Ordered Differential Fields","authors":"Pantelis E. Eleftheriou, O. Sánchez, N. Regnault","doi":"10.1215/00294527-2021-0013","DOIUrl":"https://doi.org/10.1215/00294527-2021-0013","url":null,"abstract":"Let $(R, delta)$ be a closed ordered differential field, and $C$ its field of constants. In this note, we prove that for sets definable in the pair $(R, C)$, the $delta$-dimension and the large dimension coincide. As an application, we characterize the definable sets that are internal to $C$, as those sets that are definable in $(R, C)$ and have $delta$-dimension $0$. We further show that having $delta$-dimension $0$ does not generally imply co-analyzability in $C$.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83005927","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 : 2020-02-16DOI: 10.1215/00294527-2020-0001
F. Correia, Sven Rosenkranz
Temporaryism—the view that not always everything always exists— comes in two main versions: presentism and expansionism (aka the growing block theory of time). Both versions of the view are commonly formulated using the notion of being present, which we, among others, find problematic. Expansionism is also sometimes accused of requiring extraordinary conceptual tools for its formulation. In this paper, we put forward systematic characterisations of presentism and expansionism which involve neither the notion of being present nor unfamiliar conceptual tools. These characterisations are full blown logics, each logic comprising an axiomatic proof system and an intuitive semantics with respect to which the system is both sound and complete.
{"title":"The Formalities of Temporaryism without Presentness","authors":"F. Correia, Sven Rosenkranz","doi":"10.1215/00294527-2020-0001","DOIUrl":"https://doi.org/10.1215/00294527-2020-0001","url":null,"abstract":"Temporaryism—the view that not always everything always exists— comes in two main versions: presentism and expansionism (aka the growing block theory of time). Both versions of the view are commonly formulated using the notion of being present, which we, among others, find problematic. Expansionism is also sometimes accused of requiring extraordinary conceptual tools for its formulation. In this paper, we put forward systematic characterisations of presentism and expansionism which involve neither the notion of being present nor unfamiliar conceptual tools. These characterisations are full blown logics, each logic comprising an axiomatic proof system and an intuitive semantics with respect to which the system is both sound and complete.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"22 1","pages":"181-202"},"PeriodicalIF":0.7,"publicationDate":"2020-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81035250","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 : 2020-01-01DOI: 10.1215/00294527-2019-0036
Huishan Wu
{"title":"The Complexity of Radicals and Socles of Modules","authors":"Huishan Wu","doi":"10.1215/00294527-2019-0036","DOIUrl":"https://doi.org/10.1215/00294527-2019-0036","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"6 1","pages":"141-153"},"PeriodicalIF":0.7,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87850318","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 : 2020-01-01DOI: 10.1215/00294527-2019-0035
Sergi Oms
{"title":"A Remark on Probabilistic Measures of Coherence","authors":"Sergi Oms","doi":"10.1215/00294527-2019-0035","DOIUrl":"https://doi.org/10.1215/00294527-2019-0035","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":" 7","pages":"129-140"},"PeriodicalIF":0.7,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72497727","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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