Tomáš Lávička, Tommaso Moraschini, James G. Raftery
{"title":"The algebraic significance of weak excluded middle laws","authors":"Tomáš Lávička, Tommaso Moraschini, James G. Raftery","doi":"10.1002/malq.202100046","DOIUrl":null,"url":null,"abstract":"<p>For (finitary) deductive systems, we formulate a signature-independent abstraction of the <i>weak excluded middle law</i> (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety <math>\n <semantics>\n <mi>K</mi>\n <annotation>$\\mathsf {K}$</annotation>\n </semantics></math> algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of <math>\n <semantics>\n <mi>K</mi>\n <annotation>$\\mathsf {K}$</annotation>\n </semantics></math> has a greatest proper <math>\n <semantics>\n <mi>K</mi>\n <annotation>$\\mathsf {K}$</annotation>\n </semantics></math>-congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, in a suitable form, to all protoalgebraic logics. A super-intuitionistic logic possesses a WEML iff it extends <math>\n <semantics>\n <mi>KC</mi>\n <annotation>$\\mathsf {KC}$</annotation>\n </semantics></math>. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of <math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mn>4</mn>\n </mrow>\n <annotation>$\\mathsf {S4}$</annotation>\n </semantics></math> has a global consequence relation with a WEML iff it extends <math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mn>4</mn>\n <mo>.</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$\\mathsf {S4.2}$</annotation>\n </semantics></math>, while every axiomatic extension of <math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>t</mi>\n </msup>\n <annotation>$\\mathsf {R^t}$</annotation>\n </semantics></math> with an IL has a WEML.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"79-94"},"PeriodicalIF":0.4000,"publicationDate":"2022-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Logic Quarterly","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202100046","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 2
Abstract
For (finitary) deductive systems, we formulate a signature-independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of has a greatest proper -congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, in a suitable form, to all protoalgebraic logics. A super-intuitionistic logic possesses a WEML iff it extends . We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of has a global consequence relation with a WEML iff it extends , while every axiomatic extension of with an IL has a WEML.
期刊介绍:
Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.