Logic with truth values in A linearly ordered heyting algebra

Logic with truth values in A linearly ordered heyting algebra

It is known that the theorems of the intuitionist predicate calculus are exactly those formulas which are valid in every Heyting algebra (that is, pseudo-Boolean algebra). The simplest kind of Heyting algebra is a linearly ordered set. This paper concerns the question of determining all formulas whi...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:The Journal of symbolic logic 1969-01, Vol.34 (3), p.395-408
1. Verfasser: Horn, Alfred
Format: Artikel
Sprache:eng
Schlagworte:
Algebra
Axioms
Equivalence relation
Heyting algebras
Homomorphisms
Mathematical functions
Mathematical intuitionism
Mathematical theorems
Predicate calculus
Subalgebras
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
container_end_page 408
container_issue 3
container_start_page 395
container_title The Journal of symbolic logic
container_volume 34
creator Horn, Alfred
description It is known that the theorems of the intuitionist predicate calculus are exactly those formulas which are valid in every Heyting algebra (that is, pseudo-Boolean algebra). The simplest kind of Heyting algebra is a linearly ordered set. This paper concerns the question of determining all formulas which are valid in every linearly ordered Heyting algebra. The question is of interest because it is a particularly simple case intermediate between the intuitionist and classical logics. Also the interpretation of implication is such that in general there exists no nondiscrete Hausdorff topology for which this operation is continuous.
doi_str_mv 10.2307/2270905
format Article
fullrecord <record><control><sourceid>jstor_proje</sourceid><recordid>TN_cdi_projecteuclid_primary_oai_CULeuclid_euclid_jsl_1183736852</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>2270905</jstor_id><sourcerecordid>2270905</sourcerecordid><originalsourceid>FETCH-LOGICAL-c307t-ccb0e6e50b6142c131f61fe5a5c073fe2eb7e4f3a94fd1da2e5ee243479ea0613</originalsourceid><addsrcrecordid>eNp9kE9Lw0AQxRdRsFbxKwQUPEX3_yYHQWltFQoqWgUvy2YzaTfGpO6mar-9kZYevcyD4cebNw-hY4LPKcPqglKFUyx2UI-knMUiSeQu6mFMacwTQvfRQQglxlikPOmhy0kzczb6du08av2ym1-mWkKIXB1dR5WrwfhqFTU-Bw95NIdV6-pZZKoZZN4cor3CVAGONtpH09HN8-A2ntyP7wbXk9h2idrY2gyDBIEzSTi1hJFCkgKEERYrVgCFTAEvmEl5kZPcUBAAlDOuUjBYEtZHV2vfhW9KsC0sbeVyvfDuw_iVbozTg-lks91IGSpNSMIUk4mgncXJ1uKze7DVZbP0dZdaE5piyXjKRUedrSnrmxA8FNsbBOu_evWm3o48XZNlaBv_DxavMRda-Nlixr9rqZgSWo4f9dPby_B1NHzQiv0CZUyGkA</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1290634945</pqid></control><display><type>article</type><title>Logic with truth values in A linearly ordered heyting algebra</title><source>Periodicals Index Online</source><source>JSTOR Mathematics &amp; Statistics</source><source>JSTOR Archive Collection A-Z Listing</source><creator>Horn, Alfred</creator><creatorcontrib>Horn, Alfred</creatorcontrib><description>It is known that the theorems of the intuitionist predicate calculus are exactly those formulas which are valid in every Heyting algebra (that is, pseudo-Boolean algebra). The simplest kind of Heyting algebra is a linearly ordered set. This paper concerns the question of determining all formulas which are valid in every linearly ordered Heyting algebra. The question is of interest because it is a particularly simple case intermediate between the intuitionist and classical logics. Also the interpretation of implication is such that in general there exists no nondiscrete Hausdorff topology for which this operation is continuous.</description><identifier>ISSN: 0022-4812</identifier><identifier>EISSN: 1943-5886</identifier><identifier>DOI: 10.2307/2270905</identifier><language>eng</language><publisher>New York, USA: Cambridge University Press</publisher><subject>Algebra ; Axioms ; Equivalence relation ; Heyting algebras ; Homomorphisms ; Mathematical functions ; Mathematical intuitionism ; Mathematical theorems ; Predicate calculus ; Subalgebras</subject><ispartof>The Journal of symbolic logic, 1969-01, Vol.34 (3), p.395-408</ispartof><rights>Copyright 1969 Association for Symbolic Logic</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c307t-ccb0e6e50b6142c131f61fe5a5c073fe2eb7e4f3a94fd1da2e5ee243479ea0613</citedby><cites>FETCH-LOGICAL-c307t-ccb0e6e50b6142c131f61fe5a5c073fe2eb7e4f3a94fd1da2e5ee243479ea0613</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/2270905$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/2270905$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,780,784,803,832,885,27869,27924,27925,58017,58021,58250,58254</link.rule.ids></links><search><creatorcontrib>Horn, Alfred</creatorcontrib><title>Logic with truth values in A linearly ordered heyting algebra</title><title>The Journal of symbolic logic</title><description>It is known that the theorems of the intuitionist predicate calculus are exactly those formulas which are valid in every Heyting algebra (that is, pseudo-Boolean algebra). The simplest kind of Heyting algebra is a linearly ordered set. This paper concerns the question of determining all formulas which are valid in every linearly ordered Heyting algebra. The question is of interest because it is a particularly simple case intermediate between the intuitionist and classical logics. Also the interpretation of implication is such that in general there exists no nondiscrete Hausdorff topology for which this operation is continuous.</description><subject>Algebra</subject><subject>Axioms</subject><subject>Equivalence relation</subject><subject>Heyting algebras</subject><subject>Homomorphisms</subject><subject>Mathematical functions</subject><subject>Mathematical intuitionism</subject><subject>Mathematical theorems</subject><subject>Predicate calculus</subject><subject>Subalgebras</subject><issn>0022-4812</issn><issn>1943-5886</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1969</creationdate><recordtype>article</recordtype><sourceid>K30</sourceid><recordid>eNp9kE9Lw0AQxRdRsFbxKwQUPEX3_yYHQWltFQoqWgUvy2YzaTfGpO6mar-9kZYevcyD4cebNw-hY4LPKcPqglKFUyx2UI-knMUiSeQu6mFMacwTQvfRQQglxlikPOmhy0kzczb6du08av2ym1-mWkKIXB1dR5WrwfhqFTU-Bw95NIdV6-pZZKoZZN4cor3CVAGONtpH09HN8-A2ntyP7wbXk9h2idrY2gyDBIEzSTi1hJFCkgKEERYrVgCFTAEvmEl5kZPcUBAAlDOuUjBYEtZHV2vfhW9KsC0sbeVyvfDuw_iVbozTg-lks91IGSpNSMIUk4mgncXJ1uKze7DVZbP0dZdaE5piyXjKRUedrSnrmxA8FNsbBOu_evWm3o48XZNlaBv_DxavMRda-Nlixr9rqZgSWo4f9dPby_B1NHzQiv0CZUyGkA</recordid><startdate>19690101</startdate><enddate>19690101</enddate><creator>Horn, Alfred</creator><general>Cambridge University Press</general><general>Association for Symbolic Logic, Inc</general><general>Association for Symbolic Logic</general><scope>BSCLL</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>EOLOZ</scope><scope>FKUCP</scope><scope>IOIBA</scope><scope>K30</scope><scope>PAAUG</scope><scope>PAWHS</scope><scope>PAWZZ</scope><scope>PAXOH</scope><scope>PBHAV</scope><scope>PBQSW</scope><scope>PBYQZ</scope><scope>PCIWU</scope><scope>PCMID</scope><scope>PCZJX</scope><scope>PDGRG</scope><scope>PDWWI</scope><scope>PETMR</scope><scope>PFVGT</scope><scope>PGXDX</scope><scope>PIHIL</scope><scope>PISVA</scope><scope>PJCTQ</scope><scope>PJTMS</scope><scope>PLCHJ</scope><scope>PMHAD</scope><scope>PNQDJ</scope><scope>POUND</scope><scope>PPLAD</scope><scope>PQAPC</scope><scope>PQCAN</scope><scope>PQCMW</scope><scope>PQEME</scope><scope>PQHKH</scope><scope>PQMID</scope><scope>PQNCT</scope><scope>PQNET</scope><scope>PQSCT</scope><scope>PQSET</scope><scope>PSVJG</scope><scope>PVMQY</scope><scope>PZGFC</scope></search><sort><creationdate>19690101</creationdate><title>Logic with truth values in A linearly ordered heyting algebra</title><author>Horn, Alfred</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c307t-ccb0e6e50b6142c131f61fe5a5c073fe2eb7e4f3a94fd1da2e5ee243479ea0613</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1969</creationdate><topic>Algebra</topic><topic>Axioms</topic><topic>Equivalence relation</topic><topic>Heyting algebras</topic><topic>Homomorphisms</topic><topic>Mathematical functions</topic><topic>Mathematical intuitionism</topic><topic>Mathematical theorems</topic><topic>Predicate calculus</topic><topic>Subalgebras</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Horn, Alfred</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><collection>Periodicals Index Online Segment 01</collection><collection>Periodicals Index Online Segment 04</collection><collection>Periodicals Index Online Segment 29</collection><collection>Periodicals Index Online</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - West</collection><collection>Primary Sources Access (Plan D) - International</collection><collection>Primary Sources Access &amp; Build (Plan A) - MEA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Midwest</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Northeast</collection><collection>Primary Sources Access (Plan D) - Southeast</collection><collection>Primary Sources Access (Plan D) - North Central</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Southeast</collection><collection>Primary Sources Access (Plan D) - South Central</collection><collection>Primary Sources Access &amp; Build (Plan A) - UK / I</collection><collection>Primary Sources Access (Plan D) - Canada</collection><collection>Primary Sources Access (Plan D) - EMEALA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - North Central</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - South Central</collection><collection>Primary Sources Access &amp; Build (Plan A) - International</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - International</collection><collection>Primary Sources Access (Plan D) - West</collection><collection>Periodicals Index Online Segments 1-50</collection><collection>Primary Sources Access (Plan D) - APAC</collection><collection>Primary Sources Access (Plan D) - Midwest</collection><collection>Primary Sources Access (Plan D) - MEA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Canada</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - UK / I</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - EMEALA</collection><collection>Primary Sources Access &amp; Build (Plan A) - APAC</collection><collection>Primary Sources Access &amp; Build (Plan A) - Canada</collection><collection>Primary Sources Access &amp; Build (Plan A) - West</collection><collection>Primary Sources Access &amp; Build (Plan A) - EMEALA</collection><collection>Primary Sources Access (Plan D) - Northeast</collection><collection>Primary Sources Access &amp; Build (Plan A) - Midwest</collection><collection>Primary Sources Access &amp; Build (Plan A) - North Central</collection><collection>Primary Sources Access &amp; Build (Plan A) - Northeast</collection><collection>Primary Sources Access &amp; Build (Plan A) - South Central</collection><collection>Primary Sources Access &amp; Build (Plan A) - Southeast</collection><collection>Primary Sources Access (Plan D) - UK / I</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - APAC</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - MEA</collection><jtitle>The Journal of symbolic logic</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Horn, Alfred</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Logic with truth values in A linearly ordered heyting algebra</atitle><jtitle>The Journal of symbolic logic</jtitle><date>1969-01-01</date><risdate>1969</risdate><volume>34</volume><issue>3</issue><spage>395</spage><epage>408</epage><pages>395-408</pages><issn>0022-4812</issn><eissn>1943-5886</eissn><abstract>It is known that the theorems of the intuitionist predicate calculus are exactly those formulas which are valid in every Heyting algebra (that is, pseudo-Boolean algebra). The simplest kind of Heyting algebra is a linearly ordered set. This paper concerns the question of determining all formulas which are valid in every linearly ordered Heyting algebra. The question is of interest because it is a particularly simple case intermediate between the intuitionist and classical logics. Also the interpretation of implication is such that in general there exists no nondiscrete Hausdorff topology for which this operation is continuous.</abstract><cop>New York, USA</cop><pub>Cambridge University Press</pub><doi>10.2307/2270905</doi><tpages>14</tpages></addata></record>
fulltext fulltext
identifier ISSN: 0022-4812
ispartof The Journal of symbolic logic, 1969-01, Vol.34 (3), p.395-408
issn 0022-4812
1943-5886
language eng
recordid cdi_projecteuclid_primary_oai_CULeuclid_euclid_jsl_1183736852
source Periodicals Index Online; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing
subjects Algebra
Axioms
Equivalence relation
Heyting algebras
Homomorphisms
Mathematical functions
Mathematical intuitionism
Mathematical theorems
Predicate calculus
Subalgebras
title Logic with truth values in A linearly ordered heyting algebra
url https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-30T21%3A08%3A01IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_proje&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Logic%20with%20truth%20values%20in%20A%20linearly%20ordered%20heyting%20algebra&rft.jtitle=The%20Journal%20of%20symbolic%20logic&rft.au=Horn,%20Alfred&rft.date=1969-01-01&rft.volume=34&rft.issue=3&rft.spage=395&rft.epage=408&rft.pages=395-408&rft.issn=0022-4812&rft.eissn=1943-5886&rft_id=info:doi/10.2307/2270905&rft_dat=%3Cjstor_proje%3E2270905%3C/jstor_proje%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=1290634945&rft_id=info:pmid/&rft_jstor_id=2270905&rfr_iscdi=true