{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T21:26:38Z","timestamp":1779398798128,"version":"3.53.1"},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8777,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,3]]},"abstract":"<jats:p>It is the purpose of this paper to make explicit the connection between J.-Y. Girard's \u201clinear logic\u201d [4], and certain models for the logic of quantum mechanics, namely Mulvey's \u201cquantales\u201d [9]. This will be done not only in the case of commutative linear logic, but also in the case of a version of noncommutative linear logic suggested, but not fully formalized, by Girard in lectures given at McGill University in the fall of 1987 [5], and which for reasons which will become clear later we call \u201ccyclic linear logic\u201d.<\/jats:p><jats:p>For many of our results on quantales, we rely on the work of Niefield and Rosenthal [10].<\/jats:p><jats:p>The reader should note that by \u201cthe logic of quantum mechanics\u201d we do not mean the lattice theoretic \u201cquantum logics\u201d of Birkhoff and von Neumann [1], but rather a logic involving an associative (in general noncommutative) operation \u201cand then\u201d. Logical validity is intended to embody empirical verification (whether a physical experiment, or running a program), and the validity of <jats:italic>A &amp; B<\/jats:italic> (in Mulvey's notation) is to be regarded as \u201cwe have verified <jats:italic>A<\/jats:italic>, and then we have verified <jats:italic>B<\/jats:italic>\u201d. (See M. D. Srinivas [11] for another exposition of this idea.)<\/jats:p><jats:p>This of course is precisely the view of the \u201cmultiplicative conjunction\u201d, \u2297, in the phase semantics for Girard's linear logic [4], [5]. Indeed the quantale semantics for linear logic may be regarded as an element-free version of the phase semantics.<\/jats:p>","DOI":"10.2307\/2274953","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:33:48Z","timestamp":1146940428000},"page":"41-64","source":"Crossref","is-referenced-by-count":216,"title":["Quantales and (noncommutative) linear logic"],"prefix":"10.1017","volume":"55","author":[{"given":"David N.","family":"Yetter","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026426_ref007","unstructured":"Joyal A. and Street R. , Braided monoidal categories, preprint."},{"key":"S0022481200026426_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-9351-8"},{"key":"S0022481200026426_ref008","first-page":"263","volume-title":"Computational problems in abstract algebra","author":"Knuth","year":"1970"},{"key":"S0022481200026426_ref005","volume-title":"Seminar lectures, Le Groupe Interuniversitaire en \u00c9tudes Cat\u00e9goriques","author":"Girard","year":"1987"},{"key":"S0022481200026426_ref009","first-page":"94","volume-title":"Second topology conference (Taormina, 1984), Rendiconti del Circolo Matematico di Palermo","author":"Mulvey","year":"1986"},{"key":"S0022481200026426_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(87)90045-4"},{"key":"S0022481200026426_ref010","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100065403"},{"key":"S0022481200026426_ref002","volume-title":"Advances in Mathematics","author":"Freyd"},{"key":"S0022481200026426_ref011","doi-asserted-by":"publisher","DOI":"10.1063\/1.522736"},{"key":"S0022481200026426_ref003","unstructured":"Freyd P. J. and Yetter D. N. , Coherence theorems via knot theory, preprint."},{"key":"S0022481200026426_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/1968621"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026426","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T17:51:40Z","timestamp":1558201900000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026426\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["S0022481200026426"],"URL":"https:\/\/doi.org\/10.2307\/2274953","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}