{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T18:21:27Z","timestamp":1699035687708},"reference-count":14,"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":21926,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1954,3]]},"abstract":"<jats:p>In previous papers, Post, Webb, G\u00f6tlind and the present author have described some Sheffer functions (in Swift's terminology, \u201cindependent binary generators\u201d) in <jats:italic>m<\/jats:italic>-valued logic. Professor J. Dean Swift has recently isolated the symmetric Sheffer functions of 3-valued logic. In the present paper, we will prove some properties of Sheffer functions in <jats:italic>m<\/jats:italic>-valued logic and isolate all of the Sheffer functions of 3-valued logic.<\/jats:p><jats:p>Before we proceed we will define some terms which we will find convenient. A set of functions in <jats:italic>m<\/jats:italic>-valued logic is <jats:italic>functionally complete<\/jats:italic>, if the set of the functions which can be defined explicitly from the functions of the set is exactly the set of all functions of <jats:italic>m<\/jats:italic>-valued logic. A function is <jats:italic>functionally complete<\/jats:italic>, if its unit set is functionally complete. A <jats:italic>Sheffer function<\/jats:italic> is a two-place functionally complete function. If <jats:italic>i<\/jats:italic> and <jats:italic>j<\/jats:italic> are truth values (1 <jats:italic>i<\/jats:italic>, <jats:italic>j<\/jats:italic> \u2264 <jats:italic>m<\/jats:italic>), we will say <jats:italic>i ~ j<\/jats:italic> (<jats:italic>D<\/jats:italic>), if <jats:italic>D<\/jats:italic> is a decomposition of the truth values 1, \u2026, <jats:italic>m<\/jats:italic> into 2 or more disjoint non-empty classes and <jats:italic>i<\/jats:italic> and <jats:italic>j<\/jats:italic> are elements of the same class. A binary function <jats:italic>f<\/jats:italic>(<jats:italic>p, q<\/jats:italic>) <jats:italic>satisfies the substitution law for a decomposition D<\/jats:italic>, if for any truth values <jats:italic>h, i, j, k<\/jats:italic>, whenever <jats:italic>h<\/jats:italic> ~ <jats:italic>j<\/jats:italic> (<jats:italic>D<\/jats:italic>) and <jats:italic>i~k<\/jats:italic>(<jats:italic>D<\/jats:italic>), then <jats:italic>f<\/jats:italic>(<jats:italic>h, i<\/jats:italic>) ~ <jats:italic>f<\/jats:italic>(<jats:italic>j, k<\/jats:italic>) (<jats:italic>D<\/jats:italic>). The function <jats:italic>f<\/jats:italic>(<jats:italic>p,q<\/jats:italic>) <jats:italic>satisfies the co-substitution law for D<\/jats:italic>, if for any truth values <jats:italic>h, i, j, k<\/jats:italic>, whenever <jats:italic>f<\/jats:italic>(<jats:italic>h, i<\/jats:italic>) ~ <jats:italic>f<\/jats:italic>(<jats:italic>j, k<\/jats:italic>) (<jats:italic>D<\/jats:italic>), then <jats:italic>h ~ j<\/jats:italic> (<jats:italic>D<\/jats:italic>) or <jats:italic>i ~ k<\/jats:italic> (<jats:italic>D<\/jats:italic>). We will say <jats:italic>f<\/jats:italic>(<jats:italic>p, q<\/jats:italic>) <jats:italic>has the proper substitution property<\/jats:italic>, if there is a decomposition of the truth values into less than <jats:italic>m<\/jats:italic> classes for which it satisfies the substitution law.<\/jats:p>","DOI":"10.2307\/2267650","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T15:30:20Z","timestamp":1146929420000},"page":"45-51","source":"Crossref","is-referenced-by-count":26,"title":["The Sheffer functions of 3-valued logic"],"prefix":"10.1017","volume":"19","author":[{"given":"Norman M.","family":"Martin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120008840X_ref013","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.21.5.252"},{"key":"S002248120008840X_ref008","doi-asserted-by":"publisher","DOI":"10.4064\/fm-24-1-298-301"},{"key":"S002248120008840X_ref010","unstructured":"Rose Alan , Review of [4], this Journal, vol. 16 (1951), pp. 275\u2013276."},{"key":"S002248120008840X_ref006","unstructured":"Martin Norman M. , Sheffer functions and axiom sets in m-valued propositional logic, Ph. D. thesis (University of California at Los Angeles, 1952)."},{"key":"S002248120008840X_ref005","first-page":"240","article-title":"A note on Sheffer functions in n-valued logic","volume":"3","author":"Martin","year":"1951","journal-title":"Methodos"},{"key":"S002248120008840X_ref004","first-page":"373","article-title":"Some analogues of the Sheffer stroke function in n-valued logic","volume":"12","author":"Martin","year":"1950","journal-title":"Indagationes mathematicae"},{"key":"S002248120008840X_ref011","first-page":"102","article-title":"Kryterium pe\u0142no\u015bci wielowarto\u015bciowych system\u00f3w logiki zda\u0144 (A criterion of fullness of many-valued systems of propositional logic)","volume":"32","author":"S\u0142upecki","year":"1939","journal-title":"Comptes rendus des s\u00e9ances de la Soci\u00e9t\u00e9 des Sciences et des Lettres de Varsovie"},{"key":"S002248120008840X_ref009","doi-asserted-by":"publisher","DOI":"10.2307\/2370324"},{"key":"S002248120008840X_ref007","first-page":"185","volume":"15","author":"Myhill","year":"1950","journal-title":"A complete theory of natural, rational, and real numbers"},{"key":"S002248120008840X_ref003","first-page":"42","volume-title":"O matrycach logicznych","author":"\u0141o\u015b","year":"1949"},{"key":"S002248120008840X_ref001","first-page":"141","article-title":"Some Sheffer functions in n-valued logic","volume":"11","author":"G\u00f6tlind","year":"1952","journal-title":"Portugaliae mathematicae"},{"key":"S002248120008840X_ref002","first-page":"161","volume":"17","author":"Kalicki","year":"1952","journal-title":"A test for the equality of truth-tables"},{"key":"S002248120008840X_ref014","first-page":"193","article-title":"Definition of Post's generalized negative and maximum in terms of one binary operator","volume":"58","author":"Webb","year":"1936","journal-title":"Bulletin of the American Mathematical Society"},{"key":"S002248120008840X_ref012","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1952.11988208"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120008840X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T02:02:31Z","timestamp":1559872951000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120008840X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1954,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1954,3]]}},"alternative-id":["S002248120008840X"],"URL":"https:\/\/doi.org\/10.2307\/2267650","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1954,3]]}}}