{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,31]],"date-time":"2023-08-31T08:55:08Z","timestamp":1693472108012},"reference-count":6,"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":23022,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1951,3]]},"abstract":"<jats:p>In part I of the present paper axiom schemes and rules of inference were defined for <jats:italic>m<\/jats:italic>-valued functional calculi of first order with <jats:italic>s<\/jats:italic>(<jats:italic>m<\/jats:italic> &gt; <jats:italic>s<\/jats:italic> &gt; 1) designated truth-values. A proof of plausibility was given, and it was shown that it is not difficult to extend to <jats:italic>m<\/jats:italic>-valued functional calculi of first order certain concepts that are closely analogous to the ordinary 2-valued notions of \u201cconsistency with respect to an operator\u201d and \u201cabsolute consistency.\u201d The purpose of the present paper is to show that the concept of \u201cdeductive completeness\u201d may be extended to <jats:italic>m<\/jats:italic>-valued functional calculi of first order. For this purpose we define \u201canalytic formula\u201d for the <jats:italic>m<\/jats:italic>-valued case and show that if a formula is analytic, then it is provable in our formalization of <jats:italic>m<\/jats:italic>-valued functional calculi of first order.<\/jats:p><jats:p>In proving deductive completeness for the <jats:italic>m<\/jats:italic>-valued case, it is possible to use a method which is analogous to that used by G\u00f6del in establishing the completeness of 2-valued functional calculi of first order. However, in the present paper we will indicate only very briefly how the G\u00f6del procedure may be extended to the <jats:italic>m<\/jats:italic>-valued case. Our chief concern will be the problem of extending to our formalization of <jats:italic>m<\/jats:italic>-valued functional calculi the more elegant proof of deductive completeness for the 2-valued case which has recently been developed by Leon Henkin.<\/jats:p>","DOI":"10.2307\/2268662","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T15:14:10Z","timestamp":1146928450000},"page":"22-34","source":"Crossref","is-referenced-by-count":5,"title":["Axiom schemes for <i>m<\/i>-valued functional calculi of first order. Part II"],"prefix":"10.1017","volume":"16","author":[{"given":"J. B.","family":"Rosser","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A. R.","family":"Turquette","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200102397_ref005","first-page":"159","volume":"14","author":"Henkin","journal-title":"The completeness of the first-order functional calculus"},{"key":"S0022481200102397_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF01700692"},{"key":"S0022481200102397_ref003","first-page":"76","volume-title":"Grundz\u00fcge der theoretischen Logik","author":"Hilbert"},{"key":"S0022481200102397_ref004","first-page":"66","volume-title":"Introduction to mathematical logic","author":"Church"},{"key":"S0022481200102397_ref002","first-page":"74","volume":"14","journal-title":"he G\u00f6del completeness of m-valued functional calcuof first order"},{"key":"S0022481200102397_ref006","first-page":"61","volume":"10","journal-title":"Axiom schemes for m-valued prepositional calculi"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200102397","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T06:03:01Z","timestamp":1559973781000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200102397\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1951,3]]},"references-count":6,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1951,3]]}},"alternative-id":["S0022481200102397"],"URL":"https:\/\/doi.org\/10.2307\/2268662","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1951,3]]}}}