{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T18:25:53Z","timestamp":1784139953084,"version":"3.55.0"},"reference-count":33,"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":8412,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1991,3]]},"abstract":"<jats:p>Many-valued logics in general and 3-valued logic in particular is an old subject which had its beginning in the work of \u0141ukasiewicz [\u0141uk]. Recently there is a revived interest in this topic, both for its own sake (see, for example, [Ho]), and also because of its potential applications in several areas of computer science, such as proving correctness of programs [Jo], knowledge bases [CP] and artificial intelligence [Tu]. There are, however, a huge number of 3-valued systems which logicians have studied throughout the years. The motivation behind them and their properties are not always clear, and their proof theory is frequently not well developed. This state of affairs makes both the use of 3-valued logics and doing fruitful research on them rather difficult.<\/jats:p><jats:p>Our first goal in this work is, accordingly, to identify and characterize a class of 3-valued logics which might be called <jats:italic>natural<\/jats:italic>. For this we use the general framework for characterizing and investigating logics which we have developed in [Av1]. Not many 3-valued logics appear as natural within this framework, but it turns out that those that do include some of the best known ones. These include the 3-valued logics of \u0141ukasiewicz, Kleene and Soboci\u0144ski, the logic LPF used in the VDM project, the logic <jats:italic>RM<\/jats:italic><jats:sub>3<\/jats:sub> from the relevance family and the paraconsistent 3-valued logic of [dCA]. Our presentation provides justifications for the introduction of certain connectives in these logics which are often regarded as ad hoc. It also shows that they are all closely related to each other. It is shown, for example, that \u0141ukasiewicz 3-valued logic and <jats:italic>RM<\/jats:italic><jats:sub>3<\/jats:sub> (the strongest logic in the family of relevance logics) are in a strong sense dual to each other, and that both are derivable by the same general construction from, respectively, Kleene 3-valued logic and the 3-valued paraconsistent logic.<\/jats:p>","DOI":"10.2307\/2274919","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:38:25Z","timestamp":1146940705000},"page":"276-294","source":"Crossref","is-referenced-by-count":151,"title":["Natural 3-valued logics\u2014characterization and proof theory"],"prefix":"10.1017","volume":"56","author":[{"given":"Arnon","family":"Avron","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025056_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-8297-7_12"},{"key":"S0022481200025056_ref013","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093891487"},{"key":"S0022481200025056_ref017","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(87)90045-4"},{"key":"S0022481200025056_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BF00264250"},{"key":"S0022481200025056_ref033","volume-title":"Lectures on propositional calculi","author":"W\u00f3jcicki","year":"1984"},{"key":"S0022481200025056_ref001","volume-title":"Entailment","volume":"1","author":"Anderson","year":"1975"},{"key":"S0022481200025056_ref028","first-page":"173","article-title":"On the propositional calculus p1","volume":"16","author":"Sette","year":"1973","journal-title":"Mathematica Japonica"},{"key":"S0022481200025056_ref005","unstructured":"[Av4] Avron A. , The semantics and proof theory of relevance logics and non-trivial theories containing contradictions, Thesis, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, 1984."},{"key":"S0022481200025056_ref026","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/025\/0363802"},{"key":"S0022481200025056_ref007","first-page":"334","volume":"49","author":"Avron","year":"1984","journal-title":"Relevant entailment: semantics and formal systems"},{"key":"S0022481200025056_ref002","volume-title":"Simple consequence relations","author":"Avron","year":"1987"},{"key":"S0022481200025056_ref024","volume":"48","author":"Pottinger","year":"1983","journal-title":"Uniform, cut-free formulations of T, S4 and S5"},{"key":"S0022481200025056_ref027","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19550010402"},{"key":"S0022481200025056_ref018","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(80)90001-9"},{"key":"S0022481200025056_ref029","first-page":"23","article-title":"Axiomatization of a partial system of three-value calculus of propositions","volume":"1","author":"Soboci\u0144ski","year":"1952","journal-title":"Journal of Computing Systems"},{"key":"S0022481200025056_ref014","first-page":"185","article-title":"Relations between paraconsistent logics and many-valued logic","volume":"10","author":"da Costa","year":"1981","journal-title":"Polish Academy of Sciences. Institute of Philosophy and Sociology. Bulletin of the Section of Logic"},{"key":"S0022481200025056_ref022","first-page":"1","article-title":"Construction des alg\u00e8bres de \u0141ukasiewicz trivalentes dans les alg\u00e8bres de Boole monadiques. 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