{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T03:45:17Z","timestamp":1759117517107},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11668,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,4]]},"abstract":"<jats:p>Anderson and Belnap asked in \u00a78.11 of their treatise <jats:bold><jats:italic>Entailment<\/jats:italic><\/jats:bold> [1] whether a certain pure implicational calculus, which we will call <jats:italic>P<\/jats:italic> \u2212 <jats:italic>W<\/jats:italic>, is minimal in the sense that no two distinct formulas coentail each other in this calculus. We provide a positive solution to this question, variously known as <jats:italic>The P \u2212 W problem<\/jats:italic>, or <jats:italic>Belnap's conjecture<\/jats:italic>.<\/jats:p><jats:p>We will be concerned with two systems of pure implication, formulated in a language constructed in the usual way from a set of propositional variables, with a single binary connective \u2192. We use <jats:italic>A, B,\u2026, A<jats:sub>1<\/jats:sub>, B<jats:sub>1<\/jats:sub><\/jats:italic>, \u2026, as variables ranging over formulas. Formulas are written using the bracketing conventions of Church [3].<\/jats:p><jats:p>The first system, which we call <jats:italic>S<\/jats:italic> (in honour of its evident incorporation of syllogistic principles of reasoning), has as axioms all instances of\n<jats:disp-quote><jats:p><jats:italic>(B) B \u2192 C  \u2192. A \u2192 B \u2192. A \u2192 C (prefixing)<\/jats:italic>,<\/jats:p><jats:p><jats:italic>(B) A \u2192 B \u2192. B \u2192 C \u2192. A \u2192 C (suffixing)<\/jats:italic>,<\/jats:p><\/jats:disp-quote>\nand rules\n<jats:disp-quote><jats:p><jats:italic>(BX) from B \u2192 C infer A \u2192 B \u2192. A \u2192 C (rule prefixing)<\/jats:italic>,<\/jats:p><jats:p><jats:italic>(B\u2019X) from A \u2192 B infer B \u2192 C  \u2192. A \u2192 C (rule suffixing)<\/jats:italic>,<\/jats:p><jats:p><jats:italic>(BXY) from A \u2192 B and B \u2192 C infer A \u2192 C (rule transitivity)<\/jats:italic>.<\/jats:p><\/jats:disp-quote><\/jats:p><jats:p>The second system, P \u2212 W, has in addition to the axioms and rules of <jats:italic>S<\/jats:italic> the axiom scheme\n<jats:disp-quote><jats:p><jats:italic>(I) A \u2192 A<\/jats:italic><\/jats:p><\/jats:disp-quote>\nof <jats:italic>identity<\/jats:italic>.<\/jats:p><jats:p>We write \u22a2<jats:sub><jats:italic>S<\/jats:italic><\/jats:sub><jats:italic>A<\/jats:italic> (\u22a3<jats:sub><jats:italic>S<\/jats:italic><\/jats:sub><jats:italic>A<\/jats:italic>) to mean that <jats:italic>A<\/jats:italic> is (resp. is not) a theorem of <jats:italic>S<\/jats:italic>, and similarly for <jats:italic>P \u2212 W<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273106","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:01:43Z","timestamp":1146952903000},"page":"869-887","source":"Crossref","is-referenced-by-count":38,"title":["Solution to the <i>P<\/i> \u2212 <i>W<\/i> problem"],"prefix":"10.1017","volume":"47","author":[{"given":"E.P.","family":"Martin","sequence":"first","affiliation":[]},{"given":"R.K.","family":"Meyer","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200043747_ref006","unstructured":"McRobbie M. A. , A proof theoretic investigation of relevant and modal logics, Ph. D. thesis, Australian National University, Canberra, 1979."},{"key":"S0022481200043747_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF00257480"},{"key":"S0022481200043747_ref004","volume-title":"Combinatory logic","volume":"1","author":"Curry","year":"1958"},{"key":"S0022481200043747_ref003","volume-title":"Introduction to mathematical logic","volume":"1","author":"Church","year":"1956"},{"key":"S0022481200043747_ref002","doi-asserted-by":"publisher","DOI":"10.1037\/e523212009-001"},{"key":"S0022481200043747_ref001","volume-title":"Entailment: The logic of relevance and necessity","volume":"1","author":"Anderson","year":"1975"},{"key":"S0022481200043747_ref007","unstructured":"Martin E. P. , The P \u2212 W problem, Ph.D. thesis, Australian National University, Canberra, 1978."},{"key":"S0022481200043747_ref008","first-page":"189","article-title":"A general Gentzen system for implicational calculi","volume":"1","author":"Meyer","year":"1976","journal-title":"The Relevance Logic News-letter"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200043747","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T20:18:35Z","timestamp":1558729115000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200043747\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,4]]},"references-count":8,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1982,12]]}},"alternative-id":["S0022481200043747"],"URL":"https:\/\/doi.org\/10.2307\/2273106","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,4]]}}}