{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,30]],"date-time":"2023-08-30T13:00:21Z","timestamp":1693400421871},"reference-count":2,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":26855,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1940,9]]},"abstract":"In this note I show, by means of an infinite matrix M<\/jats:italic>, that the number of irreducible modalities in Lewis's system S<\/jats:italic>2<\/jats:sub> is infinite. The result is of some interest in view of the fact that Parry has recently shown that there are but a finite number of modalities in the system S<\/jats:italic>2<\/jats:sub> (which is the next stronger system than S<\/jats:italic>2<\/jats:sub> discussed by Lewis).<\/jats:p>I begin by introducing a function \u03b8<\/jats:italic><\/jats:italic> which is defined over the class of sets of signed integers, and which assumes sets of signed integers as values. If A<\/jats:italic> is any set of signed integers, then \u03b8<\/jats:italic><\/jats:italic>(A<\/jats:italic>) is the set of all signed integers whose immediate predecessors are in A<\/jats:italic>; i.e., , so that n \u03f5 \u03b8<\/jats:italic><\/jats:italic>(A<\/jats:italic>) is true if and only if n<\/jats:italic> \u2212 1 \u03f5 A<\/jats:italic> is true.<\/jats:p>Thus, for example, \u03b8<\/jats:italic><\/jats:italic>({\u221210, \u22121, 0, 3, 14}) = {\u22129, 0, 1, 4, 15}. In particular we notice that \u03b8<\/jats:italic><\/jats:italic>(V) = V and \u03b8<\/jats:italic><\/jats:italic>(\u039b) = \u039b, where V is the set of all signed integers, and \u039b is the empty set of signed integers.<\/jats:p>It is clear that, if A<\/jats:italic> and B<\/jats:italic> are sets of signed integers, then \u03b8<\/jats:italic>(A<\/jats:italic>+B<\/jats:italic>) = \u03b8<\/jats:italic>(A<\/jats:italic>)+\u03b8<\/jats:italic>(B<\/jats:italic>).<\/jats:p>It is also easily proved that, for any set A<\/jats:italic> of signed integers we have . For, if n<\/jats:italic> is any signed integer, then<\/jats:p><\/jats:disp-formula><\/jats:p>","DOI":"10.2307\/2266864","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T14:40:00Z","timestamp":1146926400000},"page":"110-112","source":"Crossref","is-referenced-by-count":7,"title":["Proof that there are infinitely many modalities in Lewis's system *S<\/i>*_{2<\/sub>"],"prefix":"10.1017","volume":"5","author":[{"given":"J. C. C.","family":"McKinsey","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200107868_ref001","first-page":"137","volume":"4","year":"1939","journal-title":"Modalities in the Survey system of strict implication"},{"key":"S0022481200107868_ref002","first-page":"492","volume-title":"Symbolic logic","author":"Langford's"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200107868","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T15:26:04Z","timestamp":1560007564000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200107868\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1940,9]]},"references-count":2,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1940,9]]}},"alternative-id":["S0022481200107868"],"URL":"http:\/\/dx.doi.org\/10.2307\/2266864","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":["Logic","Philosophy"],"published":{"date-parts":[[1940,9]]}}}}