{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T00:38:26Z","timestamp":1783643906564,"version":"3.55.0"},"reference-count":10,"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":23446,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1950,1]]},"abstract":"<jats:p>1.1. In two previous papers a consistent theory of real numbers has been outlined by the author, using a system <jats:italic>K<\/jats:italic>\u2032. This latter system is an extension of a system <jats:italic>K<\/jats:italic>, which is \u201cbasic\u201d in the sense that every finitary (recursively enumerable) subclass of its well-formed expressions is in a certain sense represented in it. The system <jats:italic>L<\/jats:italic> described below is a further extension of <jats:italic>K<\/jats:italic>. The system <jats:italic>K<\/jats:italic>\u2032 lacks two important features possessed by <jats:italic>L<\/jats:italic>: a symbol for a special kind of implication (or \u201cconditionality\u201d) and a symbol for the modal concept \u201cnecessity.\u201d The presence of the implication symbol, and the additional assumptions that go with it, make available in <jats:italic>L<\/jats:italic> various kinds of restricted universal quantification not available in <jats:italic>K<\/jats:italic>\u2032, for example, universal quantification restricted to the real numbers of the author's theory of real numbers.<\/jats:p><jats:p>1.2. If \u2018~[a &amp; ~a]\u2019 is a theorem of <jats:italic>L<\/jats:italic>, then the proposition expressed by \u2018a\u2019 may be said to <jats:italic>L<\/jats:italic>-satisfy the principle of excluded middle. I t is always the case that \u2018<jats:italic>a<\/jats:italic>\u2019 <jats:italic>L<\/jats:italic>-satisfies the principle of excluded middle (or rather that the proposition expressed by \u2018<jats:italic>a<\/jats:italic>\u2019 does so) if and only if \u2018<jats:italic>a<\/jats:italic>\u2019 or \u2018~<jats:italic>a<\/jats:italic>\u2019 is a theorem of <jats:italic>L<\/jats:italic>. An example of a proposition that does not <jats:italic>L<\/jats:italic>-satisfy the principle of excluded middle is that expressed by \u2018<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200105031_inline1\"\/>\u2019, namely the proposition that asserts that the class of classes that are not members of themselves is a member of itself.<\/jats:p>","DOI":"10.2307\/2269228","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T19:07:57Z","timestamp":1146942477000},"page":"209-218","source":"Crossref","is-referenced-by-count":3,"title":["A further consistent extension of basic logic"],"prefix":"10.1017","volume":"14","author":[{"given":"Frederic B.","family":"Fitch","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200105031_ref004","volume-title":"On the rules of suppositions in formal logic","author":"Ja\u015bkowski","year":"1934"},{"key":"S0022481200105031_ref001","first-page":"95","volume":"13","year":"1948","journal-title":"An extension of basic logic"},{"key":"S0022481200105031_ref002","volume-title":"Logical calculus","author":"Bernays","year":"1936"},{"key":"S0022481200105031_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF01201363"},{"key":"S0022481200105031_ref005","first-page":"81","volume":"14","year":"1949","journal-title":"On natural numbers, integers, and rationals"},{"key":"S0022481200105031_ref006","volume-title":"Mathematical logic","author":"Church's","year":"1936"},{"key":"S0022481200105031_ref007","first-page":"42","volume-title":"Sitzungsberichte der Preussischen Akademie der Wissenschaften","author":"Heyting","year":"1930"},{"key":"S0022481200105031_ref008","first-page":"115","volume":"7","author":"Curry","year":"1942","journal-title":"The inconsistency of certain formal logics"},{"key":"S0022481200105031_ref009","volume-title":"Symbolic logic","author":"Langford's","year":"1932"},{"key":"S0022481200105031_ref010","first-page":"1","volume":"11","author":"Barcan","year":"1946","journal-title":"A functional calculus of first order based on strict implication"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200105031","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T10:24:06Z","timestamp":1559989446000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200105031\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1950,1]]},"references-count":10,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1950,1]]}},"alternative-id":["S0022481200105031"],"URL":"https:\/\/doi.org\/10.2307\/2269228","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1950,1]]}}}