{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T10:57:31Z","timestamp":1777892251552,"version":"3.51.4"},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":22930,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1951,6]]},"abstract":"<jats:p>We consider here the pure functional calculus of first order <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200101847_inline1\"\/> as formulated by Church.<\/jats:p><jats:p>Church, l.c., p. 79, gives the definition of the validity of a formula in a given set <jats:italic>I<\/jats:italic> of individuals and shows that a formula is provable in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200101847_inline1\"\/> if and only if it is valid in every non-empty set <jats:italic>I<\/jats:italic>. The definition of validity is preceded by the definition of a value of a formula; the notion of a value is the basic \u201csemantical\u201d notion in terms of which all other semantical notions are definable.<\/jats:p><jats:p>The notion of a value of a formula retains its meaning also in the case when the set <jats:italic>I<\/jats:italic> is empty. We have only to remember that if <jats:italic>I<\/jats:italic> is empty, then an <jats:italic>m<\/jats:italic>-ary propositional function (i.e. a function from the <jats:italic>m<\/jats:italic>-th cartesian power <jats:italic>I<jats:sup>m<\/jats:sup><\/jats:italic> to the set {<jats:italic>f, t<\/jats:italic>}) is the empty set. It then follows easily that the value of each well-formed formula with free individual variables is the empty set. The values of wffs without free variables are on the contrary either <jats:italic>f<\/jats:italic> or <jats:italic>t<\/jats:italic>. Indeed, if <jats:bold>B<\/jats:bold> has the unique free variable <jats:bold>c<\/jats:bold> and <jats:italic>\u03d5<\/jats:italic> is the value of <jats:bold>B<\/jats:bold>, then the value of <jats:bold>(c)B<\/jats:bold> according to the definition given by Church is the propositional constant <jats:italic>f<\/jats:italic> or <jats:italic>t<\/jats:italic> according as <jats:italic>\u03d5(j)<\/jats:italic> is <jats:italic>f<\/jats:italic> for at least one <jats:italic>j<\/jats:italic> in <jats:italic>I<\/jats:italic> or not. Since, however, there is no <jats:italic>j<\/jats:italic> in <jats:italic>I<\/jats:italic>, the condition <jats:italic>\u03d5(j) = t for all j in I<\/jats:italic> is vacuously satisfied and hence the value of <jats:bold>(c)B<\/jats:bold> is <jats:italic>t<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2266682","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T19:15:56Z","timestamp":1146942956000},"page":"107-111","source":"Crossref","is-referenced-by-count":25,"title":["On the rules of proof in the pure functional calculus of the first order"],"prefix":"10.1017","volume":"16","author":[{"given":"Andrzej","family":"Mostowski","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200101847_ref002","first-page":"74","volume-title":"Mathematical logic","author":"Quine","year":"1947"},{"key":"S0022481200101847_ref003","volume-title":"Studia logica","author":"Ja\u015bkowski","year":"1934"},{"key":"S0022481200101847_ref001","volume-title":"Introduction to mathematical logic","author":"Church","year":"1944"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200101847","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T07:04:32Z","timestamp":1559891072000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200101847\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1951,6]]},"references-count":3,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1951,6]]}},"alternative-id":["S0022481200101847"],"URL":"https:\/\/doi.org\/10.2307\/2266682","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1951,6]]}}}