{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T19:21:09Z","timestamp":1699039269580},"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":"We consider here the pure functional calculus of first order as formulated by Church.<\/jats:p>Church, l.c., p. 79, gives the definition of the validity of a formula in a given set I<\/jats:italic> of individuals and shows that a formula is provable in if and only if it is valid in every non-empty set 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>The notion of a value of a formula retains its meaning also in the case when the set I<\/jats:italic> is empty. We have only to remember that if I<\/jats:italic> is empty, then an m<\/jats:italic>-ary propositional function (i.e. a function from the m<\/jats:italic>-th cartesian power Im<\/jats:sup><\/jats:italic> to the set {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 f<\/jats:italic> or t<\/jats:italic>. Indeed, if B<\/jats:bold> has the unique free variable c<\/jats:bold> and \u03d5<\/jats:italic> is the value of B<\/jats:bold>, then the value of (c)B<\/jats:bold> according to the definition given by Church is the propositional constant f<\/jats:italic> or t<\/jats:italic> according as \u03d5(j)<\/jats:italic> is f<\/jats:italic> for at least one j<\/jats:italic> in I<\/jats:italic> or not. Since, however, there is no j<\/jats:italic> in I<\/jats:italic>, the condition \u03d5(j) = t for all j in I<\/jats:italic> is vacuously satisfied and hence the value of (c)B<\/jats:bold> is 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":24,"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":"http:\/\/dx.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]]}}}