{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:13:44Z","timestamp":1775837624551,"version":"3.50.1"},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13890,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1976,3]]},"abstract":"Let be a finite set of (nonlogical) predicate symbols. By an -structure, we mean a relational structure appropriate for . Let be the set of all -structures with universe {1, \u2026, n<\/jats:italic>}. For each first-order -sentence \u03c3 (with equality), let \u03bcn<\/jats:italic><\/jats:sub>(\u03c3) be the fraction of members of for which \u03c3 is true. We show that \u03bcn<\/jats:italic><\/jats:sub>(\u03c3) always converges to 0 or 1 as n<\/jats:italic> \u2192 \u221e, and that the rate of convergence is geometrically fast. In fact, if T<\/jats:italic> is a certain complete, consistent set of first-order -sentences introduced by H. Gaifman [6], then we show that, for each first-order -sentence \u03c3, \u03bcn<\/jats:italic><\/jats:sub>(\u03c3) \u2192n<\/jats:italic><\/jats:sub> 1 iff T<\/jats:italic> \u22a9 \u03c9. A surprising corollary is that each finite subset of T<\/jats:italic> has a finite model. Following H. Scholz [8], we define the spectrum of a sentence \u03c3 to be the set of cardinalities of finite models of \u03c3. Another corollary is that for each first-order -sentence a, either \u03c3 or \u02dc\u03c3 has a cofinite spectrum (in fact, either \u03c3 or \u02dc\u03c3 is \u201cnearly always\u201c true).<\/jats:p>Let be a subset of which contains for each in exactly one structure isomorphic to . For each first-order -sentence \u03c3, let \u03bdn<\/jats:italic><\/jats:sub>(\u03c3) be the fraction of members of which a is true. By making use of an asymptotic estimate [3] of the cardinality of and by our previously mentioned results, we show that v<\/jats:italic>n<\/jats:italic>(\u03c3) converges as n<\/jats:italic> \u2192 \u221e, and that limn<\/jats:italic><\/jats:sub> \u03bdn<\/jats:italic><\/jats:sub>(\u03c3) = limn<\/jats:italic><\/jats:sub> \u03bcn<\/jats:italic><\/jats:sub>(\u03c3).<\/jats:sub>If contains at least one predicate symbol which is not unary, then the rate of convergence is geometrically fast.<\/jats:p>","DOI":"10.2307\/2272945","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:40:11Z","timestamp":1146951611000},"page":"50-58","source":"Crossref","is-referenced-by-count":416,"title":["Probabilities on finite models"],"prefix":"10.1017","volume":"41","author":[{"given":"Ronald","family":"Fagin","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200051756_bib002","unstructured":"Fagin, R. , Contributions to the model theory of finite structures, Doctoral dissertation, University of California, Berkeley, 1973."},{"key":"S0022481200051756_bib007","first-page":"199","volume-title":"Contributions to Mathematical Logic","author":"Oberschelp","year":"1966"},{"key":"S0022481200051756_bib008","doi-asserted-by":"crossref","unstructured":"Scholz, H. , this Journal, vol. 17 (1952), p. 160.","DOI":"10.1017\/S0022481200100076"},{"key":"S0022481200051756_bib009","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(54)50058-2"},{"key":"S0022481200051756_bib005","first-page":"257","volume":"23","author":"Harary","year":"1958","journal-title":"Note on Carnap's relational asymptotic relative frequencies"},{"key":"S0022481200051756_bib003","volume-title":"IBM research report RC5587","author":"Fagin","year":"1975"},{"key":"S0022481200051756_bib006","doi-asserted-by":"publisher","DOI":"10.1007\/BF02759729"},{"key":"S0022481200051756_bib004","volume-title":"An introduction to probability theory and its applications. I","author":"Feller","year":"1957"},{"key":"S0022481200051756_bib001","volume-title":"Logical foundations of probability","author":"Carnap","year":"1950"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200051756","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T10:41:44Z","timestamp":1679481704000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200051756\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1976,3]]},"references-count":9,"aliases":["10.1017\/s0022481200051756","10.1017\/s0022481200051756"],"journal-issue":{"issue":"1","published-print":{"date-parts":[[1976,3]]}},"alternative-id":["S0022481200051756"],"URL":"https:\/\/doi.org\/10.2307\/2272945","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1976,3]]}}}