{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,12]],"date-time":"2024-08-12T14:29:26Z","timestamp":1723472966092},"reference-count":33,"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":13433,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1977,6]]},"abstract":"<jats:p>Let <jats:italic>L<\/jats:italic> be any finitary language. By restricting our attention to the universal Horn sentences of <jats:italic>L<\/jats:italic> and appealing to a semantical notion  of logical consequence, we can formulate the universal Horn logic of <jats:italic>L<\/jats:italic>. The present paper provides some theorems about universal Horn logic that serve to distinguish it from the full first order predicate logic. Universal Horn equivalence between structures is characterized in two ways, one resembling Kochen's ultralimit theorem. A sharp version of Beth's definability theorem is established for universal Horn logic by means of a reduced product construction. The notion of a consistency property is relativized to universal Horn logic and the corresponding model existence theorem is proven. Using the model existence theorem another proof of the definability result is presented. The relativized consistency properties also suggest a syntactical notion of proof that lies entirely within the universal Horn logic. Finally, a decision problem in universal Horn logic is discussed. It is shown that the set of universal Horn sentences preserved under the formation of homomorphic images (or direct factors) is not recursive, provided the language has at least two unary function symbols or at least one function symbol of rank more than one.<\/jats:p><jats:p>This paper begins with a discussion of how algebraic relations between structures can be used to obtain fragments of a given logic. Only two such fragments seem to be under current investigation: equational logic and universal Horn logic. Other fragments which seem interesting are pointed out.<\/jats:p>","DOI":"10.2307\/2272123","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:44:08Z","timestamp":1146951848000},"page":"221-237","source":"Crossref","is-referenced-by-count":11,"title":["Fragments of first order logic, I: universal Horn logic"],"prefix":"10.1017","volume":"42","author":[{"given":"George F.","family":"McNulty","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200050398_ref028","volume-title":"Annals of Mathematical Logic","author":"McNulty"},{"key":"S0022481200050398_ref011","unstructured":"Givant S. , Doctoral Thesis, University of California, Berkeley, 1975."},{"key":"S0022481200050398_ref005","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200050398_ref008","volume-title":"Advances in Mathematics","author":"Friedman"},{"key":"S0022481200050398_ref031","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-86718-7"},{"key":"S0022481200050398_ref017","first-page":"253","article-title":"On relations between structures","volume":"57","author":"Lindstr\u00f6m","year":"1965","journal-title":"Theoria"},{"key":"S0022481200050398_ref020","first-page":"341","article-title":"An Application of a method of Smullyan to logics on admissible sets","volume":"17","author":"Makkai","year":"1969","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences"},{"key":"S0022481200050398_ref009","first-page":"425","article-title":"On the construction of the least universal Horn class containing a given class","volume":"8","author":"Fujiwara","year":"1971","journal-title":"Osaka Journal of Mathematics"},{"key":"S0022481200050398_ref007","doi-asserted-by":"crossref","first-page":"195","DOI":"10.4064\/fm-51-3-195-228","article-title":"Reduced direct products","volume":"51","author":"Frayne","year":"1962","journal-title":"Fundamente Mathematical"},{"key":"S0022481200050398_ref018","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1959.9.143"},{"key":"S0022481200050398_ref030","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.49.6.828"},{"key":"S0022481200050398_ref021","volume-title":"Algebra Universalis","author":"Makkai"},{"key":"S0022481200050398_ref014","volume-title":"Algebra Universalis","author":"Herrmann","year":"1975"},{"key":"S0022481200050398_ref026","first-page":"186","volume":"40","author":"McKenzie","year":"1975","journal-title":"On spectra and the negative solution to the decision problem for identities having a finite nontrivial model"},{"key":"S0022481200050398_ref029","doi-asserted-by":"publisher","DOI":"10.1007\/BF02945004"},{"key":"S0022481200050398_ref015","first-page":"1","article-title":"On (m, A)-implicational classes","volume":"17","author":"Kashiwagi","year":"1972","journal-title":"Mathematica Japonicae"},{"key":"S0022481200050398_ref004","volume-title":"Word problems, decision problems and the Burnside problem in group theory","author":"Boone","year":"1973"},{"key":"S0022481200050398_ref003","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100013463"},{"key":"S0022481200050398_ref024","first-page":"2","article-title":"Tozdestvennye sootnosenija na mnogoobruzijah kvazigrupp","volume":"69","author":"Mal'cev","year":"1966","journal-title":"Matemati\u010deski\u01d0 Sbornik. 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