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It is common for a logic to be specified in the first instance by means of a Gentzen calculus, whereupon a Hilbert-style presentation \u2018for\u2019 the logic may be sought\u2014or vice versa. Where this has occurred, the word \u2018for\u2019 has taken on several different meanings, partly because the Gentzen separator \u21d2 can be interpreted intuitively in a number of ways. Here \u21d2 will be denoted less evocatively by \u22b2.<\/jats:p><jats:p>In this paper we aim to discuss some of the useful ways in which Gentzen and Hilbert systems may correspond to each other. Actually, we shall be concerned with the<jats:italic>deducibility relations<\/jats:italic>of the formal systems, as it is these that are susceptible to transformation in useful ways. To avoid potential confusion, we shall speak of Hilbert and Gentzen<jats:italic>relations<\/jats:italic>. By a<jats:italic>Hilbert relation<\/jats:italic>we mean any substitution-invariant consequence relation on<jats:italic>formulas<\/jats:italic>\u2014this comes to the same thing as the deducibility relation of a set of Hilbert-style axioms and rules. By a<jats:italic>Gentzen relation<\/jats:italic>we mean the fully fledged generalization of this notion in which<jats:italic>sequents<\/jats:italic>take the place of single formulas. In the literature, Hilbert relations are often referred to as<jats:italic>sentential logics<\/jats:italic>. Gentzen relations as defined here are their exact<jats:italic>sequential<\/jats:italic>counterparts.<\/jats:p>","DOI":"10.2178\/jsl\/1154698583","type":"journal-article","created":{"date-parts":[[2007,12,19]],"date-time":"2007-12-19T16:23:20Z","timestamp":1198081400000},"page":"903-957","source":"Crossref","is-referenced-by-count":36,"title":["Correspondences between gentzen and hilbert systems"],"prefix":"10.1017","volume":"71","author":[{"given":"J.G.","family":"Raftery","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200005983_ref059","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(97)80001-8"},{"key":"S0022481200005983_ref053","first-page":"2","volume-title":"Encyclopaedia of Mathematics, supplement III","author":"Pigozzi","year":"2001"},{"key":"S0022481200005983_ref042","doi-asserted-by":"publisher","DOI":"10.1007\/BF00370843"},{"key":"S0022481200005983_ref014","unstructured":"Blok W. 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