Most Gentzen systems arising in logic contain few axiom schemata and many rule schemata. Hilbert systems, on the other hand, usually contain few proper inference rules and possibly many axioms. Because of this, the two notions tend to serve different purposes. It is common for a logic to be specified in the first instance by means of a Gentzen calculus, whereupon a Hilbert-style presentation ‘for’ the logic may be sought—or vice versa. Where this has occurred, the word ‘for’ has taken (...) on several different meanings, partly because the Gentzen separator ⇒ can be interpreted intuitively in a number of ways. Here ⇒ will be denoted less evocatively by ⊲.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 thededucibility relationsof 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 Gentzenrelations. By aHilbert relationwe mean any substitution-invariant consequence relation onformulas—this comes to the same thing as the deducibility relation of a set of Hilbert-style axioms and rules. By aGentzen relationwe mean the fully fledged generalization of this notion in whichsequentstake the place of single formulas. In the literature, Hilbert relations are often referred to assentential logics. Gentzen relations as defined here are their exactsequentialcounterparts. (shrink)

We continue the work of Blok and Jónsson by developing the theory of structural closure operators and introducing the notion of a representation between them. Similarities and equivalences of Blok-Jónsson turn out to be bijective representations and bijective structural representations, respectively. We obtain a characterization for representations induced by a transformer. In order to obtain a similar characterization for structural representations we introduce the notions of a graduation and a graded variable of an M-set. We show that several deductive systems, (...) Gentzen systems among them, are graded M-sets having graded variables, and describe the graded variables in each case. In the last section we show that, for a sentential logic, having an algebraic semantics is equivalent to being representable in an equational consequence. This motivates the extension of the notion of having an algebraic semantics for Gentzen systems, hypersequents systems, etc. We prove that if a closure operator is representable by a transformer, then every extension of it is also representable by the same transformer. As a consequence we obtain that if one of these systems has an algebraic semantics, then so does any of its extensions with the same defining equations. (shrink)

Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be defined syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in [4] that the two definitions coincide in the case of an algebraizable sentential deductive system. A refined treatment of this equivalence was provided by W. Blok and B. Jónsson in [3]. Other authors have extended this result to the cases of (...) k-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. Our main result subsumes all existing results in the literature and reveals their common character. The proofs are of order-theoretic and categorical nature. (shrink)