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 the deducibility relations 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 relations. By a Hilbert relation we mean any substitution-invariant consequence relation on formulas—this comes to the same thing as the deducibility relation of a set of Hilbert-style axioms and rules. By a Gentzen relation we mean the fully fledged generalization of this notion in which sequents take the place of single formulas. In the literature, Hilbert relations are often referred to as sentential logics. Gentzen relations as defined here are their exact sequential counterparts.