Since a major part of the book is devoted to concrete deductive systems for hybrid logics, providing a formal definition of a deductive system is in place. We use this notion in a similar way to Indrzejczak in [53], therefore, we repeat his very general definition of a deductive system.

Definition 3.1 (Deductive system ([53])). Every deductive system may be characterized on two elementary levels of description:

· the calculus,

· the realization.

By a calculus we understand a non-empty set of schemata of rules of the form:

X1, …,Xn/Y1, …,Ym, n ≥ 0, m ≥ 1,

with a possible list of side conditions. Symbols Xi, denote some data structures being transformed into data structures Yi.

By a realization we usually understand the set of instructions of how to apply rules to perform a derivation/construct a proof.

In the book, we describe instantiations of two basic types of deductive systems: tableau calculi and sequent calculi. We dedicate to each of these types a short, introductory note. We skip all historical remarks as hardly relevant for our considerations. We also waive the idea of a profound exposition of Hilbert-style calculi, even though several such calculi are presented in this book. We assume that the notion of Hilbert-style calculus is rather familiar to most readers and we consider its thorough examination as lying outside the scope of the book. Considerations in this section are based on [53, 52].

SEQUENT CALCULI

Originally invented by Gentzen in the 1930s, sequent calculi are finite sets of primitive rules of the form:

S1, …,Sn/Sn + 1,

where Si is a sequent schema.

By a sequent we understand an ordered pair of finite sequences of formulas with a symbol ⇒ between these sequences. Every sequent is therefore of the form:

φ1, …,φn ⇒ ψ1, …,ψmn, m ≥ 0.