Chapter Headings only. About this Book. Overview. Part 1. The Fundamentals. Algebra, logic and deduction. Fundamentals of modal logic I. Fundamentals of modal logic II. Part 2. The General Theory of Modal Logic. Universal algebra and duality theory. Definability and correspondence. Reducing polymodal logic to monomodal logic. Lattices of modal logics. Part 3. Case Studies. Extensions of K4. Logics of bounded alternativity. Dynamic logic. Index. Bibliography.

Tools and Techniques in Modal Logic

I overview the work of the Tbilisi school on intuitionistic modal logics of well-founded/scattered structures and its connections with contemporary Theoretical Computer Science. Fixed-point theorems and their consequences are of particular interest.

Constructive Modalities with Provability Smack

This paper is a comparative study of the propositional intuitionistic (non-modal) and classical modal languages interpreted in the standard way on transitive frames. It shows that, when talking about these frames rather than conventional quasi-orders, the intuitionistic language displays some unusual features: its expressive power becomes weaker than that of the modal language, the induced consequence relation does not have a deduction theorem and is not protoalgebraic. Nevertheless, the paper develops a manageable model theory for this consequence and its extensions which also reveals some unexpected phenomena. The balance between the intuitionistic and modal languages is restored by adding to the former one more implication.

Speaking about Transitive Frames in Propositional Languages

We give two generic proofs for cut elimination in propositional modal logics, interpreted over coalgebras. We first investigate semantic coherence conditions between the axiomatisation of a particular logic and its coalgebraic semantics that guarantee that the cut-rule is admissible in the ensuing sequent calculus. We then independently isolate a purely syntactic property of the set of modal rules that guarantees cut elimination. Apart from the fact that cut elimination holds, our main result is that the syntactic and semantic assumptions are equivalent in case the logic is amenable to coalgebraic semantics. As applications we present a new proof of the (already known) interpolation property for coalition logic and newly establish the interpolation property for the conditional logics LCK and LCKID.

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Cut elimination in coalgebraic logics

Speaking about transitive frames in propositional languages

We give sound and complete tableau and sequent calculi for the propositional normal modal logics $4.04, K4B and G O (these logics axe the smallest normal modal logics containing K and the schemata (:)A -+ ODA, DA --+ A and D A ; DA --+ D(qA and D(D(A --+ DA) -+ A) -+ DA resp.) with the following properties: the calculi for S4.04 and G O are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the so-called analytical cut-rule. In addition we show that G 0 is not compact (and therefore not canonical), and we proof with the tableau-method that G O is chaxacterised by the class of all finite, (transi- tive) trees of degenerate or simple clusters of worlds; therefore G O is decidable and also chaxacterised by the class of all frames for G 0.

Martin Amerbauer

Papers

Cut-free Tableau Calculi for some Propositional Normal Modal Logics*