Free-variable semantic tableaux are a well-established technique for first-order theorem proving where free variables act as a meta-linguistic device for tracking the eigenvariables used during proof search. We present the theoretical foundations to extend this technique to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, and also present various techniques that improve the efficiency of the basic naive method for such tableaux.

In this paper we focus on theorem proving for conditional logics. First, we give a detailed description of CondLean, a theorem prover for some standard conditional logics. CondLean is a SICStus Prolog implementation of some labeled sequent calculi for conditional logics recently introduced. It is inspired to the so called “lean” methodology, even if it does not fit this style in a rigorous manner. CondLean also comprises a graphical interface written in Java. Furthermore, we introduce a goal-directed proof search mechanism, (...) derived from the above mentioned sequent calculi based on the notion of uniform proofs. Finally, we describe GOALDUCK, a simple SICStus Prolog implementation of the goal-directed calculus mentioned here above. Both the programs CondLean and GOALDUCK, together with their source code, are available for free download at. (shrink)

We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert- and Natural Deduction-style proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed -calculus metalanguage of the Logical Frameworks. These formalizations yield readily proof-editors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO.