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By Matthias Baaz (auth.), Uwe Egly, Chritian G. Fermüller (eds.)

This ebook constitutes the refereed lawsuits of the overseas convention on automatic Reasoning with Analytic Tableaux and comparable tools, TABLEAUX 2002, held in Copenhagen, Denmark, in July/August 2002.
The 20 revised complete papers and process descriptions awarded including invited contributions have been conscientiously reviewed and chosen for inclusion within the e-book. All present matters surrounding the mechanization of logical reasoning with tableaux and related equipment are addressed. one of the common sense calculi investigated are linear common sense, temporal good judgment, modal logics, hybrid common sense, multi-modal logics, fuzzy logics, Goedel good judgment, Lukasiewicz good judgment, intermediate logics, quantified boolean good judgment, and, in fact, classical first-order common sense.

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The cases involving the remaining logical rules as well as the cut rule are analogous. Suppose the last rule applied in d is (com) and d ends as follows ·· d · 1 G ∪ {Γ, Σ ⇒ A} ·· d · 2 G ∪ {Γ, Σ ⇒ B} G ∪ {Γ ⇒ A} ∪ {Σ ⇒ B} (com) By induction hypothesis one can find two proofs d1 and d2 in HIF with the required properties of the 1-1-reduced hypersequents (G | Γ, Σ ⇒ A)# and (G | Γ, Σ ⇒ B)# . Applying the (com) rule to them one obtains the hypersequent G# | Γ # ⇒ A | Σ # ⇒ B in which there can be at most two pairwise equal components (if both the components Γ # ⇒ A and Σ # ⇒ B are in G# ).

In the existential rules, c is a parameter which is new to the branch. As parameters are never quantified over, the substitution [c/x] is free for the formula φ(x). In the universal rules, t is any grounded term on the branch (thus either a first-order constant, a parameter or a grounded definite description). A grounded definite description is a term @n q for n a nominal and q a non-rigid designator from IC. Existential rules @s ∃xφ(x) ¬@s ∀xφ(x) @s φ(c) ¬@s φ(c) Universal rules @s ∀xφ(x) ¬@s ∃xφ(x) @s φ(t) ¬@s φ(t) Besides Fitting’s [4] Reflexivity (Ref) and Replacement (RR) rules, there are three extra rules for equality.

2. A. Avron: Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Artificial Intelligence, 4: 225–248, 1991. 3. A. Avron: The Method of Hypersequents in the Proof Theory of Propositional Nonclassical Logics. In W. Hodges, M. Hyland, C. Steinhorn and J. Truss editors, Logic: from Foundations to Applications, European Logic Colloquium Oxford Science Publications. Clarendon Press. Oxford. 1–32. 1996. 4. M. Baaz, A. Ciabattoni, C. Ferm¨ uller: Cut-Elimination in a Sequents-of-Relations Calculus for G¨ odel Logic.

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