By V. Michele Abrusci (auth.), Marta Cialdea Mayer, Fiora Pirri (eds.)
This booklet constitutes the refereed court cases of the overseas convention on computerized Reasoning with Analytic Tableaux and comparable equipment, TABLEAUX 2003, held in Rome, Italy in September 2003.
The 20 revised complete papers provided have been rigorously reviewed and chosen for inclusion within the publication. All present concerns surrounding the mechanization of logical reasoning with tableaux and comparable equipment are addressed within the context of a large number of good judgment calculi.
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Additional resources for Automated Reasoning with Analytic Tableaux and Related Methods : International Conference, TABLEAUX 2003, Rome, Italy, September 2003. Proceedings
1 can be extended in the presence of this extra rule. 2 Lukasiewicz Logics We start by deﬁning the inﬁnite-valued Lukasiewicz logic L, noting that in this work we identify theoremhood in a logic with derivability in the corresponding Hilbert-style system. Deﬁnition 1 (Lukasiewicz Inﬁnite-Valued Logic, L). A Hilbert-style system for L, using the connectives ⊃ and ⊥, consists of the rule: (mp) A ⊃ B, A B 2 together with the axioms: Note that a single sequent calculus for L has also been deﬁned in  34 Agata Ciabattoni and George Metcalfe L1 A ⊃ (B ⊃ A) L2 (A ⊃ B) ⊃ ((B ⊃ C) ⊃ (A ⊃ C)) L3 ((A ⊃ B) ⊃ B) ⊃ ((B ⊃ A) ⊃ A) L4 ((A ⊃ ⊥) ⊃ (B ⊃ ⊥)) ⊃ (B ⊃ A) Other connectives are deﬁned as follows: ¬A =def A ⊃ ⊥, A⊕B =def ¬A ⊃ B and A B =def ¬(A ⊃ ¬B).
K Δk . is derivable in GLBn . Hence Q is derivable in GLBn by (nC). ✷ We arrive at the following cut-elimination theorem for GLBn . Theorem 8. If G|Γ, A in GLBn . Δ, A is derivable in GLBn , then G|Γ Δ is derivable References  A. Avron. A constructive analysis of RM. J. of Symbolic Logic, 52:939–951, 1987. 33, 37  C. C. Chang. Algebraic analysis of many valued logics. Trans. Amer. Math. , 88:467–490, 1958. 32, 35  A. Ciabattoni. Bounded contraction in systems with linearity. In Proceedings of TABLEAUX 99, volume 1617 of LNAI, pages 113–127.
K Δk |Γk , λk B Δk , λk A is derivable in GL. We proceed by induction on h, the height of a proof of Q. We assume λi > 0 for some i, 1 ≤ i ≤ k, as otherwise the proof is trivial. If h = 0 then Q = A ⊃ B A ⊃ B and Q = A ⊃ B|B A ⊃ B, A which is derivable in GL. For h > 0 there are several cases for the last rule r applied. – r is (W L) and we step to: Γ1 , (λ1 − 1)(A ⊃ B) Δ1 | . . |Γk , λk (A ⊃ B) Δk so by the induction hypothesis: Γ1 Δ1 |Γ1 , (λ1 − 1)B Δ1 , (λ1 − 1)A| . . |Γk Δk |Γk , λk B Δk , λk A is derivable in GL; hence Q is derivable by repeatedly applying (M ) and (S) (λ1 − 1) times together with (EC) and (EW ) as necessary.