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Cut-free Sequent Calculus and Natural Deduction for the Tetravalent Modal Logic
Studia Logica ( IF 0.6 ) Pub Date : 2021-05-30 , DOI: 10.1007/s11225-021-09944-3
Martín Figallo

The tetravalent modal logic (\({\mathcal {TML}}\)) is one of the two logics defined by Font and Rius (J Symb Log 65(2):481–518, 2000) (the other is the normal tetravalent modal logic \({{\mathcal {TML}}}^N\)) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character (tetravalence) with a modal character. In fact, \({\mathcal {TML}}\) is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic \({\mathcal {TML}}\) and the algebras is not so good as in \({{\mathcal {TML}}}^N\), but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus (see Font and Rius in J Symb Log 65(2):481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property. Then, using a general method proposed by Avron et al. (Log Univ 1:41–69, 2006), we provide a sequent calculus for \({\mathcal {TML}}\) with the cut-elimination property. Finally, inspired by the latter, we present a natural deduction system, sound and complete with respect to the tetravalent modal logic.



中文翻译:

四价模态逻辑的免割序演演算与自然演绎

所述四价模态逻辑\({\ mathcal {TML}} \) )是由字体和里乌斯定义的两个逻辑中的一个(j SYMB日志65(2):481-518,2000)(另一种是正常的四价模态逻辑 \({{\mathcal {TML}}}^N\) ) 与 Monteiro 的四价模态代数有关。这些逻辑是著名的Belnap-Dunn 四值逻辑的扩展,将多值字符(四价)与模态字符组合在一起。事实上,\({\mathcal {TML}}\)是保留关于四价模态代数的真实度的逻辑。正如 Font 和 Rius 所观察到的,逻辑\({\mathcal {TML}}\)和代数之间的联系并不像在\({{\mathcal {TML}}}^N\),但是,作为补偿,它具有更好的证明理论行为,因为它具有非常充分的 Gentzen 演算(参见 J Symb Log 65( 2):481–518, 2000)。在这项工作中,我们证明了由 Font 和 Rius 给出的序列演算不具有消减特性。然后,使用 Avron 等人提出的一般方法。(Log Univ 1:41–69, 2006),我们为\({\mathcal {TML}}\)提供了一个具有切割消除特性的连续演算。最后,受后者的启发,我们提出了一个自然的演绎系统,相对于四价模态逻辑来说是健全和完整的。

更新日期:2021-05-30
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