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.

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。