In [J. Andrikonis, Loop-free calculus for modal logic S4. I, Lith. Math. J., 52(1):1-12, 2012], loop-free calculus for modal logic S4 is presented. The calculus uses several types of indexes to avoid loops and obtain termination of derivation search. Although the mentioned article proves that derivation search in the calculus is finite, the proof of soundness and completeness is omitted and, therefore, is presented in this paper. Moreover, this paper presents loop-free calculus for modal logics K4, which is obtained by modifying the calculus for S4. Finally, some remarks for programming the given calculi are offered.
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