In the paper by Riečan and Markechová (Fuzzy Sets Syst. 96, 1998), some fuzzy modifications of Shannon’s and Kolmogorov-Sinai’s entropy were studied and the general scheme involving the presented models was introduced. Our aim in this contribution is to provide analogies of these results for the case of the logical entropy. We define the logical entropy and logical mutual information of finite partitions on the appropriate algebraic structure and prove basic properties of these measures. It is shown that, as a special case, we obtain the logical entropy of fuzzy partitions studied by Markechová and Riečan (Entropy 18, 2016). Finally, using the suggested concept of entropy of partitions we define the logical entropy of a dynamical system and prove that it is the same for two dynamical systems that are isomorphic.