The article is the second part of the author’s article under the same title, published in the previous issue of this journal. The article discusses the problem of formation of logical competences of future teachers of mathematics both at the level of bachelor’s degree and at the level of master’s degree. In the article, logic is considered in three aspects – classical Aristotelian logic, modern mathematical logic and its application to Aristotelian logic and non-classical logics. A number of measures related to the content of logical training of prospective teachers of mathematics and aimed at improving this training in order to develop the logical and general professional competence of teachers of mathematics produced by pedagogical and classical universities are proposed. The article characterizes the knowledge and skills in the field of logic and mathematics teaching methods, which are necessary for future math teachers to teach mathematics to their future students most effectively and develop in them logical thinking skills. The most important skills are understanding the logical structure of mathematical propositions (definitions and theorems), understanding the essence of proving mathematical theorems, knowing methods of proving them, understanding the structure of mathematical theories, and the ability to apply logical knowledge in teaching mathematics.