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In this work, with the help of a calculus system syllogistic logic with Carroll's diagrams (SLCD), we construct a useful algorithm for the possible deductions of polysyllogisms (soriteses). This algorithm makes a general deduction in categorical syllogisms with the help of diagrams to depict each proposition of polysyllogisms. The developed calculus system PolySLCD (PSLCD) is used to allow a formal deduction from premises set by comprising synchronically biliteral and triliteral diagrammatical appearance and simple algorithmic nature. This algorithm can be used to deduce new conclusions, step by step, through recursive conclusion sets that are obtained from premises of categorical polysyllogisms. The fundamental contributions of this paper are accurately deducing conclusions from sets corresponding to given premises as exact human reasoning using a single algorithm and designing this algorithm based on SLCD. Therefore, it is more suitable for computer-aided solution. Since the algorithm is set-based, it is a novel algorithm in the literature and it can easily contribute to the researchers using polysyllogisms in different scientific branches, such as computer science, decision-making systems and artificial intelligence. [ABSTRACT FROM AUTHOR]