Hajhosseini's Non-Truth-functional Logic

Mortaza Hajhosseini in the second edition of his book Two Non-Classical Logic Systems, A new Outlook on Elements of Logic has introduced four non-classical logics: truth-functional, non-truth-functional, and combinations of the two, which are naturally extensions of the former two. In another article, we have examined Hajhosseini’s truth-functional logic, and in this article, we will discuss the non-truth-functional logic and its extension. In this article, we will only deal with formal-mathematical objections, and we will leave philosophical and non-formal objections as well as related historical materials to another article. In addition to some common flaws between Hajhosseini’s truth-functional and non-truth-functional logics, such as the vicious circle in the definition of the natural deduction system, the lack of an example for the condition of "normality of arguments" in semantics, incompleteness, and the inaccuracy of extra-problems, there are other flaws in the non-truth-functional logic. A formal problem is that many of the main rules in this system can be proven with the help of other main rules and thus are redundant. Another formal problem of the non-truth-functional logic is that it has a rule called " Hajhosseini’s rule" which causes every propositional variable in this system to be a theorem and the whole system becomes trivial. The third objection is that some forms of the distributivity rule in the expansion of the non-truth-functional logic of this book reduce the whole system to the classical logic of Frege and Russell.