عنوان مقاله [English]
In other articles, we have examined Mortaza Hajhosseini's natural deduction systems as presented in the second edition of his book, "Two Non-Classical Logic Systems: A New Outlook on Elements of Logic." In this article, our focus shifts to the semantics of these systems and an analysis of the issues surrounding soundness and completeness, as proven by Hajhosseini. We find that the book lacks clear definitions or definite specifications for several semantical terms used, including 'world,' 'supposed,' 'supposible,' 'basic situation,' and 'basic model.' Consequently, the semantic concepts of "validity" and "truth-preserving" in this context can be subject to multiple interpretations, some of which render certain relevant inferences irrelevant while others deem irrelevant inferences valid.
Furthermore, we demonstrate that this semantics framework is incompatible with the proof theory presented in the book. The meta-theorems of soundness and completeness, as formulated, are erroneous and can be refuted with counterexamples. For instance, within the truth-functional system's semantics outlined in the book, a logical truth implies any other logical truth. However, the proofs provided in the book's natural deduction system are deemed 'invalid' because they violate Dag Prawitz's "normality" condition.