Document Type : Research Paper
Author
Department of Logic, Iranian Institute of Philosophy, Tehran,, Iran.
Abstract
Mortaza Hajhosseini introduced two truth-functional and non-truth-functional systems in the second edition of his book Two Non-Classical Logic Systems, A New Outlook on Elements of Logic. In other articles, I have reviewed the natural deduction and the semantics of the truth-functional system. In this paper, I will address the semantics of the non-truth-functional system and the meta-theorems of soundness and completeness that he claimed to have proven. I demonstrate that: 1. This semantics is based on a new set theory that has not yet been formulated. 2. The definition of ‘model’ in this book defines ‘complete models,’ while all introduced models are ‘incomplete models’. 3. The truth conditions of non-truth-functional formulas are reminiscent of similar formula truth conditions in logic, as emphasized by C.I. Lewis, but are valid in Libnizian semantics. 4. This semantics is not consistent with the proof theory of the book. 5. Therefore, the validity and completeness consequences are actually incorrect and have counterexamples. 6. Negation operators in this non-truth-functional system have been considered truth-functional, while they should be non-truth-functional. 7. A finite number of relationships (exactly five) have been introduced in this semantics, while this number is infinite. 8. Therefore, non-truth-functional logic must be undecidable, while the non-truth-functional system has been declared decidable
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