خلاصة:
In modern logic, the logical implication between premises and conclusion in any proof, as well as the implication between the negation of the conclusion and the contradiction resulting from it in indirect proofs, is of the material implication type; however, in ancient logic, it is of the necessary implication type. Furthermore, in indirect proofs, after assuming this necessary relationship, it is assumed that the conclusion itself is accidentally conditioned by its own negation, and this latter relationship is not the basis of Reductio ad Absurdum; rather, first, it is superfluous and there is no need to assume it, and second, its reasoning is flawed. We will also show that the scope of application of Reductio ad Absurdum in ancient logic is broader than assumed. Additionally, first, its application in various formal systems depends on proving the consistency and completeness of that system. Second, since this proof is based on the principle of excluded middle, it will not be applicable in multi-valued formal systems.
ملخص الجهاز:
The relationship between the negation of the conclusion and the contradiction resulting from it in the form of indirect proofs In modern logic, the validity of a proof form based on the method founded on truth and falsity (semantic method) means that it is impossible for the premises to be true and the conclusion to be false.
، 2A، 1A and we wish to prove it by reductio ad absurdum, we say: If the inference of B from the premises 1A to nA is not correct, then the inference of its negation, based on the principle of the law of excluded middle, must be correct; then, if the negation of the conclusion, in combination with the premises whose truth is assumed, leads to a contradiction, one can rule the truth of the conclusion itself based on the principle of the impossibility of contradiction, and its general form will be as follows: As can be observed, the symbol in line "1+m" is of the type of material implication.
Second: In the continuation of their analysis of reductio ad absurdum, they have once again applied the conditional reason and inferred the truth of the conclusion from the assumption of its negation; and in analyzing how the truth of "If B, then not B" occurs—considering that in a necessary implication, a false antecedent cannot result in anything other than a false consequent—they did not consider it to be of the necessary type.
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