One allows disjunction introduction, and one allows disjunctive syllogism.
2.
One approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity.
3.
The third Idea is based on the disjunctive syllogism.
4.
Another approach is to reject disjunctive syllogism.
5.
From the perspective of dialetheism, it makes perfect sense that disjunctive syllogism should fail.
6.
Disjunctive syllogism holds in classical propositional logic and intuitionistic logic, but not in some paraconsistent logics.
7.
Disjunctive syllogism is closely related and similar to hypothetical syllogism, in that it is also type of syllogism, and also the name of a rule of inference.
8.
Proof-theoretic paraconsistent logics usually deny the validity of one of the steps necessary for deriving an explosion, typically including disjunctive syllogism, disjunction introduction, and reductio ad absurdum.
9.
In classical logic, "'disjunctive syllogism "'( historically known as "'modus tollendo ponens "') is a disjunctive statement for one of its premises.
10.
Unlike modus ponendo ponens and modus ponendo tollens, with which it should not be confused, disjunctive syllogism is often not made an explicit rule or axiom of logical systems, as the above arguments can be proven with a ( slightly devious ) combination of reductio ad absurdum and disjunction elimination.
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