Alleged Impossibility of Its Proof or Denial
As is true of all axioms of logic, the law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the grounds that any proof or disproof must use the law itself prior to reaching the conclusion. In other words, in order to verify or falsify the laws of logic one must resort to logic as a weapon, an act which would essentially be self-defeating. Since the early 20th century, certain logicians have proposed logics that deny the validity of the law. Collectively, these logics are known as "paraconsistent" or "inconsistency-tolerant" logics. But not all paraconsistent logics deny the law, since they are not necessarily completely agnostic to inconsistencies in general. Graham Priest advances the strongest thesis of this sort, which he calls "dialetheism".
In several axiomatic derivations of logic, this is effectively resolved by showing that (P ∨ ¬P) and its negation are constants, and simply defining TRUE as (P ∨ ¬P) and FALSE as ¬(P ∨ ¬P), without taking a position as to the principle of bivalence or the law of excluded middle.
Some, such as David Lewis, have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true. A related objection is that "negation" in paraconsistent logic is not really negation; it is merely a subcontrary-forming operator.
Read more about this topic: Law Of Noncontradiction
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