Difficulties With The Use of Vacuous Truth
- All pink rhinoceros are carnivores.
- All pink rhinoceros are herbivores.
Both of these seemingly contradictory statements are true using classical or two-valued logic – so long as the set of pink rhinoceros remains empty. (See also Present King of France.)
One fundamental problem with such “demonstrations” is the uncertainty of the truth-value of any of the statements which follow (or even whether they do follow) when our initial supposition is false. Stated another way, we should ask ourselves which rules of mathematics or inference should still be applicable if we suppose that pi is an integer (which it is not).
The problem occurs when it is not immediately obvious that we are dealing with a vacuous truth. For example, if we have two propositions, neither of which implies the other, then we can reasonably conclude that they are different; counter-intuitively, we can also conclude that the two propositions are the same. The reason for this is that is a tautology in classical logic, so every assertion that is made about "two propositions, neither of which implies the other" is an assertion about nothing, hence vacuously true. Although such a fact that "two propositions, neither of which implies the other, are both different and the same" poses no theoretical problems, it can easily be disturbing to the human mind.
Avoidance of such paradox is the impetus behind the development of non-classical systems of logic relevant logic and paraconsistent logic which refuse to admit the validity of one or two of the axioms of classical logic. Unfortunately the resulting systems are often too weak to prove anything but the most trivial of truths.
Read more about this topic: Vacuous Truth
Famous quotes containing the words difficulties and/or truth:
“It appeared that he had once represented his tribe at Augusta, and also once at Washington, where he had met some Western chiefs. He had been consulted at Augusta, and gave advice, which he said was followed, respecting the eastern boundary of Maine, as determined by highlands and streams, at the time of the difficulties on that side. He was employed with the surveyors on the line. Also he called on Daniel Webster in Boston, at the time of his Bunker Hill oration.”
—Henry David Thoreau (1817–1862)
“I am sorry to think that you do not get a man’s most effective criticism until you provoke him. Severe truth is expressed with some bitterness.”
—Henry David Thoreau (1817–1862)