Scope of The Concept
A statement is “vacuously true” if it resembles the statement, where is known to be false.
Statements that can be reduced (with suitable transformations) to this basic form include the following:
- , where it is the case that .
- , where the set is empty.
- , where the symbol is restricted to a type that has no representatives.
Vacuous truth is usually applied in classical logic, which in particular is two-valued, and most of the arguments in the next section will be based on this assumption. However, vacuous truth also appears in, for example, intuitionistic logic in the same situations given above. Indeed, the first two forms above will yield vacuous truth in any logic that uses material conditional, but there are other logics which do not.
Read more about this topic: Vacuous Truth
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“For it is not the bare words but the scope of the writer that gives the true light, by which any writing is to be interpreted; and they that insist upon single texts, without considering the main design, can derive no thing from them clearly.”
—Thomas Hobbes (1579–1688)
“Happy is that mother whose ability to help her children continues on from babyhood and manhood into maturity. Blessed is the son who need not leave his mother at the threshold of the world’s activities, but may always and everywhere have her blessing and her help. Thrice blessed are the son and the mother between whom there exists an association not only physical and affectional, but spiritual and intellectual, and broad and wise as is the scope of each being.”
—Lydia Hoyt Farmer (1842–1903)
“As the creative adult needs to toy with ideas, the child, to form his ideas, needs toys—and plenty of leisure and scope to play with them as he likes, and not just the way adults think proper. This is why he must be given this freedom for his play to be successful and truly serve him well.”
—Bruno Bettelheim (20th century)
“There is a concept that is the corrupter and destroyer of all others. I speak not of Evil, whose limited empire is that of ethics; I speak of the infinite.”
—Jorge Luis Borges (1899–1986)