3
$\begingroup$

Wikipedia states "a vacuous truth is a statement that asserts that all members of the empty set have a certain property". Clearly the statement: 'all elements of said (empty) set possess said property' is vacuously true. However, one could argue that the negation of the statement: 'no elements in said set posses said property' is also true. Shouldn't that mean that the statement is both true and false. I understand there may be slightly different definitions of what constitutes a vacuous statement, but I suppose this particular issue will show up nevertheless.

$\endgroup$
  • $\begingroup$ "Vacuous truth" means a statement that is true but mathematically inconsequential. It comes up with more than empty sets. "If 9 is even, then it is prime". That's true, but (arguably) never useful. That's why it is called vacuous. $\endgroup$ – DanielV Dec 2 '16 at 9:28
6
$\begingroup$

You did not negate the statement "all elements of a set S have property X" correctly. The opposite of "all elements of a set S have property X" is not "no elements of set S have property X".

The opposite of "all elements of a set S have property X" is "some element of S does not have property X". If S is empty, "some element of S does not have property X" is definitely not true.

$\endgroup$
  • 3
    $\begingroup$ Your negation is also missing a "not" as it should be the existence of an element without the property. $\endgroup$ – Tobias Kildetoft Dec 2 '16 at 9:14
  • $\begingroup$ Yes, of course. $\endgroup$ – Tuneer Chakraborty Dec 2 '16 at 9:36
  • $\begingroup$ @TobiasKildetoft Corrected. $\endgroup$ – Ted Dec 2 '16 at 16:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.