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.
2026-03-29 14:00:53.1774792853
Aren't vacuous statements True and False simultaneously?
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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.