Take the proposition "it is not sufficient for the monkey to dance in order for me to get an A on the test"
m = the monkey dances
a = I get an A on the test
It makes sense why I can translate the statement "it IS sufficient for the monkey to dance in order for me to get an A on the test" to m → a, because if m and a are both true the proposition is true, and if m is false I can understand why the proposition is vacuously true.
Where my understanding falls apart is when you negate the statement:
| m | a | m → a |
¬(m → a) |
|---|---|---|---|
| T | T | T | F |
| T | F | F | T |
| F | T | T | F |
| F | F | T | F |
It makes sense why ¬(m → a) is equivalent to "it is not sufficient for the monkey to dance in order for me to get an A on the test", because it is a negation of m → a.
But it doesn't make sense that m and a both being true makes "it is not sufficient for the monkey to dance in order for me to get an A on the test" false. If the monkey dances, and I get an A on the test, wouldn't that make this proposition vacuously true? This proposition claims that the monkey dancing isn't sufficient, but just because both happen, it doesn't contradict the proposition. Is it just "vacuously false"?
No, in your specified interpretation,
¬(m→a)is neither vacuously true nor vacuously false ($m$ being false would make it “vacuously false” though), just synthetically false, and just, regardless of meaning assignments, having the flipped truth value of(m→a).In your specified interpretation,
¬(m→a)is a descriptive statement, so the translation “the monkey dancing is not sufficient for me getting an A on the test”, which sounds less analytic, is more accurate.I've boldfaced your key observation: $(m,a)=(T,T)$ indeed does not make
¬(m→a)a contradiction (logically false). On the other hand, if $a$ was actually a compound proposition of the form $(q\lor\lnot q),$ then¬(m→a)would be logically false, i.e., $m$ would be logically not sufficient for $a,$ which is a stronger assertion than merely that¬(m→a)is false. For a fuller explanation, please refer to Not necessarily implies.The truth table claiming that
¬(m→a)is false for $(m,a)=(T,T)$ is consistent with the claims that “(x=y or x≠y) is not sufficient for (x=y or x≠y)” and¬(T→T)are false.Addendum to address the OP's above comments:
Your argument here is in the same vein as, “$(m,a)=(T,T)$ does not prove that
(m→a)is true, since, after all, you might have gotten an A even if the monkey does not dance”.So, the root of your question is actually not about non-sufficiency per se, but about what it means for
(m→a)to be true (or to be false). Well, in your specified interpretation,(m→a)and¬(m→a)are neither logically nor universally nor analytically true/false, and the truth value of $a$ can be determined only empirically and using the logical operations (or truth table) involved; in particular, no future tense is being suggested.However, adding the axiom “The test result is capped at grade B” to the sytem, then $m$ is sufficient for $a$ precisely when the monkey does not dance, and then you can make predictions like, “Since I know that the monkey dancing will be sufficient for scoring A, I know that the monkey won't dance” and “Since I know that the monkey will dance, I know that the monkey dancing won't be sufficient for scoring A”.
You are using the phrase “vacuously true/false” wrongly (it means that the falseness of $m$ immediately gives the truth values of
(m→a)and¬(m→a)), and your question is, rather, about the various levels of truth/‘implies’.