6
m a m → a ¬(m → a)
T T T F
T F F T
F T T F
F F T F

$m$ := the monkey dances
$a$ := I get an A on the test
$Q$ := the monkey dancing IS SUFFICIENT for me to get an A on the test
$\lnot Q$ := the monkey dancing IS NOT SUFFICIENT for me to get an A on the test

It makes sense that I can translate the statement m → a to the statement $Q,$ because if $m$ is false I can understand why this statement is vacuously true, while if $m$ is true then $a$ is true. And it makes sense that its negation ¬(m → a) is equivalent to the statement $\lnot Q.$

But it doesn't make sense that $m$ and $a$ both being true should contradict $\lnot Q.$ The monkey dancing together with me getting an A doesn't actually disprove the monkey dancing is NOT SUFFICIENT for me to get an A on the test, since I could have gotten an A regardless of whether the monkey danced. In this case, shouldn't $\lnot Q$ be vacuously true (contrary to what the truth table says) in the same way that a conditional with a false premise is vacuously true? Or does $m$ and $a$ both being true make $\lnot Q$ just "vacuously false"?

ryang
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Ben
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  • Since $m\to a$ is false only in the case that $m$ is true but $a$ is false, (regardless of what $m$ and $a$ mean), then that means that $\neg(m\to a)$ can only be true if $m$ is true but $a$ is false. So the statement $\neg(m\to a)$ is equivalent to the monkey having danced, and you not getting an A on your test (thus $m$ is not sufficient for $a$). – Graviton Mar 06 '23 at 03:07
  • But if someone were to say: "the monkey dancing isn't sufficient for you to get an A", and then the monkey dances and you get an A, the statement is not disproven (because you could've gotten an A regardless of whether the monkey danced), so shouldn't it be vacuously true in the same way that in if the premise of a conditional is false it is vacuously true? – Ben Mar 06 '23 at 03:21
  • "because you could've gotten an A regardless of whether the monkey danced" is exactly what it means to be not sufficient. Also, I think you are using the term vacuous truth in an informal way. For conditional statements like $A\to B$, we may say $A\to B$ is a vacuous truth if it is true without needing to consider all the truth values of $A$ and $B$. For example, the statement "If the sky is green, then I can fly" is a vacuous truth, because the sky is not green, so whether I can fly or not does not even need to be considered. – Graviton Mar 06 '23 at 05:21
  • ""because you could've gotten an A regardless of whether the monkey danced" is exactly what it means to be not sufficient." Yeah that's the problem. That would make the statement "it is not sufficient" true when the truth table says it should be false. – Ben Mar 06 '23 at 05:27
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    I get the feeling that you would have to go to either model theory or modal logic to get a formalization of what you're trying to get at. For model theory: you set up some classes of a model of a situation involving the monkey dancing and grading of a test, and then show some model in which the monkey dances yet you don't get an A, so you don't have entailment: monkey dances $\not\models$ get an A on the test. For modal logic, use the "many worlds" interpretation and show maybe something like $\lnot\square(MonkeyDances\rightarrow grade(test) = A)$ though I'm fuzzier on that. – Daniel Schepler Mar 06 '23 at 18:23
  • And then to show that, or the equivalent $\Diamond (MonkeyDances \land grade(test) \ne A)$, you are invoking the existence of a "world state" in which the monkey dances but still you don't get an A. – Daniel Schepler Mar 06 '23 at 18:26

2 Answers2

2

The statement ¬(X→Y) is “vacuously false” iff $X$ is false. Your main question is not related to vacuous truth/falsity at all.

$X$ := the monkey dances
$Y$ := I get an A on the test

In the framework of propositional logic, absent axioms, the assertion X is insufficient for Y, that is, ¬(X→Y), just descriptively and non-analytically corresponds to its truth table's specific row regarding the specified or tacit context. The statement ¬(X→Y) is neither logically nor universally true/false, and for the particular case $(X,Y)=(\text{true,true}),$ is simply synthetically false.

It's worth noting that X is insufficient for Y (equivalently: Y is unnecessary for X) is equivalent to $X ∧ ¬Y,\tag*{}$ and specifically asserts that $Y$ is false. If this is surprising or outright feels unreasonable, it's because propositional logic is simply not the correct tool to richly investigate sufficiency and necessity over quantification and possible worlds.

In contrast, X(t) is insufficient for Y(t) (in first-order logic), usefully and without unintuitiveness, doesn't imply that Y(t) must be false, since it means $\exists t \,\big(X(t) \land \lnot Y(t)\big),\tag*{}$ since X(t) is sufficient for Y(t) means $\forall t \,\big(X(t) \to Y(t)\big).\tag*{}$ Related: The various types of ‘implies’.

ryang
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0

Your statement is, "It is not the case that if the monkey dances then I get A grade in test." Using symbols, it is $P:\sim(m\implies a)$.
The case of vacuous truth arises if the truth value of a conditional $p\implies q$ is $T$ whenever $p$ is $F$. The given statement $P$ being the negation of a conditional statement $m\implies a$, so the case of vacuous truth doesn't arise here.

Nitin Uniyal
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