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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"?

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
  • 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

1

$m$ = the monkey dances

$a$ = I get an A on the test

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. The proposition claims that the monkey dancing isn't sufficient, but just because both happen, it doesn't contradict the proposition. If the monkey dances, and I get an A on the test, wouldn't that make the proposition vacuously true? 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).

  1. 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.

  2. 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:

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),

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”.

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?

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’.

ryang
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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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