(6:05) “If you are vaccinated and worried about this virus, go get your booster.”
Logically this implies that if you have gotten your booster, you should not be worried about this virus.
Interesting.
It does not. “go get your booster” is clearly short for “you should get your booster assuming you don’t already have one”. The negation of that isn’t “you already have a booster”, it’s “you should not get your booster even if you don’t already have one”. So the contrapositive of this statement is “if you shouldn’t get your booster even though you don’t already have one, then you shouldn’t be worried about the virus”.
(You can also see that something went wrong with the implication you’ve drawn, seeing as you’ve derived an unreasonable statement from a reasonable one. The contrapositve is logically equivalent, it can’t be less reasonable.)
Saying “if you are worried, you should get your booster” is not at all equivalent to “if you have gotten your booster, you should not be worried about this virus”.
The first statement permits that one can still be worried after having gotten one’s booster (just a little less), which the second statement does not permit. Therefore, those two statements cannot be logically equivalent.
If you had written “if you have gotten your booster, you should not be as worried about this virus [as before]”, that would have been fine.
Flag: even being less worried is an additional inference not demanded by the form of the contrapositive of “if you’re worried, you should get your booster” . But that’s just a nitpick over how literally we’re meaning logical equivalence.
Presumably, a more complete statement would be “If you are vaccinated and worried about this virus, go get your booster. If (or once) you have your booster, if you’re still worried, increase ventilation and limit social contact, particularly in poorly ventilated spaces. Also, lose some weight, fatty.”
It does not. “go get your booster” is clearly short for “you should get your booster assuming you don’t already have one”. The negation of that isn’t “you already have a booster”, it’s “you should not get your booster even if you don’t already have one”. So the contrapositive of this statement is “if you shouldn’t get your booster even though you don’t already have one, then you shouldn’t be worried about the virus”.
(You can also see that something went wrong with the implication you’ve drawn, seeing as you’ve derived an unreasonable statement from a reasonable one. The contrapositve is logically equivalent, it can’t be less reasonable.)
I disagree that the derived statement is unreasonable. I also disagree that it’s obviously saying something other than what it literally says.
Saying “if you are worried, you should get your booster” is not at all equivalent to “if you have gotten your booster, you should not be worried about this virus”.
The first statement permits that one can still be worried after having gotten one’s booster (just a little less), which the second statement does not permit. Therefore, those two statements cannot be logically equivalent.
If you had written “if you have gotten your booster, you should not be as worried about this virus [as before]”, that would have been fine.
Flag: even being less worried is an additional inference not demanded by the form of the contrapositive of “if you’re worried, you should get your booster” . But that’s just a nitpick over how literally we’re meaning logical equivalence.
Presumably, a more complete statement would be “If you are vaccinated and worried about this virus, go get your booster. If (or once) you have your booster, if you’re still worried, increase ventilation and limit social contact, particularly in poorly ventilated spaces. Also, lose some weight, fatty.”