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If slavery is not wrong, nothing is wrong. -- Abraham Lincoln I've been trying to get an intuition (make myself feel good) about the material conditional log

If slavery is not wrong, nothing is wrong. -- Abraham Lincoln I've been trying to get an intuition (make myself feel good) about the material conditional logical connective and this seems to satisfy the case for a false antecedent and false consequent (¬p→¬q). The equally true statement seems to check out: If slavery is not wrong, something is wrong. (¬p→q) However this also evaluates to True which annoys me: If slavery is wrong, something is wrong. (p→q) And this evaluates to False: If slavery is wrong, nothing is wrong. (p→¬q) I get that the formal definition of → is different from English's if/then but that just makes understanding proofs so much harder. Perhaps someone could better paraphrase Lincoln in first-order logic?

mikehall314 9y ago
If at least one thing is wrong, slavery is wrong?
matt4077 9y ago
I'm not sure if you're trying to make a joke. In any case, quotes such as this are ill-suited for learning formal logic, because they rely on a lot more than their literal meaning. In this case, you cannot see the statement as being about boolean true/false values. It only works on an ordinal scale, and our calibration of the cutoff for "wrongness". Slavery here is assumed axiomatically to be "really really bad". In fact, among all the bad things, it is the worst. But is it wrong? That depends on how bad something must be to be wrong. But if you do not consider slavery to be wrong, but merely bad, nothing can be wrong, because everything is less bad than slavery. So this is basically Lincoln trying to argue for AUC as a better metric than sensitivity/specificity.
luhn 9y ago
Applying logic to pithy quotes is generally going to be painful, because there's tends to be a lot of implicit meaning. But let's have some fun and break this down. (Bear with me, it's been a while since I've done rigorous proofs.) Let's say there's a function R that outputs a numerical value of an actions morality. Let's define an action as wrong if R(action) is below some threshold M. So what Lincoln is saying is: R(slavery) >= M -> R(action) >= M for all actions If we assume Lincoln's statement to be true, that must mean that R(slavery) <= R(action) for all actions. Which can be reduced to R(slavery) = min(R(actions)) Translated to English, Lincoln is saying that slavery is most morally despicable act.

mikehall314 9y ago

If at least one thing is wrong, slavery is wrong?

matt4077 9y ago

I'm not sure if you're trying to make a joke. In any case, quotes such as this are ill-suited for learning formal logic, because they rely on a lot more than their literal meaning. In this case, you cannot see the statement as being about boolean true/false values. It only works on an ordinal scale, and our calibration of the cutoff for "wrongness". Slavery here is assumed axiomatically to be "really really bad". In fact, among all the bad things, it is the worst. But is it wrong? That depends on how bad something must be to be wrong. But if you do not consider slavery to be wrong, but merely bad, nothing can be wrong, because everything is less bad than slavery. So this is basically Lincoln trying to argue for AUC as a better metric than sensitivity/specificity.

luhn 9y ago

Applying logic to pithy quotes is generally going to be painful, because there's tends to be a lot of implicit meaning. But let's have some fun and break this down. (Bear with me, it's been a while since I've done rigorous proofs.) Let's say there's a function R that outputs a numerical value of an actions morality. Let's define an action as wrong if R(action) is below some threshold M. So what Lincoln is saying is: R(slavery) >= M -> R(action) >= M for all actions If we assume Lincoln's statement to be true, that must mean that R(slavery) <= R(action) for all actions. Which can be reduced to R(slavery) = min(R(actions)) Translated to English, Lincoln is saying that slavery is most morally despicable act.