“A implies B” means if A is true then B must be true; if A is false, then B can be anything. In other words, the only state not allowed is A being true and B being false. Therefore, the only “hole” is the part of A that doesn’t include B.
Yup, that’s my interpretation too. It just doesn’t sit well with all the other operators.
All the others are phrased as direct questions about the values of A and B:
A AND B = “Are A and B both true?”
A OR B = “Are either A or B true, or both?”
A NAND B = “Is (A AND B) not true?”
A IMPLIES B = “Is it possible, hypothetically speaking, for it to be the case that A implies B, given the current actual values of A and B?”
You see the issue?
Edit: looking online, some people see it as: “If A is true, take the value of B.” A implies that you should take the value of B. But if A is false, you shouldn’t take the value of B, instead you should use the default value which is inexplicably defined to be true for this operation.
This is slightly more satisfying but I still don’t like it. The implication (ha) that true is the default value for a boolean doesn’t sit right with me. I don’t even feel comfortable with a boolean having a default value, let alone it being true instead of false which would be more natural.
Consider the implication to be some claim, for example, “When it’s raining (A), it’s wet (B)”. The value of the implication tells us whether we should call the claimant a liar or. So in case it’s raining (A = true) and is is not wet (B = false) the claim turns out to be false, so the value of the implication is false.
Now, supposing it is not raining (A = false). It doesn’t matter whether it’s wet or not, we can’t call the claim false because there just isn’t enough information.
It’s about falsifiability (or lack thereof, in case A is never true).
I never got why “implies” is called that. How does the phrase “A implies B” relate to the output’s truth table?
I have my own “head canon” to remember it but I’ll share it later, want to hear someone else’s first.
“A → B” is true in any variable assignment where B is true if A is true.
It has always been mostly obvious to me.
“A implies B” means if A is true then B must be true; if A is false, then B can be anything. In other words, the only state not allowed is A being true and B being false. Therefore, the only “hole” is the part of A that doesn’t include B.
I think ‘implies’ asks whether it’s possible that A causes B to be true. In other words, it is false if there is evidence that A does not cause B.
So:
If A is true and B is false, then the result is false, since A could not cause B to be true.
If A and B are both true, then the result is true, since A could cause B.
If A is false and B is true, then the result is true since A could or could not make B true (but another factor could also be making B true)
If A and B are both false we don’t have any evidence about the relationship between A and B, so the result is true.
I don’t know for sure, though. I’m not a mathematician.
Yup, that’s my interpretation too. It just doesn’t sit well with all the other operators.
All the others are phrased as direct questions about the values of A and B:
You see the issue?
Edit: looking online, some people see it as: “If A is true, take the value of B.” A implies that you should take the value of B. But if A is false, you shouldn’t take the value of B, instead you should use the default value which is inexplicably defined to be
truefor this operation.This is slightly more satisfying but I still don’t like it. The implication (ha) that
trueis the default value for a boolean doesn’t sit right with me. I don’t even feel comfortable with a boolean having a default value, let alone it beingtrueinstead offalsewhich would be more natural.Edit 2: fixed a brain fart for A NAND B
Consider the implication to be some claim, for example, “When it’s raining (A), it’s wet (B)”. The value of the implication tells us whether we should call the claimant a liar or. So in case it’s raining (A = true) and is is not wet (B = false) the claim turns out to be false, so the value of the implication is false.
Now, supposing it is not raining (A = false). It doesn’t matter whether it’s wet or not, we can’t call the claim false because there just isn’t enough information.
It’s about falsifiability (or lack thereof, in case A is never true).