That would be the symbol\operation called TRUE or TOP or “tautology” which is always true. They’re actually missing quite a few of the weirder ops, including implication and biconditional\iff\if-and-only-if. (Edit: Actually I think XNOR is also the biconditional. I guess pretend like I said “material implication” and “reverse implication”. Fricken booleans man!)
I will save this picture because I think it’s important and will look at it while scrolling my phone a couple of years from now and wonder why I never used it all that time.
I truly have no idea and wish I did, haha. It looks like a shorthand for which operation is being followed, maybe like a group theory thing, but I really don’t know.
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).
That’s interesting. I’ll have to read up on that. You’re right, I am thinking about boolean algebra.
In the mean time though, I’ll note that Boolean algebra on Wikipedia also refers to this operation, so I’m not alone:
Material conditional
The first operation, x → y, or Cxy, is called material implication. If x is true, then the result of expression x → y is taken to be that of y (e.g. if x is true and y is false, then x → y is also false). But if x is false, then the value of y can be ignored; however, the operation must return some Boolean value and there are only two choices. So by definition, x → y is true when x is false (relevance logic rejects this definition, by viewing an implication with a false premise as something other than either true or false).
It also uses the second interpretation that I mentioned in my earlier comment (4 above this one), with true being default, rather than the one we’ve been discussing.
Yeah, that one’s always bothered me too. I think the difference in meaning from colloquial “implication” and logical “material implication” are also involved in the raven paradox.
So the statement that “all ravens are black” can be taken as “if RAVEN then BLACK”. Is this statement true? If you see a black raven then trivially yes, if you see a white raven then trivially false (via counter example).
However if you see a non-raven, it is evidence for the truth of the statement because it doesn’t go against it: not-ravens being black-or-not-black both reduce the universe of possible objects without proving not-black-ravens exist.
Or something like that, I think it’s stupid too. Trinary logic can adopt a more sensible (IMHO) definition of implication that makes A being false always lead to the third value (usually defined as indeterminate or neither-true-nor-false).
“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.
That would be the symbol\operation called TRUE or TOP or “tautology” which is always true. They’re actually missing quite a few of the weirder ops, including implication and
biconditional\iff\if-and-only-if. (Edit: Actually I think XNOR is also the biconditional. I guess pretend like I said “material implication” and “reverse implication”. Fricken booleans man!)I will save this picture because I think it’s important and will look at it while scrolling my phone a couple of years from now and wonder why I never used it all that time.
/c/justadhdthings
Samesies
This cheat sheet needs a cheat sheet. What do the numbers with 3 numbers mean?
I truly have no idea and wish I did, haha. It looks like a shorthand for which operation is being followed, maybe like a group theory thing, but I really don’t know.
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.
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).
The comment you replied to is my response to this. It’s the only boolean operation that works this way. All the others are straightforward.
I think the problem is that you’re thinking in terms of boolean algebra, while implication being implication comes from propositional logic.
That’s interesting. I’ll have to read up on that. You’re right, I am thinking about boolean algebra.
In the mean time though, I’ll note that Boolean algebra on Wikipedia also refers to this operation, so I’m not alone:
It also uses the second interpretation that I mentioned in my earlier comment (4 above this one), with
truebeing default, rather than the one we’ve been discussing.Yeah, that one’s always bothered me too. I think the difference in meaning from colloquial “implication” and logical “material implication” are also involved in the raven paradox.
So the statement that “all ravens are black” can be taken as “if RAVEN then BLACK”. Is this statement true? If you see a black raven then trivially yes, if you see a white raven then trivially false (via counter example).
However if you see a non-raven, it is evidence for the truth of the statement because it doesn’t go against it: not-ravens being black-or-not-black both reduce the universe of possible objects without proving not-black-ravens exist.
Or something like that, I think it’s stupid too. Trinary logic can adopt a more sensible (IMHO) definition of implication that makes A being false always lead to the third value (usually defined as indeterminate or neither-true-nor-false).
“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.
“A → B” is true in any variable assignment where B is true if A is true.
It has always been mostly obvious to me.
Yeah, that kinda works but I don’t like it. See my reply to the other comment.