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 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.
“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.
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).
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!)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.
“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.
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).