A card is black if and only if it is the ace of spades.
There are other conditions under which B (a card is black) can happen, so the second statement is not true.
A conclusion that would be correct is “If a card is not black, it is not the ace of spades.”. The condition is that if A is true B will also always be true, so if B is false we can be sure that A is false as well - i.e. “If not B, not A”.
I think I understand now, but what has left me scratching my nose (metaphorically):
Why is it called “B if and only if A”, if what it really means is “B only if A and vice versa”? (Am I correct in thinking that’s what it means?)
I just don’t understand how that translates grammatically. To me, “B if and only if A” sounds the same as “B only if A”. I can accept that they mean different things in the context of logic, just like I can assign any meaning to any label, like I could say that “dog” now means “kite” in a certain context. But it seems unintuitive and doesn’t really make sense to me. Does that make sense?
“B if and only if A” is a shorthand way to write “B if A and B only if A”. It’s like how “He is young and thin” means “He is young and he is thin”. We could write it the second way without trouble, but the first way is shorter, we agree that they mean the same thing, and we prefer to conserve energy when writing.
The form “if and only if” is merely a convenient shorthand. Shorthand is usually more convenient for the writer than for the reader. 🤷♂️
Imagine these natural language sentences and analyze how they are different:
I’ll go outside if it’s not snowing
I’ll go outside only if it’s not snowing
Hint: what do you do in each case when it’s snowing?
if youre doing homework, i recommend writing out truth tables for the statements and comparing, gives you a bit more insight into the statement truth conditions
The first statement only tells you when B is true. It says nothing about when it is false. The second statement both tells you when B is true (if A) and when it is not (only if A). Therefore, the two statements cannot be equal.
I just saw a video on all the logical fallacies that exist, and this was one of them but my shit-ass memory can’t recall what the name of the fallacy was.
Edit: I see the error in my below response. I leave wrong answers for conversational completeness
That’s not equivalent either. “if not b, then not a” works if it’s a sequence but doesn’t work for options in which multiple inputs can lead to the same output. If you get pizza every Tuesday and Friday, then answering “what’s for lunch” with “if Tuesday, then pizza” and “if Friday, then pizza” doesn’t let it work in reverse. “what day is it” can’t be answered with “if pizza lunch, then Tuesday”
You’d have to firm up your definition of car and vehicle before you could decide that one. Does a hot wheels car count as a car? Does a vehicle have to be large enough to move people or freight?
I just figured with Lemmy’s interest in politics it seemed like an obvious example. I threw in the car because I didn’t want to be that guy who makes everything about nazis…
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