An example of why this is incorrrect.
If a card is the ace of spades, it is black.
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”.
You’ve have some examples, but in case they are not clear enough:
If [you have AIDS] then [you are unwell]
[You are unwell] if and only if [you have AIDS]
The first one is not the same as the second. Why? There are plenty of ways to be unwell, without necessary developing AIDS.
The first statement only defines one possible path to B, not all of them.
Not just HIV, but full blown AIDS?
Actually a good example:
- If you have AIDs (A) then you have HIV (B). True
- You have HIV (B) if, and only if, you have AIDS (A). Not true
- If you don’t have HIV (B), then you don’t have AIDs (A). True, and the actual inverse of “If A then B”; which is “If not B, then not A”
It’s important to stress the “full blown” modifier in any example.
If A, then B
If Not B, then Not A
If it’s raining then the grass is wet, but you can’t tell if it’s raining if the grass is wet, because of say, a hose or sprinkler.
All that you can tell is that if the grass is dry, then it is not raining, and I that’s called a contrapositive.
No.
No
Is “If B then A” equal to “B if and only if A”?
No. They are effectively the same statement.
(A <=> B ) = (A=>B AND B=> A)
Wait. If they are effectively the same statement, wouldn’t that mean they ARE equal?
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
“If X is cat, then X is mammal” =?> “X is mammal if and only if X is cat”
Obviously doesn’t hold: What if X doge?
Great analogy!
if I brake, my Car will stop.
will my Car only stop if i brake?
Brake*
shhhhhhh. Nobody saw, nobody knows.
Also It’s 00:30 over here, cut me some Slack.
A => B is not the same as B <=> A
No.
B iff A is defined as “If B then A and if A then B”.
If that doesn’t make it clear enough for you, then try writing out the truth table for both statements.
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 A then B” means if A is true, then B is guaranteed to be true. Note that if B is true and A is false, “if A then B” is still true.
“B only if A” means the only way for B to be true is for A to be true. It’s weird, but it has the inverse truth table as “(not A) and B”.
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.
It’s Cunningham’s Fallacy.
Nope. The first statement doesn’t exclude any paths to B
NP = PThe 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.
No. It is equal to “if not B, then not A.” You’re welcome for doing your logic 101 homework for you.
First thing I thought lmao. Somebody is taking logic
