Does a if and only if b imply b if and only if a?

Question

I was told that a if and only if b implies b if and only if a. I am not sure I believe this because I can think of many examples where this seems to be false.

The animal is a human if and only if it is a mammal (True). The animal is a mammal if and only if it is a human (False). Therefore a iff b does not imply b iff a.

"The animal is a mammal if and only if it is a human" is clearly false because there are many mammals that are not humans.

"The animal is a human if and only if it is a mammal". Every animal must belong to a taxonomic class. An animal can not be without a class. Every human is a mammal. Therefore the animal is a human if and only if it is a mammal. It does not suffice to say that the animal is a human because it has 10 fingers. We can not ignore or avoid assigning taxonomic class.

Answer 1

"If and only if" is the biconditional connection; a statement of material equivalence.

$A\leftrightarrow B$ is equivalent to $\underbrace{(A\leftarrow B)}_\text{if}\underbrace{\wedge}_\text{and} \underbrace{(A\to B)}_\text{only if}$, and you can see the symmetry there in.

That is that "$A$ if and only if $B$" means "$A$ if $B$, and $A$ only if $B$".

So your proposed counterexample of "An animal is human if and only if it is a mammal" means "An animal is human if it is a mammal, and an animal is human only if it is a mammal."   Which is false, and thus actually equivalent to "An animal is a mammal if and only if it is human," which is also false.

Answer 2

Your statement, "The animal is a human if and only if it is a mammal" is equivalent to both of the following:

1. If the animal is a human, then it is a mammal. (aka "is human only if is mammal")
2. If the animal is a mammal, then it is a human. (aka "is human if is mammal")

As you have pointed out, while 1 is true, 2 is not. Hence, the biconditional is false (at least in general). And this is the case regardless of the order you put them in - the statement "The animal is a mammal if and only if it is a human" is logically equivalent, and thus equally false.

Some statements that can be written as true biconditionals might include:

• I get paid if and only if I go to work. (If I go to work, I get paid. If I don't go to work, I don't get paid.)
• You get this promotional toy if and only if you spend more than \$30 in the store. (You can only get the toy by spending over \$30, and you always get the toy when you spend over \$30.)
• The animal is a duck if and only if it walks like a duck, quacks like a duck, and swims like a duck. (If any of those three conditions is not true, the animal is not a duck, and if the animal is a duck, then all three conditions is true.)

Answer 3

You are failing to appreciate the distinction between "if" statements and "if and only if" statements.

The "counter examples" you are using are if statements true in only one direction. If they were true both ways, they would be "if and only if" statements.

Answer 4

$a\iff b$ is equivalent to saying $b\implies a$ and $a\implies b$. $b\iff a$ is equivalent to saying $b\implies a$ and $a\implies b$. So, clearly, as statements they are the same.

Be cautious about confusing if statements with if and only if statements.

Answer 5

a if b... means if b then a. Or whenever b then a. If you have b you can't not have a. The only way you can have b is if you also have a. So b only if a

So "a if b" = "if b then a " = " b $\implies $ a" = "a $\impliedby$ b" = "b only if a".

a only if b ... means a can only happen if b. If you have a then you must have b be a use if you didn't have be you wouldn't be able to have a. In other words: "a $\implies $ b" and "b $\impliedby $ a".

So "a if and only if b" means "a $\impliedby $ b" and "a $\implies $ b". We write this as $a \iff b $ meaning both $a \implies b $ and $b \implies a $.

So "a if and only if b" means either both a and b are true or neither a nor b is true.

obviously the order of a and b can be reversed as both must be true or both must be false.