Originally Posted by thedog123123

banks will replace the bill IF and only if the serial number is intact.

The biconditional doesn't work here because it would imply that every single bill with an intact serial number would be replaced. You only want the "only if" part.

i never understood IFF i always just thought it meant the same as "only if this was true can this occur" or w.e... im intrigued as to what the diff is.. care to explain with examples haha

Originally Posted by krup92

i never understood IFF i always just thought it meant the same as "only if this was true can this occur" or w.e... im intrigued as to what the diff is.. care to explain with examples haha

IFF means it's the only condition in which the subject is true or false.

"banks will replace the bill IF and only if the serial number is intact."

In this example, the condition is whether or not the serial number is intact. The bill will only be replaced in the single circumstance that the serial number is intact, and there are no exceptions, such as an alternate identification method.

Once my dad found a 10 bill in a soccer field, ripped in half. He went home, taped it back together and took us out for ice cream.It was awesome.

Hopefully the bank will help you out - the serial number is probably to make sure its real.

Well, in English, we take 'Q if P' to mean that Q follows from P. For example, 'if a number is positive, then it is greater than zero' to mean 'the fact that a number is greater than zero follows from its positivity'. We then call P a sufficient condition for Q, and Q a necessary condition for P, since if Q was false, then P couldn't possibly be true (example: 'if you buy three pizzas, the fourth is free'. If the fourth pizza isn't free, then you haven't bought three pizzas, so Q is necessary for P).

Now, when we say 'Q only if P' we mean 'if Q then P'. For example, 'free Coke only if you buy a pizza' means that if you got a free Coke, you bought a pizza - in other words, P follows from Q.

IFF is the combination of these two - namely, 'P iff Q' means that P follows from Q and that Q follows from P. Or that Q is a necessary and sufficient condition for P (and vice-versa).

Of course, I'm not suggesting that English works like this. In the construction of this logical language (called 'first-order logic'), people took particular examples from the natural languages they spoke (English, German, French, etc,) and used those ideas to represent abstract properties of deduction or rules of inference that they thought were our "laws of thought". The question of whether such systems ('calculi') are actually representative of our own deductive processes is one addressed in philosophy of language, as well as logic - it also forms a fundamental part of artificial intelligence research.

Originally Posted by justinsftw

The bill will only be replaced in the single circumstance that the serial number is intact, and there are no exceptions, such as an alternate identification method.

Yes...this is Only If. "The bill will be replaced only if the number is intact." Logically, there is no such thing as an exception, so that's a moot point.

The "If part" of the statement says "If the serial number is intact, then the bill will be replaced" but this is false because I have a wallet full of bills whose serial numbers are intact, which aren't being replaced.

In other words, the serial number being intact is a sufficient but not necessary condition. (The IFF statement is logically equivalent to it being necessary and sufficient.)


Now note that we can 'correct' this by making the statement "Serial number is intact" + "Want a bill replacement" <=> "Bill replaced." While the extra condition of wanting a bill replaced is implied, it wasn't explicitly stated (so 'logically' the statement is incorrect).

In other words: