🚀 go-pugleaf

I am surprised at the amount of comment this has caused - but (at the risk
of making a blooper myself - it's always possible and frequently probable)
here is my attempt to kill the thread:

There are at least three different phrases one must distinguish:
  A if B
  A only if B
  A if and only if B

Further, keep in mind that 
  A --> B  =  -B --> -A
 (and that B <-- A is often used as an alternate notation for A --> B
  - it is not as rare as one person suggested, at least in the crowds I
  frequent, and in certain non-commutative logics it is necessary to have
  two such implications - where of course <-- and --> would be different,
  and only equivalent if the model is commutative.
  It has the convenience that B <-- A then reads "B if A" - though that is
  one of the points of this note I guess, the other being that A --> B is
  read "A only if B" or "A implies B".)

Clearly "A if B" is just B --> A.  The thing to remember is that "A only if B"
is the converse (remember all the proofs in your undergraduate maths classes
of theorems of the form "A iff B" where half way through the prof. said "Now,
conversely ..."?) - So "A only if B" is A --> B

So we get the table:

  A if B             =  B --> A  =   -A --> -B
  A only if B        =  A --> B  =   -B --> -A
  A if and only if B =  B --> A and A --> B  =  -A --> -B and A --> B

(Use the backwards arrow if you like to get more equivalences...)

So when Mom says "You must have a cookie only if you wash your hands", she
means that she wants the universe so ordered that if she sees you eating a
cookie, she must be able to conclude that you have washed your hands.  
(ie if you don't wash your hands, don't go for the cookie tin...)  But if
you aren't eating a cookie, she won't know if you've washed or not...
(She - as a good logician - will be sceptical, and will probably check!)  
On the other hand, if she says "you must have a cookie if you wash your 
hands" you only have to avoid eating a cookie in order to keep your
hands dirty.  (But just because your hands are dirty, she cannot 
conclude you're not eating the cookie.  You might be, which is
why most Moms really mean to put the "only" in the sentence, and if they
forget, you can raid the cookie tin in good conscience!)

If you wish to point out that "common English" often confuses these three
meanings, I can only agree - but that surely wasn't the point of the
original posting. 

Hope this is clear.  As someone else said, if you read "A only if B" as 
"A {\huge ONLY} {\tiny if} B" it might seem more reasonable...

-rags