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- From: rags@triples.math.mcgill.ca (Robert A. G. Seely)
- Subject: "A only if B" vs "A if and only if B"
- Message-ID: <1992Oct13.224541.3542@thunder.mcrcim.mcgill.edu>
- Sender: news@thunder.mcrcim.mcgill.edu
- Nntp-Posting-Host: triples.math.mcgill.ca
- Organization: Dept Of Mathematics and Statistics
- Date: Tue, 13 Oct 92 22:45:41 GMT
- Lines: 56
-
-
- 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
-