How is "p implies q" same as "q unless not p"? I want to know how is "p implies q" same as "q unless not p"?
ie how is "$p\Rightarrow q$" same as "$q$ unless $\neg p$" ?
 A: It doesn't! 
"p implies q", $p\to q$ is equivalent to "q or not p", $q\vee \neg p$.   This is an inclusive or; it does not exclude the possibility that $q$ and $\neg p$ may be both true.
That is not the same thing as "q unless not p", which would be an exclusive or.
A: First, "unless" is not formal mathematical language. It is loose terminology.
Read it longer as "$q$ is true unless $p$ is false." This is read as "either $q$ is true or $p$ is not true."
If this is true, and $p$ is true, it in necessarily the case that $q$ is true.
A: Sorry, but I disagree with the last example (given by Aditya Raj).  I would interpret "She finds a good job unless she does not study math" as saying that the only circumstance that would cause her not to find a good job is failing to study math.  But this is not at all implied by the original conditional statement which does guarantee that studying math will lead to a good job, but allows for other possibilities (does not make it a requirement).
A: Suppose: "If $p$, then $q$."
There are two cases: $p$ can be true, or it can be false. If $p$ is true, then so is $q$. So it follows that $q$ holds unless $p$ is false. Ergo, $q$ unless not $p$.          
Now suppose: "$q$ unless $\neg p$."
We're trying to prove "If $p$, then $q$." So assume $p$. Since its not the case that $\neg p$ holds, hence from "$q$ unless $\neg p$" we deduce that $q$ holds. Ergo, if $p$, then $q$.
A: I teach this class out of Rosen and disagree with this statement as well.  I believe it should be not p unless q.  If you want further evidence, take a look at the very first problem in the following section of 1.2 in which Rosen suddenly conforms to this himself. In this problem, e: "You can edit a protected wikipedia entry" and a: "You are an administrator."  The provided answer to the translation of "You cannot edit a protected wikipedia entry unless you are an administrator" is e implies a.
For another example not from the book, let p: "It rains" and q: "It is wet."  Let us assume p implies q is a truthful conditional statement.  It would make sense to say "It did not rain unless it is wet" (not p unless q).  However, it would NOT make sense to say "It is wet unless it did not rain" (q unless not p)...that is nonsense as the original conditional statement would allow for other ways for it to be wet!
Maybe some will disagree with this, but it is what I go with.
A: Let's take an example. You can find the same in Rosen's book
Conditional statement: if she studies math (p), then she will find a good job (q)
This is of the type: If p, then q.
q unless p:
She finds a good job unless she does not study math.
Here q is true only is ~p (not p) is false. If ~p is true then q will become false.
Hope this helped!
