Difference between only if and if and only if

[ainster31]

Difference between "only if" and "if and only if"

$$1.\quad p\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$2.\quad p\quad only\quad if\quad q\\ \equiv if\quad p\quad then\quad q\\ \equiv p\rightarrow q\\ \\$$$$3.\quad p\quad only\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$4.\quad p\quad iff\quad q\\ \equiv (p\rightarrow q)\wedge (q\rightarrow p)$$

I think #3 is wrong but I'm not sure why.

 

[1MileCrash]

Yes, #3 is wrong. The second line says if q, then p, but we know that p may only occur when q does from the first line.

If q, then p allows p to occur without q.

 

[jedishrfu]

Here's a reference that may help you understand it better:

http://en.wikipedia.org/wiki/Only_if

 

[ssd]

p only if q
⇔ q if p.
So the first line in (3) does not imply the second line.

 

[FactChecker]

I watch the "Teletubbies" show only if I have a TV. I have a TV, but I would rather tear out my eyes than watch Teletubbies. So p only if q is not the same as q implies p.