# Is there a difference between 'only if' and 'if and only if'

So am reading a book titled 'an introduction to logic', and the topic at hand is Sentential logic > Biconditionals. At one point, the author gives examples of 3 sentences that draw upon three similar though different logical arguments:

''24. The figure on the board is a triangle only if it has exactly three sides.

25. The figure on the board is a triangle if it has exactly three sides.

26. The figure on the board is a triangle if and only if it has exactly three sides.

Let T mean ‘The figure is a triangle’ and S mean ‘The figure has three sides. ''

Source: 'An introduction to Logic' by P.D Magnun - Page 27 https://textbookequity.org/Textbooks/Magnus_forallx.pdf

So basically, from what I read, 24 is the same as T -> S, 25. Is the the same as S -> T, and so basically, if 24 is 'if T then S', and 25 is 'if S then T', then where 26 is saying 'if and only if, then it's a combination of statements 24 and 25, so T implies S, S implies T, thus S and T are synonymous.

Here's the thing though, I would of thought that since the statement 'only if A, then B' infers that if A occurs, B follows, and if B occurs, then A had to have occurred prior B. So, personally, I think 24 and 26 are the same, only that 26 is a stronger statement than 24 (not logically, just persuasively, but of course I could be wrong)

So if i'm playing a game where I win 'only if I score 100 points',then if I score 100, I win, if I win, I scored 100 points.

P.S, I don't know whether this question is suited for philosophy, I've read that logic is tied into philosophy, though am also aware that this type of reasoning is also employed in mathematical proofs. If you think it's more suited for the latter, please don't downvote, but merely comment asking me to move this question.

• "I win only if I score 100 points" doesn't mean there cannot be additional conditions that must be met for winning. In table tennis, you only win a set if you get 21 points. But you can get 21 points and lose (21 to 23). – gnasher729 Feb 24 '16 at 15:51
• math.stackexchange.com/questions/68293/… The question is answered here. – Matt Feb 24 '16 at 15:55

Yes, sure.

"P only if Q" is translated in symbiols with the conditional: P → Q, while "P if and only if Q" needs the bi-conditional: P ↔ Q.

See :

All three statements are true, but they don't have the same meaning. I'll try to make the difference more obvious by changing the left hand.

1. The figure on the board is a triangle or square only if it has exactly three sides.

2. The figure on the board is a triangle or square if it has exactly three sides.

3. The figure on the board is a triangle or square if and only if it has exactly three sides.

Number 24 is wrong. A figure with four sides may also be a triangle or square (to be more precise, it may be a square), but isn't necessarily so. The "only if" is wrong.

Number 25 is correct. If a figure has three sides then it is a triangle or square. Actually it is a triangle, but that doesn't make the statement wrong, because every triangle is a triangle or square.

Number 26 is wrong. Squares don't have three sides. Correct would be "a figure is a triangle or square if it has either three sides, or four sides of equal length at right angle towards each other".