# If and only if, which direction is which?

I can never figure out (because the English language is imprecise) which part of "if and only if" means which implication.

($A$ if and only if $B$) = $(A \iff B)$, but is the following correct:

($A$ only if $B$) = $(A \implies B)$

($A$ if $B$) = $(A \impliedby B)$

The trouble is, one never comes into contact with "$A$ if $B$" or "$A$ only if $B$" using those constructions in everyday common speech.

• I'm not sure if this is a genuine question or not, but in case it is: A (must hold) if B (does). A (can hold) only if B (does). This makes it pretty unambiguous. – Matt E Aug 1 '10 at 4:36
• The number of people who got this wrong when I marked introductory logic exams was unbelievable... – Seamus Sep 8 '10 at 12:21
• Unambiguous, I think. But surely, confusing! – Srivatsan Sep 3 '11 at 0:49
• @MattE What leads you to not be sure if this a genuine question? – Al Jebr Sep 26 '18 at 16:13

I wouldn't say that I never come into contact with those phrasings--they are certainly rare in technical use, but perhaps more common in plain language. Below is a table of equivalent phrasings of p=>q, from UCSMP Precalculus and Discrete Mathematics, 3rd ed., © 2010 Wright Group/McGraw Hill (Lesson 1-5).

table http://www.imgftw.net/img/171340741.png

• Certainly the last box of the right column is not true! I intend to accept this answer when it becomes possible to do so, however. – user126 Jul 29 '10 at 11:01
• The last box of the right column is logically equivalent to the rest of the statements in the column--the relationship between necessary/sufficient conditions and if/only-if can be difficult to get a handle on. – Isaac Jul 29 '10 at 11:07
• Ah, I was replacing "condition" with "motivation" in the two bottom right boxes. – user126 Jul 29 '10 at 11:18
• Some of those statements are not a way that anyone would speak or would expected to understand. To make it (a bit) more natural, I would have written the following. For "p only if q", I would use: "You have applied early only if you have increased your chance of receiving aid". For q is a necessary condition for p, I would write: "Having increased your chance of receiving aid is a necessary condition for having applied early" or "is a necessary consequence of having applied early". – Casebash Jul 29 '10 at 13:54
• @Casebash: While it's less of an issue in the table, in the context of the lesson as a whole, the change in tense would have been problematic. (Also, given that this is in Chapter 1 of a high school book, applying early and financial aid are both in the future for the student.) – Isaac Jul 29 '10 at 14:04

This example may be more clear, because apples ⊂ fruits is more obvious:

"This is an apple if it is a fruit" is false.
"This is an apple only if it is a fruit" is true.
"This is a fruit if it is an apple" is true.
"This is a fruit only if it is an apple" is false.

A is an apple => A is a fruit

• '"This is an apple only if it is a fruit" is true.' well being a fruit isn't the only thing that makes an apple an apple. – user10389 Aug 9 '12 at 22:00
• @user10389 You're confusing "only if" with "if and only if". The former implies the possibility of further neccessary requirements, while the latter is considered an equivalence. – Arthur Aug 11 '12 at 1:00

It's easier to work out if you have a specific example:

Let A:I am a parent B:I have a child

I am a parent if and only if I have a child has two parts:

I am a parent if I have a child can be rephrased: If I have a child, then I am a parent. B => A

I am a parent only if I have a child can be understood to mean: if I do not have a child, then I am not a parent: ~B -> ~A But this is logically equivalent to if I am a parent, then I have a child: A=> B

So the "if and only if" locution implicitly involves some grammatical transformations. The meaning may not be immediately obvious, but it can be worked out.

The explanation in this link clearly and briefly differentiates the meanings and the inference direction of "if" and "only if". In summary, $$A \text{ if and only if } B$$ is mathematically interpreted as follows:

• '$$A \text{ if } B$$' : '$$A \Leftarrow B$$'
• '$$A \text{ only if } B$$' : '$$\neg A \Leftarrow \neg B$$' which is the contrapositive (hence, logical equivalent) of $$A \Rightarrow B$$