# 'If…then…' and '…if…' and '…only if…' and 'If… only then…' statements?

Suppose you have two statements A and B and "If A then B". I am trying to think of what this implies and alternative ways of writing this.

I think

"If A then B"

= A$\rightarrow$B

= "A is sufficient but not necessary for B. B is neither necessary nor sufficient for A"

= "If not B then not A"

= B'$\rightarrow$ A'

= "B' is sufficient for A' but not necessary"

And it seems to me that 'B if A' is equivalent to 'If A then B' (please correct me if I am wrong!)

When it comes to only if, I think "B only if A" is equivalent to "If A only then B"

I think

"B only if A"

= B$\rightarrow$A

= "B is sufficient for A but not necessary for A. A is necessary for B but not sufficient for B"

="If not A then not B"

=A'$\rightarrow$B'

= "Not A is sufficient for not B and not A is necessary for not B"

I see that I must have made mistakes somewhere because several things are not consistsent. Firstly, I am pretty sure my last statement is wrong but this is how I interpret "If not A then not B". Also, I don't understand how if the '...if...' cases, single headed arrows only implied sufficiency, and here for the '...only if...'single headed arrows seem to be implying something about necessity as well...

Thank you in advance for any clarifications, and also if anyone has a link to a good explanation of these statements I would be very grateful. I am trying to understand them in the context if proofs and proving statements the right way around...

EDIT: Thank you for all the answers and comments so far. Jut thought I would add something that helped me in case someone else comes across my question and also requires help with if/iff/necessary/sufficient etc. I found it easier to visualise the cases. 'B if A' can be represented as a circle A within a circle B. Automatically, if A then B, so A is sufficient but not necessary for B, but B is necessary for A and not B implies not A. However A does not imply B. In a similar way, 'B only if A' is a circle B within a circle A, because B implies A- it is sufficient, but not necessary for A, and A is necessary but not sufficient for B. 'B if and only if A' is the double headed arrow because A and B are the same ring. One is both necessary and sufficient for the other, and one implies the other....

• Yes : $A \to B$ is : "if A then B" and also "B if A" ans "A is a sufficient condition for B". $A \to B$ is also "B is a necessary condition for A" and "A only if B". – Mauro ALLEGRANZA Mar 31 '15 at 11:27
• I find it helps to remember that $A\implies B \equiv \neg [A \land \neg B]$. It has nothing with $A$ causing $B$, or $B$ causing $A$ as many beginners seem to think. There is no causality in mathematics. – Dan Christensen Mar 31 '15 at 15:52

$A\to B$ means "$A$ implies $B$", "$A$ is sufficient for $B$", "if $A$, then $B$", "$A$ only if $B$", and such.

$A\leftarrow B$ means "$A$ is implied by $B$", "$A$ is necessary for $B$", "$A$ whenever $B$", "$A$ if $B$", and such

$A\leftrightarrow B$ means "$A$ is necessary and sufficient for $B$", "$A$ if and only if $B$".

Note: $A\to B$ means "$A$ is sufficient for $B$ and may or may not be necessary for $B$".   It neither affirms nor denies the necessity.

We can also say,

$A\to B$ means "$B$ if $A$", "$B$ whenever $A$", and "$B$ is necessary for $A$",

$A\leftarrow B$ means "$B$ only if $A$", "if $B$, then $A$", and "$B$ is sufficient for $A$".

$A\wedge\neg B$ means "$A$ is not sufficient for $B$", and $\neg A\wedge B$ means "$A$ is not necessary for $B$"

• Brilliant answer! – Konstantin Feb 11 '17 at 17:04

Your thinking is a little off. If $A$ is sufficient for $B$, then indeed $B$ is necessary for $A$. This is because if $B$ is not true, then $A$ is not true.

Long comment

There are no "inconsistency" ...

$A \to B$ is : "if $A$, then $B$" and also "$B$, if $A$" and also "$A$ is a sufficient condition for $B$".

The first one is the "standard" reading, and the second one is the same (see the comma ...).

$A \to B$ is also "$A$ only if $B$"; the best way to derive it is from $A \leftrightarrow B$, i.e. "$A$ if and only if $B$".

This is "($A$, if $B$) and ($A$ only if $B$)", that translates $(B \to A) \land (A \to B)$.

Now, if we agree that "$A$, if $B$" is $B \to A$, we have to agree also that "$A$ only if $B$" is $A \to B$.

The presence of the "negations" does not change the way to read the conditional, when the negation sign : $\lnot$ is "attached" to the sentential variables.

Thus, $\lnot A \to \lnot B$ is : "if not-$A$, then not-$B$", and so on.