# How is it that 'if A then B' can be equivalent to 'A only if B'?

I've seen this stated multiple times that the two are equal. However, I think it's actually logically impossible for the two terms to be the same, as either one of the statements seems to preclude the other.

I know that's a bold statement, but consider:

If A then B, means that A happens first, B happens second, consequently.

So, 'A only if B', should thus be the same as 'only if B, then A. This means 'only if B then A' = 'if A then B'. However, A and B cannot happen as a consequent of each other, as that's literally impossible (well... I guess there's that whole chicken or the egg thing, but still).

• Mathematical logic, at least the basic stuff every mathematician knows, has no notion of causality, and indeed is only weakly linked to a person's every day experience. Nothing "happens" at all, and we add interpretation to the predicates at our own risk. Mar 6, 2016 at 0:09
• But we even call it antecedents and consequents, plus I know what your saying, but logic is also applied extensively throughout science as well as maths, where a very strong sense of causality is inferred. Infact causality is one of the core concepts within science, for e.g when we observe how independent variables react to changes in dependent variables. Mar 6, 2016 at 1:11
• Mar 6, 2016 at 8:34

I think you might misunderstand what "A only if B" means. We consider "only if" as a phrase on its own, independent of "if."

What "A only if B" means is that the only way that A can possibly hold is if B holds. That is clearly the same as saying "If B does not hold, there is no way that A can hold." That is the same as saying "If A holds, then B holds." The last that I mentioned is the principle of contraposition (https://en.wikipedia.org/wiki/Contraposition).

• Could one say that, 'A only if B' is the same as 'B, only if A had occurred'? Also thanks ^.^ Mar 6, 2016 at 0:02
• @user108262 No; switch A and B. Mar 6, 2016 at 0:13

Both expressions mean that $B$ is necessary condition (with respect to $A$), or equivalently that $A$ is a sufficient condition (with respect to $B$). They are both denoted as $$A\Rightarrow B$$ and are both equivalent to "$A$ implies $B$".

The expression "if $A$ then $B$" stresses the sufficiency of $A$ while the expression "$A$ only if $B$" stresses the necessity of $B$, in the following sense: It says that $A$ holds true only if $B$ does. This becomes evident, since $A$ implies $B$, so when $A$ holds true then it is an automatic consequence that $B$ also does. If $B$ weren't true then neither would $A$ be.

The only way to refute "if A then B" is to find a case where A is true, yet B is false.

Which is exactly what "A onlyif B" says -- A is supposed to hold only if B does also, which holds except if A is true yet B is false.

Note that neither "if A then B" nor "A onlyif B" say anything about a sequence of events. Classical logic has no concept of "before" and "after". It doesn't even really have the concept of a "cause". In fact, "if A then B" (in classical logic) is usually defined as being an abbreviation for "either B or not A". Which, again, says that for "if A then B" to hold, in all cases either B must be true, or if not, then A must be false as well.

This definition of "if A then B" is responsible for quite a few of the counter-intuitive properties of classical logic. It is, for example, responsible for "ex falso quodlibet", meaning if A is never true, then "if A then B" is always true. Therefore, if you can prove that "earth is not a disc", then in classical logic, you can also prove "if earth is a disc, santa exists". Such sentences are calles vacuous truths, because while logically true, they don't provide any actual information (in this case about the existance of santa)