Is there a name for the logical scenario where A does not necessarily imply B, but B implies A?

Question

A real life example of this is the 'Active' status on Facebook Messenger. (For those interested see this article here, and some Quora answers here for details.)

When you are actively using Facebook Messenger, your status is guaranteed to be 'Active.' However, if your status says 'Active,' it doesn't necessarily imply that you're actually using Facebook Messenger.

Say

A = Active Status

B = Actually Using Facebook Messenger

Therefore,

A⊬B

but

B→A

Is there a name to this logical scenario? When trying to explain this to people, I'd like to sum up the situation in a couple words, or a statement of logic.

Answer 1

In context, if $B$ implies $A$ but $A$ does not necessarily imply $B$, we typically say: $B$ implies $A$, but the converse does not hold.

Answer 2

However, if your status says 'Active,' it doesn't necessarily imply that you're actually using Facebook Messenger.

What you are trying to describe is a lack of an assumption. The lack of an assumption $X$ is not the same thing as assuming $\lnot X$. Formally describing an absent assumption is difficult. However, $A \not \vdash B$ and $A \not \implies B$ are both incorrect. Both lead to theorems that would otherwise not be inferrable from the simple absence of the assumption $A \implies B$.

What does it mean for a statement to be absent from assumptions? It could mean 2 things. One is literal: the exact text of the assumption is not present. If you have a set $\Gamma$ representing your assumptions, you could say $「A \implies B」 \not \in \Gamma$. But even this is somewhat vague: it doesn't necessarily exclude something like $「\lnot \lnot (A \implies B)」 \in \Gamma$.

The other meaning is more subtle: it could mean that the statement is not entailed by the assumptions. That is, $\Gamma \not \vdash 「A \implies B」$. When you want to describe an absent assumption, you have to decide in which sense you mean it, this meaning or the former.

Consider the following scenario of two assumptions: (1) every person named Jack is tall, (2) the pilot is a person named Jack. In that scenario are we assuming the pilot is tall? No, in the sense that this exact assumption is not present. But yes, in the sense that "the pilot is tall" is not independent of our assumptions.

In practice, to avoid this problem, assumptions are usually just written out explicitly, and the reader is trusted not to accidently add extra assumptions. It would be common to say "we assume that use implies active" and trust the reader not to incorrectly assume the converse as an assumption until told. If the absent statement is truly independent of assumptions, then you could be kind and say "we are not assuming that active implies use".