Is there a proper term and/or symbol for an "agnostic" conclusion?

Question

My question stems from the material conditional:

$p \rightarrow q\\p\\\therefore\space q$

However, if $\bar p$ then the conditional is silent. I would like a way to represent this fact using, if possible, accepted symbols and terminology.

While I don't like neologisms, I have informally been referring to this as an "agnostic" conclusion (I know, oxymoron) and using $\bar\therefore$ so that in the above syllogism we would have:

$p \rightarrow q\\ \bar p\\ \bar\therefore\space q$

Answer 1

You can find many such symbols and conceptual variants of this sort of thing in Priest's An Introduction to Non-Classical Logic: From If to Is.

Answer 2

This doesn't really involve symbols, but the first thing that comes to mind is "vacuous truth". If $p\Rightarrow q$ but $\neg p$ ($\overline p$ if you prefer), then we say that the implication is "vacuously true". I feel like "vacuousness" in this context corresponds to your "the conditional is silent".

In such a situation, without more information, we "can't say anything" about the truth of $q$ (see a google search showing that this is a decently common phrase).