What is the difference of “speak at” and “speak of”?

  1. They allow him to speak at every hazard.

This sentence is ambiguous.

In theory, it could mean either of the following:

  • They allow him to speak whenever they encounter a hazard.
    → "Now that we've encountered another hazard, you can take your gag out if you have something to say."

  • They allow him to address himself to every hazard they encounter.
    → "Hi, hazard. I'm glad to meet you.")

But neither of those interpretations mean the same thing as the following:

  1. They allow him to speak of every hazard.
  • They allow him to orate on the topic of every hazard they will or might encounter.
    → "On this journey, we'll encounter many hazards. The first is well documented . . .")