I agree with Jack that (a) is the correct interpretation, and this is because (1) contains a "that"-content clause embedded in a relative clause. Notice how these sentences are equivalent:
- He didn't know THAT the things he said were untrue = He said things he didn't know were untrue.
For interpretation (b) to be possible, these should be an "if"- or "whether"- clause, and, unlike "that," these conjunctions cannot be tacit if that is the meaning that needs to be conveyed:
- He didn't know IF the things he said were untrue = He said things he didn't know if they were untrue (grammatically dubious)
Do you agree, David, that "if"-content clauses don't work as well as "that"-clauses if embedded in a relative clause?