witnesses for his propensity to be a liar.

Here's the rule (this is the Federal Rule - there are some minor variations from State to State):

(a) Reputation or Opinion Evidence. A witness’s credibility may be attacked or supported by testimony about the witness’s reputation for having a character for truthfulness or untruthfulness, or by testimony in the form of an opinion about that character. But evidence of truthful character is admissible only after the witness’s character for truthfulness has been attacked.

(b) Specific Instances of Conduct. Except for a criminal conviction under Rule 609, extrinsic evidence is not admissible to prove specific instances of a witness’s conduct in order to attack or support the witness’s character for truthfulness. But the court may, on cross-examination, allow them to be inquired into if they are probative of the character for truthfulness or untruthfulness of:

(1) the witness; or

(2) another witness whose character the witness being cross-examined has testified about.

By testifying on another matter, a witness does not waive any privilege against self-incrimination for testimony that relates only to the witness’s character for truthfulness.