This question has been raised several times (here, here and here), and several users proposed in their comments to synonymize (or merge) them, but nobody concretely suggested them as synonymous on Mathematics.
Thus, I formally proposed to synonymize them here, you can upvote or downvote as you prefer.
- predicate-logic and first-order-logic have already a considerable overlap (at the time of this post, 619 questions contain both tags out of 2267 questions in predicate-logic);
- almost all the questions tagged with predicate-logic without first-order-logic could coherently and meaningfully be tagged with first-order-logic;
- the really few questions tagged with predicate-logic and that concern second or higher order logic could be tagged with second-order-logic or higher-order-logic, plus possibly first-order-logic.