I have been informed here What is the difference between function and predicate? that in formal semantics, predicates are always functions that map from individuals (i.e. arguments) to truth values. There are other types of functions, though, namely those that map to individuals instead of to truth values. My understanding of this issue is therefore that in formal semantics, every predicate is a function, but there are functions that are not predicates. Function is therefore the broader concept, broader than predicate.
In syntax, content verbs, adjectives, and predicative expressions of various sorts are predicates, and many of these predicates cannot be construed as mapping to truth values, at least not in a straightforward way. This is often the case with embedded predicates, i.e. with predicates that are subordinate to the matrix predicate, e.g.
(1) Hank said Susan laughed.
(2) Hank considers Susan entertaining.
In sentence (1), there are two predicates, said and laughed. Only said can be construed as mapping to a truth value in a straightforward way. The sentence can be true or false regardless of whether or not Susan actually did or did not laugh. The situation is similar in sentence (2); there are two predicates there, considers and entertaining. The sentence can be true or false regardless of whether Susan is or is not entertaining.
I have been informed that intensionality is at play in sentences like (1) and (2), the implication being that intensionality allows one to maintain the position that all predicates map to truth values. I am confused about how this works, though. It would make sense to me if one assumes a third truth value, namely undecided. There would hence be three truth values (true 1, false 0, and undecided ?). Predicates like laughed in (1) and entertaining in (2) could then be construed as mapping to truth values, but in these cases, the truth value they map to would be undecided.
Thanks for your time!