Is this a well-formed formula of FOL?

Question

Define the following atomic well-formed formulae:

M(x) - "x is a man"

H(x) - "x is a hero"

B(x) - "x is a building"

(Forgive me for the randomness)

Am I right in saying that M(x) → H(x) ↔ B(x) is not a well-formed formula since the lack of brackets gives rise to ambiguities? OR am I wrong and is this a well-formed formula?

Thanks

Answer 1

Unless you have made a prior definition of priority this is ambiguous (this is sometimes done to avoid the use of too many parentheses, usually operators like 'and' are then assigned to be the strongest in order etc.). The pure FOL doesn't do this however, so yes it is not clear what that formula means.

Note that M(x) → [ H(x) ↔ B(x) ] is not the same as [M(x) → H(x)] ↔ B(x).

Ambiguous formulae are never well-formed.