# Is this a well-formed formula of FOL?

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

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.

• And due to the ambiguity, it is not a well formed-formula of FOL, right? Commented Feb 15, 2016 at 12:38
• I'd assume, any formula that is ambiguous would not be well-formed, as (to me) those states are opposites of each other. Commented Feb 15, 2016 at 13:00
• Yes, ambiguous statements are not well-formed.
– MM8
Commented Feb 15, 2016 at 18:12