Final Answer:
The expression ∀a(∀x∃z(P(m) & Q(x,z,a)) || R(a,b)) is a well-formed formula (WFF) in First-Order Logic (FOL) with quantifiers properly nested and connected by logical operators. If the quantifier ∀a is removed, the resulting expression, (∀x∃z(P(m) & Q(x,z,a)) || R(a,b)), still constitutes a well-formed formula within FOL. However, the expression Q(a,b,c,x,y,z) alone does not qualify as a well-formed formula in FOL as it lacks logical connectives or quantifiers.
Step-by-step explanation:
In First-Order Logic (FOL), a well-formed formula (WFF) adheres to specific syntax rules involving the correct arrangement of quantifiers, predicates, variables, and logical connectives. The expression ∀a(∀x∃z(P(m) & Q(x,z,a)) || R(a,b)) is a WFF as it comprises nested quantifiers ∀ and ∃, logical connectives (&&), and predicates (P, Q, R) combined in accordance with the rules of FOL. Removing the outermost quantifier (∀a) doesn't disrupt the well-formed structure of the expression, leaving it still a valid WFF.
However, the expression Q(a,b,c,x,y,z) in isolation doesn't meet the criteria to be considered a well-formed formula in FOL. It lacks logical operators, quantifiers, or predicates properly structured to form a valid statement within FOL. Without these essential components or logical structure, it doesn't conform to the syntax rules of First-Order Logic and therefore cannot be classified as a well-formed formula in FOL. To be a WFF, it needs the inclusion of quantifiers or logical connectives to establish a meaningful logical statement or proposition.