Which one of the following is the most appropriate logical formula to…

Correct formalization: ∀x((G(x) ∨ S(x)) → P(x))

Read in words: For every object x, if x is a gold ornament or x is a silver ornament, then x is precious.

• Identify predicates: G(x) = x is a gold ornament; S(x) = x is a silver ornament; P(x) = x is precious.

• Use a universal quantifier because the sentence makes a general claim about all such ornaments.

• Use a disjunction in the antecedent (G(x) ∨ S(x)) because the sentence refers to gold or silver ornaments.

• Combine as an implication: if an object is gold or silver then it is precious, yielding ∀x((G(x) ∨ S(x)) → P(x)).

Why the other formulas are incorrect:

• ∀x(P(x)→(G(x)∧S(x))) incorrectly says every precious object must be both gold and silver, reversing the intended implication.

• ∀x((G(x)∧S(x))→P(x)) only guarantees objects that are both gold and silver are precious, and does not cover objects that are only gold or only silver.

• ∃x((G(x)∧S(x))→P(x)) is existential and far too weak for the universal claim; as written it is also missing a closing parenthesis.