Identify the correct translation into logical notation of the following…

Correct translation: (∃x)(boy(x) ∧ (∀y)(girl(y) → taller(x,y)))

Explanation: There exists an individual x who is a boy, and for every individual y, if y is a girl then x is taller than y.

• Why the outer conjunction is required: Using (∃x)(boy(x) → ...) would allow a non-boy to satisfy the existential by making the implication true vacuously, so it would not guarantee the existence of a boy with the property.

• Why the inner implication is required: (∀y)(girl(y) → taller(x,y)) says for every y, being a girl is sufficient to require x be taller than y. Using (girl(y) ∧ taller(x,y)) inside the universal would incorrectly require that every object y is a girl and that x is taller than every object.

• Common incorrect forms and why they fail:

• Example incorrect formula: (∃x)(boy(x) → (∀y)(girl(y) → taller(x,y))).

• Why it fails: The existential could be witnessed by an object that is not a boy, making the implication true without any boy being taller than all girls.

• Example incorrect formula: (∃x)(boy(x) ∧ (∀y)(girl(y) ∧ taller(x,y))).

• Why it fails: The universal conjunct (girl(y) ∧ taller(x,y)) asserts every object y is a girl and that x is taller than every object, which is stronger than the intended meaning.

Conclusion: The formula (∃x)(boy(x) ∧ (∀y)(girl(y) → taller(x,y))) precisely captures 'Some boys in the class are taller than all the girls.'