Texas A&M University (TAMU) PHIL 240 Introduction to Logic Exam 1 Practice

The statement tests how negating a universal claim works. “It is not the case that every student passed” means there exists at least one student who did not pass. In predicate logic this is written as the negation of a universal statement: ¬∀x (Student(x) → Passed(x)). Using standard logical equivalences, this is equivalent to ∃x (Student(x) ∧ ¬Passed(x)). So the correct form is ¬∀x (Student(x) → Passed(x)).

The other forms don’t match the intended meaning: ∃x (Student(x) ∧ Passed(x)) says there exists someone who passed (not the negation of “every student passed”), ∀x (Student(x) → ¬Passed(x)) says every student failed (stronger than the negation), and ¬∃x (Student(x) ∧ Passed(x)) says no one passed (also stronger).