Universal Statements Quiz: Interpret General Logical Claims

The property “all primes greater than 2 are odd” does not claim that every prime is odd — only those primes beyond 2. So the logical form must state: if x is prime and x > 2, then x is odd. This ensures 2 is not incorrectly forced to be odd while correctly covering all other primes.

From a universal statement ∀x(P(x) → Q(x)), we can apply universal instantiation to obtain P(c) → Q(c) for the specific element c.

Given the premise P(c), modus ponens allows us to conclude Q(c). This is a standard rule of inference.

Statements of the form ∀x(P(x) → Q(x)) fail only if we find some x where P(x) is true but Q(x) is false. Such an x is called a counterexample. If no such x exists, the universal conditional is true.

When we have ∀xP(x), we may substitute any particular constant c from the domain.

The result P(c) is guaranteed to be true because the universal statement covers every element.

The formula ¬∀xP(x) means “it is not true that P holds for all x.”

This is equivalent to saying “there exists some x for which P does not hold,” written as ∃x¬P(x).

This is a core De Morgan–style rule for quantifiers.

In ∀x∃yP(x,y), the existential quantifier is inside the universal, meaning:

for each x, you only need at least one y that works — and this y may depend on x.

There is no requirement for a single y to work for all x.

Both 1 and 2 satisfy x>0. With domain {1, 2}, checking ∀x(x > 0) requires confirming the predicate for each element.

Since 1 > 0 and 2 > 0, the universal statement is true.

Negating a universal yields an existential counterexample. When we negate “all chickens can fly,” we claim that there exists at least one chicken that cannot fly.

Thus ¬∀xP(x) becomes ∃x¬P(x).

Negating universals always introduces the idea of a counterexample.

Negate existential by switching to universal with negated predicate. ¬∃xP(x) means “there is no x such that P(x) is true.”

This is equivalent to “for every x, P(x) is false,” written as ∀x¬P(x).

This conversion is another fundamental quantifier negation law.

To say “not all birds can fly” is to assert that at least one bird is an exception.

Thus formalization requires an existential of the form ∃x(Bird(x) ∧ ¬Fly(x)).

Law of excluded middle: every element satisfies P or not P. The universal statement ∀x(P(x) ∨ ¬P(x)) is always true in classical logic because, for any x,

a statement is either true or false — no middle case exists.

By chaining implications, P leads to R universally. Given ∀x(P(x) → Q(x)) and ∀x(Q(x) → R(x)), we can combine them to conclude:

for each x, P(x) implies Q(x), and Q(x) implies R(x),

so overall P(x) must imply R(x) for every x.

Negate nested universals by producing nested existentials with negation. The negation of ∀x∀yP(x,y) states that it is not the case that P(x,y) holds for every pair.

This means: there exists some pair (x, y) where P(x,y) fails → ∃x∃y¬P(x,y).

2 is a counterexample: prime but not odd. To disprove “all primes are odd,” we only need one prime number that is not odd.

The prime number 2 satisfies primeness but violates oddness, making the universal sentence false.

An existential states existence of at least one x with both properties. The formula ∃x(Cat(x) ∧ Black(x)) means there is at least one object in the domain that is both a cat and black.

This matches the English statement “some cats are black.”

A universal requires all true; one counterexample falsifies. Universal claims ∀xP(x) demand that P(x) holds for every x.

If even one object violates P(x), the entire universal claim becomes false.

Negation: there exists an x making x²≥0 false, which means x²


To negate ∀x(x² ≥ 0), we require a specific x for which the inequality fails.

Failing ≥0 means being
(Over real numbers this is unsatisfiable, showing the original statement is universally true.)

Every student must satisfy Passed(x) if a student. The universal implication ∀x(Student(x) → Passed(x)) requires that whenever x is a student, x is also someone who passed.

It does not assert that every x is a student — only that students are among the passers.

Vacuous truth: if antecedent never holds, conditional is always true. In ∀x(P(x) → Q(x)), if P(x) is false for every x, then P(x) → Q(x) is automatically true for all x.

Therefore, the universal is considered true even when P never occurs.

The domain determines the set of elements for which P must hold. Changing the domain changes which elements x must satisfy P(x).

For example, ∀x(x > 0) over ℕ is true, but over ℤ it is false.

Thus the meaning and truth of a universal statement depend critically on the domain.