Truth Conditions, Counterexamples, and Universal Claims Quiz

The formula ∀x(P(x) → Q(x)) asserts that for every element of the domain the implication P(x) → Q(x) holds; by the semantics of the universal quantifier this means there is no x for which P(x) is true and Q(x) false. Concretely, to verify the universal statement you check each x and confirm that whenever P(x) holds then Q(x) also holds; if this conditional relation always succeeds (including vacuously when P(x) is false), the whole universal formula is true.

A universal statement quantifies over all members of a domain; the phrase “Every dog barks” explicitly applies a property to every dog and therefore must be formalized with a universal quantifier (for example ∀x(Dog(x) → Barks(x))). The other options use existential language (“some”, “there exists”, “a few”), which marks them as existential rather than universal claims.

The expression ∀x Dog(x) lacks any implication or restriction and states that for every object x in the domain the predicate Dog(x) holds; that is exactly the claim that all objects in the domain are dogs. If the intended meaning were “for all x, if x is in some set then …” an implication would appear, but here the bare universal quantifier applied to Dog(x) makes the unconditional assertion that every element is a dog.

A universal statement ∀x(x > 0) claims that every element satisfies x > 0; it fails precisely when you can produce a counterexample, i.e., an x for which x > 0 is false. Thus the presence of at least one nonpositive element is sufficient to falsify the universal claim. Testing some elements or finding positive examples cannot prove the universal truth; only the absence of counterexamples would.

A proof of a universal statement must establish the property for an arbitrary but otherwise unconstrained element of the domain; in practice one picks an arbitrary element a and proves P(a) without using any special features of a. Demonstrating P for one particular instance or for some subset does not suffice; the logical force of the universal quantifier requires showing the property holds for every possible element.

The universal claim “All primes are odd” is refuted by a single counterexample; the integer 2 is both prime and even, which directly contradicts the asserted property that every prime is odd. Stating “some primes are odd” is true but it doesn’t provide the precise minimal information that falsifies the universal statement.

The quantifier ∀x ranges over a particular domain of discourse, and the truth of ∀xP(x) depends on which objects are included in that domain; changing the domain can turn a true statement into a false one or vice versa because different elements are being tested against P. The domain does not itself alter the predicate’s internal semantics, but it determines the set of x values for which P(x) must hold.

The formula x = x expresses reflexivity of equality and holds for every object in any interpretation of classical first-order logic; applying the universal quantifier simply states that every object equals itself, which is true in all domains and under standard interpretations of equality. This is a logical truth—true solely by virtue of the meanings of equality and the universal quantifier.

Negation of a universal quantifier converts it into an existential quantifier with the predicate negated: ¬∀x φ(x) is logically equivalent to ∃x ¬φ(x). Applying that pattern to φ(x) ≡ (x > 0) yields ∃x ¬(x > 0), which can be written naturally as ∃x(x ≤ 0). This states the existence of at least one element that fails the original > 0 condition and is exactly the logical negation of the universal.

The universally quantified implication is falsified by producing a single witness x for which the antecedent P(x) is true and the consequent Q(x) is false; formally ¬∀x(P(x) → Q(x)) ≡ ∃x ¬(P(x) → Q(x)), and since ¬(P → Q) ≡ P ∧ ¬Q, the negation becomes ∃x(P(x) ∧ ¬Q(x)). This existential-conjunctive form captures the exact counterexample that invalidates the universal implication.

The ordinary-language phrase “not every” is the negation of a universal, so it asserts that there exists at least one student who did not pass; put formally ¬∀x Passed(x) ≡ ∃x ¬Passed(x). That existential statement is the minimal negation of the universal claim and is precisely captured by “some student failed.”

The falsity of a universal statement means there is at least one counterexample; negating the universal yields an existential claim about the negated predicate. Concretely, ¬∀xP(x) is equivalent to ∃x¬P(x), so whenever the universal fails there must exist some x such that P(x) is false. Other options either overreach (claiming all are ¬P) or confuse existential and universal statuses.

Denying the universal “All humans are mortal” produces the existential statement that there exists a human who is not mortal. This follows the standard quantifier-negation equivalence ¬∀x Mortal(x) ≡ ∃x ¬Mortal(x). The correct negation is not the sweeping claim that no humans are mortal, nor the paradoxical claim that all are immortal; it is simply the existence of at least one non-mortal human.

Negation pushes through nested universal quantifiers one at a time and turns each universal into an existential while negating the matrix: ¬∀x∀yP(x,y) ≡ ∃x¬∀yP(x,y) ≡ ∃x∃y¬P(x,y). The result states that there exist specific x and y such that P(x,y) fails, which is the direct existential counterpart to the original universal claim over all ordered pairs.

The negation of “all cats like fish” is that there exists at least one cat that does not like fish. Formally ¬∀x LikesFish(x) ≡ ∃x ¬LikesFish(x). This existential interpretation is weaker than “no cats like fish” and simply asserts the presence of at least one non-fish-loving cat rather than describing the tastes of the entire population.