Reasoning with Universal Quantifiers and Logical Inference Quiz

The natural-language statement restricts attention to primes that are greater than 2 and then asserts they are odd. The correct formalization places the conjunction Prime(x) ∧ x > 2 in the antecedent of an implication whose consequent is Odd(x), and then universally quantifies over x. Writing the condition as ∀x(Prime(x) ∧ x > 2 → Odd(x)) precisely captures “for every x, if x is prime and x > 2, then x is odd.” The other forms either assert every object is both prime and odd or merely assert existence rather than the required universal conditional.

From the universal statement ∀x(P(x) → Q(x)), by universal instantiation we can derive P(c) → Q(c). Given that P(c) is true, we can apply modus ponens to conclude Q(c) must be true. This is a fundamental pattern of reasoning with universally quantified conditionals.

A universally quantified implication requires only that no x make P true and Q false simultaneously. It places no restriction on elements for which P(x) is false; such elements make the implication true vacuously and they may or may not satisfy Q(x). Hence it is perfectly consistent with ∀x(P → Q) that there exist x with ¬P(x) ∧ Q(x). Option B would contradict the universal implication, option C is an independent stronger claim, and option D is unrelated and possibly compatible but not what the implication specifically allows as a typical instance.

Universal instantiation is a standard rule: from ∀xP(x) we may substitute any specific object c from the domain to conclude P(c). This is a direct application of the meaning of the universal quantifier. While ∀xP(x) also entails ∃xP(x) in any nonempty domain, the immediate and precise inference from the universal assumption and an arbitrary constant c is P(c).

In any nonempty domain, a true universal statement ∀xP(x) guarantees that at least one element satisfies P, hence ∃xP(x) follows. This relies on the domain being nonempty; the standard meta-assumption in most mathematical contexts is a nonempty domain, and under that assumption ∀xP(x) implies ∃xP(x). If the logic explicitly allows empty domains, then one must treat this implication with the empty-domain caveat, but in ordinary treatments the existential follows from the universal.

Conjoining two universal statements is equivalent to a single universal statement whose matrix is the conjunction of the two predicates: if every x satisfies P and every x satisfies Q, then every x satisfies both P and Q, and conversely if every x satisfies P ∧ Q then each of ∀xP and ∀xQ holds. This equivalence follows from the distributive behavior of quantifiers over logical conjunction.

If either every x satisfies P or every x satisfies Q, then in either case every x satisfies the disjunction P ∨ Q. Formally, from ∀xP(x) we get ∀x(P(x) ∨ Q(x)) by weakening, and similarly from ∀xQ(x) we get the same result; thus the disjunction of the universals yields ∀x(P ∨ Q). Note that the converse need not hold: ∀x(P ∨ Q) does not imply one of the universals must hold.

The quantifier order matters: ∀x∃yP(x,y) asserts that for every x in the domain we can find (possibly depending on x) a y such that P(x,y) holds. This is different from ∃y∀xP(x,y) which would assert a single y that works for all x. The correct reading matches the indicated nesting: universal over x, existential over y.

With domain {1,2} both elements satisfy x > 0, so the universal claim holds because every element in the finite domain meets the predicate. Finite-domain universals reduce to finite conjunctions: here x > 0 must hold for x = 1 and x = 2, which it does, so the universally quantified statement is true.

From the universal implication we may instantiate to get P(c) → Q(c). Given P(c) as well, modus ponens yields Q(c). This is the standard pattern of reasoning: universal instantiation followed by modus ponens on the specific instance produces the consequent for that particular object.

The plain reading of “all respondents” is universal: every individual who responded to the survey is included and is asserted to have liked the product. This differs from “some” or “most,” and it equivalently means there are no respondents who did not like the product.

The phrase “all citizens” denotes universality over the set of citizens; the legal formulation intends to assert that every individual who qualifies as a citizen holds rights. The statement attributes rights to each member of the specified class, not merely to some or most.

For any proposition P(x), the disjunction P(x) ∨ ¬P(x) is an instance of the law of excluded middle, which is classically true for each x. Universal quantification over x preserves that truth, yielding a formula true for every x in any classical interpretation. Hence the whole universally quantified disjunction is a logical truth (always true) in classical logic.

The statement “All squares are rectangles” asserts that for every individual x, if x is a square then x is a rectangle. This is most naturally and directly formalized as ∀x (Square(x) → Rectangle(x)). The existential options do not express the universal claim, and stating that no squares exist contradicts the intended meaning. The universal implication is the correct representation.

An implication P(x) → Q(x) is always true when P(x) is false, regardless of Q(x)’s truth value. This is known as vacuous truth. Therefore, for any element x for which P(x) fails, the implication automatically evaluates to true. The universal statement requires that all such implications be true, and vacuous truth ensures this requirement is met for any x with false antecedent.