UGCNET Questions

Modus tollens is the valid rule that allows us to infer the negation of the antecedent from a conditional and the negation of its consequent. In symbolic form, from P → Q and ¬Q we conclude ¬P. This pattern preserves validity, because if P had been true, Q would have had to be true as well, so the falsity of Q forces the falsity of P.

Option A:
Option A is modus ponens, which affirms the antecedent to derive the consequent, a different valid pattern.

Option B:
Option B commits the fallacy of denying the antecedent, which is not generally valid.

Option C:
Option C correctly captures the modus tollens form, using the falsity of the consequent to infer the falsity of the antecedent.

Option D:
Option D is the fallacy of affirming the consequent, which is invalid in general.

A contradiction is a statement that is false under every possible assignment of truth values to its variables. The expression P ∧ ¬P asserts that P is both true and not true at the same time, which cannot occur. In any valuation, either P is true or P is false; in neither case can the conjunction of P and its negation be true. Hence this expression is always false and is a standard example of contradiction.

Option A:
Option A directly combines a proposition with its own negation using “and,” ensuring that no scenario makes the whole statement true.

Option B:
Option B, P ∨ ¬P, is actually a tautology, as it is always true regardless of P’s value.

Option C:
Option C can be true or false depending on the values of P and Q, so it is a contingency, not a permanent contradiction.

Option D:
Option D, P → Q, is also contingent; its truth depends on how P and Q are valued.

De Morgan’s laws state that the negation of a disjunction is equivalent to the conjunction of the negations: Not (P or Q) is the same as (Not P) and (Not Q). Intuitively, saying “it is not the case that P or Q” means that neither P is true nor Q is true. The resulting statement is stronger than simply denying one of them; it requires both to be false. This equivalence holds for all combinations of truth values of P and Q.

Option A:
Option A correctly replaces the outer negated “or” with an “and” of the separate negations, in line with the formal statement of De Morgan’s law for disjunction.

Option B:
Option B would express the negation of a conjunction, Not (P and Q), rather than Not (P or Q), and is therefore the wrong transformation.

Option C:
Option C adds a biconditional between the negations, which is not equivalent to negating the disjunction.

Option D:
Option D introduces an implication relation between the negations, which again has a different truth table from the negation of P or Q.

The contrapositive of a conditional “If P then Q” is “If not Q then not P.” It is logically equivalent to the original statement. Here, P is “it is a bird” and Q is “it can fly.” Negating Q gives “it cannot fly,” and negating P gives “it is not a bird.” Therefore, the contrapositive is “If it cannot fly, then it is not a bird,” which preserves the truth conditions of the original conditional.

Option A:
Option A follows the correct procedure of reversing and negating both the antecedent and consequent. It captures the idea that anything that fails the flight condition cannot belong to the bird category under this rule.

Option B:
Option B is the converse, “If Q then P,” which is not guaranteed to be equivalent; many non birds can fly.

Option C:
Option C is the inverse, “If not P then not Q,” which is also not equivalent and can be false even if the original is true.

Option D:
Option D contradicts the original conditional by stating that non birds can fly as a rule, which is unrelated to the given statement.

A tautology is a compound proposition that evaluates to true under every possible assignment of truth values to its component propositions. It is logically true by virtue of its form rather than specific factual content. In a full truth table, the column for a tautology’s final result contains only Ts and no Fs. Because of this feature, tautologies play an important role in testing argument forms and deriving equivalences.

Option A:
Option A matches the formal definition of a tautology as a statement that never turns out false in any row of its truth table. It captures the idea of logical necessity independent of empirical facts.

Option B:
Option B describes a contradiction, which is always false and has only Fs in its truth table, the exact opposite of a tautology.

Option C:
Option C refers to a contingency, which is true on some valuations and false on others, so it is neither tautological nor contradictory.

Option D:
Option D suggests that the statement cannot be tested, which is incorrect because tautologies are routinely checked using truth tables and formal methods.

Given a conditional “If P then Q,” the inverse is formed by negating both the antecedent and the consequent while keeping their order: “If not P then not Q.” Here, P is “the student is sincere” and Q is “the student passes the exam.” The inverse is therefore “If a student is not sincere, then the student does not pass the exam.”

Option A:
Option A is the converse, reversing the roles of antecedent and consequent to “If Q then P.”

Option B:
Option B correctly negates both antecedent and consequent without swapping them, matching the standard definition of the inverse.

Option C:
Option C negates only the antecedent while leaving the consequent positive, which does not correspond to any standard paired form (it is neither converse, inverse, nor contrapositive of the original).

Option D:
Option D is the contrapositive, “If not Q then not P,” which is logically equivalent to the original conditional, not its inverse.

A statement that is always true under every possible assignment of truth values to its components is called a tautology. The disjunction “p or not p” covers all possibilities: if p is true, the first part is true; if p is false, not p is true. Thus, in all cases the compound statement evaluates to true. This makes “p or not p” a classic example of a tautology.

Option A:
Option A, “p and not p,” is instead a contradiction because it asserts that p is both true and false simultaneously. There is no assignment of truth values that can make both p and not p true together, so this statement is never true.

Option B:
Option B, “p or not p,” correctly spans all truth possibilities for p. Since one of p or not p must be true, their disjunction is always true, demonstrating tautological status.

Option C:
Option C, “p → not p,” is not always true. If p is true, the implication would require not p to be true as well, which is impossible, so the statement can be false.

Option D:
Option D, “not p → p,” also fails to be always true; there are assignments where it becomes false, so it is not a tautology.

In propositional logic, the connective “and” corresponds to conjunction. A conjunction p and q is true only when both component statements p and q are true simultaneously. Here p says the number is even and q says it is divisible by 3. Putting them together with “and” requires both properties to hold. Thus, the combined statement means that the number must be even and divisible by 3.

Option A:
Option A uses “or” instead of “and,” which corresponds to a different connective (disjunction). It allows numbers that are only even or only divisible by 3, and hence does not capture the meaning of “p and q.”

Option B:
Option B mixes “either” with a negation in a way that is not equivalent to the original conjunction. It introduces “not divisible by 3,” which is not part of q.

Option C:
Option C reflects the standard interpretation of a logical conjunction: both conditions must be simultaneously satisfied. It directly translates “p and q” into ordinary language as “even and divisible by 3.”

Option D:
Option D makes both properties false, which would correspond to the negation of p and q, not the conjunction of p and q.

Negation in propositional logic is often represented by the symbol ¬ placed before a proposition. It expresses the logical opposite of the statement’s truth value: if the original proposition is true, its negation is false, and vice versa. This operator is fundamental in constructing complex statements and in defining other connectives using truth tables.

Option A:
Option A, ¬, is the standard modern symbol for “not,” used widely in logic texts and formal reasoning. It unambiguously marks the negation of whatever proposition follows it.

Option B:
Option B, ∧, denotes conjunction and is used for “and,” not for negation.

Option C:
Option C, ∨, is the symbol for disjunction, corresponding to “or,” and has a different truth function.

Option D:
Option D, ↔, stands for biconditional, “if and only if,” which expresses equivalence rather than negation.