Fundamentals of Contradictions in Propositional Logic Quiz

A contradiction is a statement that can never be true under any assignment of truth values. The expression P ∧ ¬P requires P to be true and not true at the same time. This is impossible because a proposition cannot simultaneously hold both a truth value and its negation. In contrast, the other expressions are always true (tautologies), because P → P, P ↔ P, and P ∨ ¬P all evaluate to true regardless of the truth value of P. Therefore, P ∧ ¬P is the only expression in the list that is always false, making it a contradiction.

A contradiction is defined as a statement that cannot be satisfied under any truth assignment. No matter what truth value is given to the variables involved, the statement evaluates to false. This is different from contingent statements, which can be true or false depending on the inputs, and tautologies, which are always true. Because contradictions necessarily evaluate to false in all interpretations, their truth value is always false.

This statement asserts two mutually incompatible conditions: that it is raining and simultaneously not raining. Since a condition and its negation cannot both be true at the same time, the statement can never be true in any possible scenario. The first option is a tautology since a statement or its negation is always true. The third and fourth options are conditionals that can be true in many situations. Only the claim that it is raining and not raining is structurally impossible, making it a contradiction.

The expression P ∨ ¬P is a tautology because for any truth value of P, either P is true or its negation is true. Taking the negation of a tautology results in a contradiction. Applying De Morgan’s Law to ¬(P ∨ ¬P), we obtain ¬P ∧ ¬¬P. Double negation simplifies ¬¬P to P, so the whole expression becomes ¬P ∧ P, which is simply P ∧ ¬P. This cannot be true for any truth value of P, confirming that the negation of a tautology is indeed a contradiction.

The statement P ∨ ¬P is a tautology, meaning it is always true regardless of the truth value of P. This makes it the only expression listed that is not a contradiction. The other expressions all force mutually incompatible conditions. P ∧ ¬P explicitly requires P to be true and false. The expression (P → Q) ∧ (P ∧ ¬Q) demands that P implies Q while also requiring P to be true and Q to be false, which makes the implication false. The expression (P ↔ Q) ∧ (P ↔ ¬Q) demands that Q and ¬Q both match P, which is impossible. Therefore, P ∨ ¬P is the only non-contradiction among the choices.

A tautology is true under every possible truth assignment. Negating such a statement flips every truth value in its truth table. Because a tautology never evaluates to false, its negation will never evaluate to true. A statement that is false for all truth assignments is, by definition, a contradiction. This shows that tautologies and contradictions are logical opposites. Therefore, the negation of a tautology must be a contradiction.

P ∧ ¬P asserts that P is both true and false, which is impossible. Therefore the expression is false in every truth assignment, which means it is a contradiction. The other options are not contradictions. P → P is always true, making it a tautology. P ∧ P is equivalent to P, which may be true or false depending on P, making it contingent instead of contradictory. P ∨ P is also equivalent to P, likewise contingent. Only P ∧ ¬P is always false.

The expression Q ∧ ¬Q is itself a contradiction because it requires Q to be both true and false simultaneously. Once any portion of a conjunction is false, the entire conjunction becomes false, no matter what the other components are. Therefore, P ∧ (Q ∧ ¬Q) becomes P ∧ False, which is simply False. This means the entire expression cannot be true in any interpretation and is therefore a contradiction.

A contradiction is a statement that is false under all possible truth assignments. A statement is satisfiable if there exists at least one interpretation under which it becomes true. Since contradictions never become true, they have no satisfying truth assignment, meaning they are unsatisfiable by definition. They are not always true, and they are not contingent because contingent statements can be true sometimes and false sometimes. Contradictions are always false and thus unsatisfiable.

The statement P ↔ P is a tautology because a proposition is always logically equivalent to itself. It is true under every truth assignment. Negating a tautology yields a contradiction. To determine the explicit form, recall that P ↔ P means (P → P) ∧ (P → P), which is always true. Negating it means the result must always be false. The standard contradiction involving a single variable is P ∧ ¬P, which evaluates to false for all truth values. Therefore ¬(P ↔ P) is equivalent to P ∧ ¬P.

P ∨ ¬P is the law of excluded middle, a tautology that is always true. Negating it must produce a contradiction. Using De Morgan’s Law, ¬(P ∨ ¬P) becomes ¬P ∧ ¬¬P. Double negation reduces ¬¬P to P, giving ¬P ∧ P, which is the same as P ∧ ¬P. Since this expression is false in every case, it represents the contradiction obtained by negating a tautology.

A number cannot be both even and odd simultaneously. Even numbers are divisible by 2, while odd numbers are not. These conditions are mutually exclusive, so the statement that a number is both even and odd is impossible under all interpretations, making it a contradiction. The first option is a tautology because every integer is either even or odd. The remaining statements are conditionals or biconditionals that can be true in valid contexts. Only the claim of being both even and odd is inherently contradictory.

To evaluate (P → Q) ∧ (¬P → Q), we must consider all possible truth assignments. The first implication P → Q states that if P is true, then Q must be true. The second implication ¬P → Q states that if P is false, then Q must be true. Together, these implications cover all possibilities: either P is true (forcing Q true by the first implication) or P is false (forcing Q true by the second implication). Since P must be either true or false, and both cases require Q to be true, the conjunction is satisfied exactly when Q is true. We can verify this with a truth table: when Q is true, both implications are automatically true regardless of P's value, making the conjunction true. When Q is false, at least one implication fails, making the conjunction false. Therefore, the expression is logically equivalent to Q, which is a contingent statement, not a contradiction.

The inner part ¬P∧Q can sometimes be true if P is false and Q is true. However, the entire expression includes P ∧ (¬P ∧ Q), meaning P must be true at the same time that ¬P ∧ Q is true. Since P and ¬P cannot both be true, this forces a contradiction. No truth assignment can make the whole expression true. The outer conjunction guarantees that even if Q is true, the requirement that P and ¬P both hold eliminates all possibilities.

A contradiction is false under every possible truth assignment. Negating such a statement flips the false values to true, resulting in a statement that is true under every interpretation. Because a tautology is defined as a statement that is always true, the negation of a contradiction must be a tautology. Thus contradictions and tautologies are logical opposites: negating one yields the other.