Contradictions Quiz: Test Statement Consistency and Breakdown

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| Questions: 20 | Updated: Dec 17, 2025
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1) What is (P ∧ ¬P) ∨ (P ∧ ¬P) equivalent to?

Explanation

The conjunction P ∧ ¬P cannot ever be true because a statement and its negation cannot hold simultaneously. Since both sides of the OR contain the impossible conjunction, each side evaluates to false, and combining two false values with OR keeps the whole expression false in all cases. This makes the expression a contradiction.

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About This Quiz
Contradictions Quiz: Test Statement Consistency And Breakdown - Quiz

Some statements collapse the moment you check them carefully. In this contradictions practice quiz, you’ll test assumptions, push statements to their limits, and learn how contradictions reveal hidden issues in reasoning. It’s engaging, hands-on practice that strengthens your ability to evaluate claims with confidence.

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2) The expression P ∧ (P → ¬P) ∧ P is always a contradiction.

Explanation

The implication P→¬P states that whenever P is true, ¬P must also be true, which is impossible; but when evaluating it inside a larger expression, the moment P is assumed true, the implication forces ¬P, creating the impossible combination P ∧ ¬P. This contradiction collapses the entire expression into something that cannot be satisfied.

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3) ¬(P ∧ ¬P) ∨ Q simplifies to:

Explanation

Since P ∧ ¬P is false in every possible truth assignment, negating it yields ¬(P ∧ ¬P), which is always true. Once a disjunction contains a true component, the entire OR expression becomes true regardless of other terms. Therefore, the expression is a tautology.

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4) Fill in the result: Q ∧ (P ∧ ¬P) = _______.

Explanation

Whenever an expression includes P ∧ ¬P as part of an AND chain, that contradiction immediately forces the entire conjunction to be false. Even if Q is true or false, the presence of the impossible P ∧ ¬P means the final result can never be true.

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5) Which statement is a contradiction?

Explanation

The claim combines mutually exclusive properties; "positive" and "negative" cannot both apply to the same number at the same time. This creates the real-world analog of P ∧ ¬P, showing why the statement is inherently contradictory.

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6) (P ∧ Q) ∧ (¬P ∧ Q) is a contradiction because it forces P and ¬P simultaneously.

Explanation

Any time an expression requires P and also requires ¬P as part of the same condition, it becomes unsatisfiable. No truth assignment can satisfy both simultaneously, so the expression is a contradiction.

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7) What is (P ∧ ¬P) ↔ Q equivalent to?

Explanation

In an equivalence X ↔ Q, the biconditional is true when X and Q share the same truth value. Here, X is P ∧ ¬P, which is always false, so the biconditional becomes false ↔ Q. This expression is true only if Q is also false.

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8) Negation of ¬(P ∨ ¬P) is _______.

Explanation

P ∨ ¬P is always true for every P. Negating this gives ¬(P ∨ ¬P), which is always false, and negating again restores the original always-true expression. Double negation of a tautology leaves it a tautology.

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9) (P ∧ ¬P) ∧ (Q ∧ ¬Q) is always a contradiction.

Explanation

When each part of an AND consists of a statement like P ∧ ¬P, each portion evaluates to false. A conjunction of two false values remains false in all truth assignments, leaving no way for the expression to ever evaluate as true.

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10) P ∧ (¬P → Q) simplifies to:

Explanation

If P is true, ¬P becomes false. An implication with a false antecedent (¬P→Q) is automatically true regardless of Q. This means the combined expression reduces to true ∧ P, which simplifies to P itself.

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11) Fill in: (P ∧ ¬P) → (Q ∧ ¬Q) always evaluates to _______.

Explanation

In implication logic, X→Y is true whenever X is false. Therefore, in an expression involving ¬P→Q, if ¬P is false (meaning P is true), the implication is trivially true. If ¬P is true (meaning P is false), the truth of the implication depends solely on Q. But the key rule is that a false antecedent guarantees a true implication.

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12) Which biconditional is a contradiction?

Explanation

For P ↔ ¬P to be true, P and ¬P would need to share the same truth value, which is impossible; one is always the opposite of the other. Therefore, the biconditional between them evaluates to false under every assignment, becoming a contradiction.

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13) The expression (P ∧ ¬P) ∧ Q is always a contradiction.

Explanation

If one part of a conjunction is always false (such as P ∧ ¬P), then combining it with any other statement using AND yields a result that can never be true. The contradiction dominates the conjunction.

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14) What is the result of ¬(P ∧ ¬P)?

Explanation

Negating the contradiction yields ¬(P ∧ ¬P), which is logically equivalent to P ∨ ¬P by De Morgan’s law. This disjunction is always true, since P must be either true or false, making the final expression a tautology.

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15) Negation of a contradiction P ∧ ¬P is _______.

Explanation

Because contradictions evaluate to false in every scenario, applying negation flips all those outcomes into true ones. Therefore, ¬(contradiction) is guaranteed to be true no matter what, forming a tautology.

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16) Which expression is equivalent to a contradiction?

Explanation

The first part demands that neither P nor Q be true, while the second demands that at least one must be true. These requirements are mutually exclusive, so the conjunction can never be satisfied, making it a contradiction.

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17) The statement 'The door is open and closed' is a contradiction.

Explanation

Statements that impose incompatible conditions mirror P ∧ ¬P in logic. Physical examples like “the door is open and closed at the same time” illustrate why impossibilities in logic are contradictions: they require simultaneous and conflicting states.

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18) P ∧ (Q → ¬P) is equivalent to:

Explanation

The expression combines P with Q→¬P. If Q is true, then ¬P must follow from the implication, but the conjunction requires P as well, resulting in P ∧ ¬P, which is impossible. Even if Q is false, the contradiction appears in all assignments that satisfy P, making the overall expression unsatisfiable.

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19) Fill in the equivalence: (P ↔ Q) ∧ (P ↔ ¬Q) is _______.

Explanation

If an expression contains both Q ↔ P and Q ↔ ¬P, then for these to hold together, Q would need the same truth value as P and simultaneously the opposite truth value of P. This can never happen, so no truth assignment satisfies both conditions at once.

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20) What is the negation of a contradiction?

Explanation

Since contradictions evaluate to false under every possible truth assignment, negating them makes the statement true in every scenario. This is the definition of a tautology—always true, no exceptions.

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What is (P ∧ ¬P) ∨ (P ∧ ¬P) equivalent to?
The expression P ∧ (P → ¬P) ∧ P is always a contradiction.
¬(P ∧ ¬P) ∨ Q simplifies to:
Fill in the result: Q ∧ (P ∧ ¬P) = _______.
Which statement is a contradiction?
(P ∧ Q) ∧ (¬P ∧ Q) is a contradiction because it forces P and...
What is (P ∧ ¬P) ↔ Q equivalent to?
Negation of ¬(P ∨ ¬P) is _______.
(P ∧ ¬P) ∧ (Q ∧ ¬Q) is always a contradiction.
P ∧ (¬P → Q) simplifies to:
Fill in: (P ∧ ¬P) → (Q ∧ ¬Q) always evaluates to _______.
Which biconditional is a contradiction?
The expression (P ∧ ¬P) ∧ Q is always a contradiction.
What is the result of ¬(P ∧ ¬P)?
Negation of a contradiction P ∧ ¬P is _______.
Which expression is equivalent to a contradiction?
The statement 'The door is open and closed' is a contradiction.
P ∧ (Q → ¬P) is equivalent to:
Fill in the equivalence: (P ↔ Q) ∧ (P ↔ ¬Q) is _______.
What is the negation of a contradiction?
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