Contradictions Quiz: Test Statement Consistency and Breakdown

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.