Contradictions, Tautologies, and Satisfiability Quiz

The expression P ∧ ¬P is already a contradiction because it requires P to be true and not true at the same time. Once a conjunction contains a false component, the entire conjunction becomes false regardless of any additional components. Thus (P ∧ ¬P) ∧ Q becomes False ∧ Q, which is false for every truth value of Q. Because no assignment can make the expression true, it is equivalent to a contradiction.

Begin with the expression P ∧ ¬P, which is a contradiction. Taking the negation ¬(P ∧ ¬P) and applying De Morgan’s Law produces ¬P ∨ ¬¬P. The double negation simplifies ¬¬P to P, leaving ¬P ∨ P, which is exactly the same as P ∨ ¬P. This expression is always true regardless of the truth value of P, and therefore the negation of a contradiction yields a tautology.

A contradiction describes a situation that cannot logically occur. The statement that the light is on and off simultaneously requires a condition and its negation to be true at the same time. Since these states exclude each other, the statement can never be true. The other options express either tautologies or conditionals that may be true in realistic scenarios. Only the assertion of simultaneous on and off states is inherently impossible, making it a contradiction.

The first part, P ↔ Q, states that P and Q share the same truth value. The second part, P ↔ ¬Q, states that P is the opposite truth value of Q. For both of these to hold at once, Q would need to be both equal to P and opposite to P, which is impossible. When expanded, the expression demands Q ∧ ¬Q or ¬Q ∧ Q at some stage, which cannot occur. Therefore the conjunction cannot be satisfied by any truth assignment and is a contradiction.

Expressions of the form X ∧ ¬X are always contradictions because they require a statement and its negation to be true simultaneously. Choices A, B, and D all directly contain such a structure. The remaining expression (P → Q) ∧ (¬P → Q) is different: whenever Q is true, both implications are automatically true regardless of the value of P. This means the expression is satisfiable and therefore not a contradiction. It is logically equivalent to Q, making it true whenever Q is true.

The part ¬P ∧ ¬Q requires P and Q to both be false. However, the part P ∨ Q requires at least one of them to be true. These conditions are incompatible: P ∨ Q cannot be true at the same time that both P and Q are false. No truth assignment satisfies both conditions simultaneously. Therefore the entire conjunction is always false, making it a contradiction.

The expression ¬(P ∧ ¬P) negates a contradiction. The negation of a negation simply restores the original expression: negating ¬(P ∧ ¬P) results in P ∧ ¬P. There is no need for further simplification. The final expression is again a contradiction because it requires P to be true and false at the same time. Thus the negation of ¬(P ∧ ¬P) is P ∧ ¬P.

The expression (P ∧ Q) ∧ ¬P requires three conditions to hold simultaneously: P must be true, Q must be true, and P must be false. Since P cannot be both true and false at the same time, this expression can never be satisfied. No matter what truth values we assign to P and Q, the expression will always evaluate to false. Choice B simplifies to ¬P ∨ Q (contingent), choice C simplifies to Q ∧ ¬P (contingent), and choice D simplifies to ¬P ∨ P ∨ Q, which simplifies further to a tautology. Only choice A creates an impossible condition by explicitly requiring P and ¬P simultaneously within a conjunction.

P ∨ ¬P is a tautology because it is always true. Negating a tautology yields a contradiction, so ¬(P ∨ ¬P) is always false. A conjunction with a false component is always false regardless of the truth value of the other component. Thus ¬(P ∨ ¬P) ∧ Q is false for all truth assignments and is therefore a contradiction.

The door cannot be simultaneously open and closed, so the statement asserts two mutually exclusive conditions. Because the conditions cannot be satisfied together, the statement is always false. The first choice is a tautology because the door must be in one of the two states. The remaining choices are conditional forms that may be true or false depending on circumstances. Only the assertion that the door is both open and closed creates an impossible situation, making it a contradiction.

To evaluate P ∧ (Q → ¬P), we must analyze the conditional Q → ¬P under different truth assignments. When P is true, ¬P is false. The conditional Q → ¬P is equivalent to ¬Q ∨ ¬P by the material conditional definition. Substituting, we get P ∧ (¬Q ∨ ¬P). Distributing P over the disjunction yields (P ∧ ¬Q) ∨ (P ∧ ¬P). Since P ∧ ¬P is a contradiction (always false), this simplifies to (P ∧ ¬Q) ∨ False, which equals P ∧ ¬Q. We can verify with a truth table: the expression is true only when P is true and Q is false. This makes it a contingent statement equivalent to P ∧ ¬Q, not a contradiction, because it can be satisfied under at least one truth assignment.

To evaluate this expression, we must determine when both components can be true simultaneously. The component ¬P ∧ Q requires that P is false and Q is true. If these conditions hold, let us check whether P ∨ Q is also true: when P is false and Q is true, P ∨ Q evaluates to False ∨ True, which equals True. Therefore, both parts of the conjunction can be satisfied when P is false and Q is true, meaning the entire expression evaluates to true under this assignment. Under all other truth assignments, at least one component of the conjunction fails. When we simplify, the expression (P ∨ Q) ∧ (¬P ∧ Q) is logically equivalent to ¬P ∧ Q. This is a contingent statement, not a contradiction, because there exists at least one truth assignment that makes it true.

Logical statements are categorized by their truth tables. A tautology is true in every row (always true). A contingency is true in at least one row and false in at least one row (depends on the input). A contradiction is unique because it yields a value of False for every single possible assignment of truth values to its variables. Therefore, the defining feature is that the final column of its truth table contains only False.

Applying De Morgan’s Law to ¬(P ∧ ¬P) yields ¬P ∨ ¬¬P. Simplifying the double negation gives ¬P ∨ P, which is a tautology. Since P ∧ ¬P is always false, its negation must be always true. Therefore the negation of the contradiction P ∧ ¬P is the tautology P ∨ ¬P.

Each biconditional places a different requirement on the relationships between P and Q. The first requires P and Q to have the same truth value. The second requires P and Q to have opposite truth values. The third reverses the relation again. These conditions cannot all be satisfied simultaneously. At least two of them will disagree for any assignment of P and Q, producing an unavoidable contradiction. No truth assignment satisfies all three biconditionals at once.