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P¬PP ∧ ¬P
FTF
TFF

This represents the simplest contradiction in logic: a proposition and its negation cannot both be true simultaneously. This form is sometimes called the law of non-contradiction and is a fundamental principle in classical logic. No matter what truth value is assigned to P, the formula evaluates to false because a statement cannot be both true and false at the same time.

Learn more:


What Is a Contradiction?
Contradictions in Proofs and Reasoning
P ∧ ¬P: The Law of Non-Contradiction
(P → Q) ∧ (P → ¬Q) ∧ P: Inconsistent Implications
(P ∨ Q) ∧ ¬P ∧ ¬Q: Denied Disjunction
P ↔ ¬P: Self-Contradictory Equivalence
(P ∧ Q) ∧ (¬P ∨ ¬Q): Conjunction Meets De Morgan
(P → Q) ∧ (Q → R) ∧ P ∧ ¬R: Broken Implication Chain
(P → Q) ∧ (Q → R) ∧ ¬R ∧ P: Reordered Chain Contradiction
(P ↔ Q) ∧ (P ↔ ¬Q): Conflicting Equivalences
(P ∧ Q) ∧ (P ∧ ¬Q): Asserting Q and ¬Q
(¬(P → Q)) ∧ (P ∧ Q): Negated Implication Conflict


What Is a Contradiction?

A contradiction is a compound proposition that is false under every possible assignment of truth values to its variables. No matter what combination of true and false you assign, the formula always evaluates to false. In a truth table, the final column contains only F values.

Contradictions are also called unsatisfiable formulas — there is no interpretation that makes them true. They are the logical opposite of tautologies: the negation of a contradiction is a tautology, and the negation of a tautology is a contradiction.

The simplest contradiction is P ∧ ¬P — a proposition and its negation conjoined. This expresses the law of non-contradiction, one of the three classical laws of thought. More complex contradictions arise from inconsistent sets of premises, broken implication chains, and conflicting equivalences.

Contradictions in Proofs and Reasoning

Contradictions are not just logical curiosities — they are a powerful proof tool. Proof by contradiction (reductio ad absurdum) works by assuming the negation of the desired conclusion and showing that this assumption leads to a contradiction. Since contradictions cannot be true, the assumption must be false, and the original statement must be true.

Contradictions also serve as a diagnostic tool for consistency. If a set of premises conjoined produces a contradiction, the premises are mutually inconsistent — they cannot all be true simultaneously. Identifying the source of the contradiction reveals which premises conflict.

In formal systems, the principle of explosion (ex falso quodlibet) states that from a contradiction, any statement can be derived. This makes contradictions maximally dangerous in a logical system: a single contradiction makes the entire system trivially provable and therefore useless.

P ∧ ¬P: The Law of Non-Contradiction

P ∧ ¬P is the simplest possible contradiction: it asserts that a proposition P is both true and false at the same time. The truth table has only two rows, and both evaluate to false.

When P is true, ¬P is false, so the conjunction is false. When P is false, ¬P is true, but P itself is false, so the conjunction is again false. There is no escape — the formula is unsatisfiable.

The law of non-contradiction (that P ∧ ¬P is always false) is one of the foundational principles of classical logic. Together with the law of excluded middle (P ∨ ¬P is always true) and the law of identity, it forms the three classical laws of thought attributed to Aristotle.

(P → Q) ∧ (P → ¬Q) ∧ P: Inconsistent Implications

This contradiction arises from asserting three mutually inconsistent claims: P implies Q, P implies ¬Q, and P is true.

When P is true, the first implication forces Q to be true. But the second implication forces Q to be false. Q cannot be both true and false, so no truth assignment satisfies all three conjuncts.

When P is false, both implications are vacuously true, but the third conjunct (P itself) is false, making the whole conjunction false. Either way, the formula is false.

This pattern illustrates a general principle: if a set of premises forces a variable to take contradictory values, the premises are inconsistent.

(P ∨ Q) ∧ ¬P ∧ ¬Q: Denied Disjunction

This contradiction asserts a disjunction (at least one of P or Q is true) while simultaneously denying both disjuncts (P is false and Q is false).

The disjunction P ∨ Q requires at least one of its components to be true. But ¬P and ¬Q together assert that both are false. These requirements are mutually exclusive — if both P and Q are false, the disjunction P ∨ Q is false, contradicting the first conjunct.

By De Morgan's law, ¬P ∧ ¬Q is equivalent to ¬(P ∨ Q). So the formula reduces to (P ∨ Q) ∧ ¬(P ∨ Q), which is the basic contradiction pattern X ∧ ¬X.

P ↔ ¬P: Self-Contradictory Equivalence

The biconditional P ↔ ¬P asserts that P and its negation have the same truth value. But by definition, P and ¬P always have opposite truth values — when one is true, the other is false.

When P is true, ¬P is false, and the biconditional T ↔ F evaluates to false. When P is false, ¬P is true, and F ↔ T evaluates to false. Both rows are false — this is a contradiction.

Compare this with the tautology P ↔ ¬¬P (double negation), where the two sides always agree. Here the two sides always disagree, producing the opposite result.

(P ∧ Q) ∧ (¬P ∨ ¬Q): Conjunction Meets De Morgan

This contradiction asserts that both P and Q are true (via P ∧ Q) while also asserting that at least one of them is false (via ¬P ∨ ¬Q).

By De Morgan's law, ¬P ∨ ¬Q is equivalent to ¬(P ∧ Q). So the formula is equivalent to (P ∧ Q) ∧ ¬(P ∧ Q) — the basic contradiction pattern X ∧ ¬X where X = P ∧ Q.

The truth table has four rows, and the result is false in all of them. In the only row where P ∧ Q is true (T, T), ¬P ∨ ¬Q is false. In all other rows, P ∧ Q is already false.

(P → Q) ∧ (Q → R) ∧ P ∧ ¬R: Broken Implication Chain

This contradiction constructs a chain of implications (P → Q → R) while asserting the start of the chain (P is true) and denying its end (R is false).

The chain forces a cascade: P is true, so Q must be true (by P → Q). Q is true, so R must be true (by Q → R). But the formula also asserts ¬R — R is false. This is irreconcilable.

The truth table has eight rows (three variables), and all evaluate to false. This pattern generalizes: any implication chain P → Q₁ → Q₂ → ··· → Qₙ combined with P ∧ ¬Qₙ is a contradiction, because the chain forces Qₙ to be true while ¬Qₙ asserts it is false.

(P → Q) ∧ (Q → R) ∧ ¬R ∧ P: Reordered Chain Contradiction

This formula is logically identical to the previous one — the conjuncts ¬R and P are simply reordered. Conjunction is commutative: the order of conjuncts does not affect the truth value.

The contradiction arises for exactly the same reason: the implication chain P → Q → R forces R to be true when P is true, but ¬R asserts R is false. Both formulas are included to illustrate that reordering conjuncts does not resolve inconsistency.

In practice, recognizing that two seemingly different formulas are logically equivalent (differing only in conjunct order) is an important skill in logical analysis and proof checking.

(P ↔ Q) ∧ (P ↔ ¬Q): Conflicting Equivalences

This contradiction asserts that P is equivalent to Q and also equivalent to ¬Q. If P ↔ Q holds, then P and Q have the same truth value. If P ↔ ¬Q also holds, then P and ¬Q have the same truth value. But Q and ¬Q always differ, so P cannot match both.

When P and Q are both true: P ↔ Q is true, but P ↔ ¬Q becomes T ↔ F = false. When P is true and Q is false: P ↔ Q becomes T ↔ F = false. Every row fails at least one biconditional.

This pattern generalizes: asserting equivalence with both a proposition and its negation is always contradictory.

(P ∧ Q) ∧ (P ∧ ¬Q): Asserting Q and ¬Q

This contradiction directly asserts Q and ¬Q simultaneously (under the assumption that P is true). The first conjunct says P and Q are both true. The second says P is true and Q is false.

Extracting Q from the first conjunct and ¬Q from the second gives Q ∧ ¬Q — the basic contradiction. The presence of P in both conjuncts is a red herring; the real conflict is between Q and ¬Q.

The formula can be rewritten as P ∧ (Q ∧ ¬Q) by factoring out P. Since Q ∧ ¬Q is always false, the entire conjunction is always false regardless of P.

(¬(P → Q)) ∧ (P ∧ Q): Negated Implication Conflict

This contradiction exploits the definition of material implication. The negation ¬(P → Q) is true only when P is true and Q is false (since P → Q ≡ ¬P ∨ Q, its negation is P ∧ ¬Q).

But the second conjunct P ∧ Q asserts that both P and Q are true. The first conjunct requires Q to be false; the second requires Q to be true. These conditions are mutually exclusive.

The formula reduces to (P ∧ ¬Q) ∧ (P ∧ Q) = P ∧ (¬Q ∧ Q) = P ∧ ⊥ = ⊥. The conflict between Q and ¬Q makes the formula unsatisfiable.