Contradictions Truth Tables

P	¬P	P ∧ ¬P
F	T	F
T	F	F

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

Common Pitfalls

Contradictions FAQ

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.

Pitfall: Calling a contingent formula a contradiction

Seeing an F in the column of P → Q and calling it a contradiction stops too early: P → Q is F only at P = T, Q = F and T in the other three rows. A contradiction is F in every row; P → Q is contingent.

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

Fact: Logics that survive a contradiction

Paraconsistent logics, first built by Stanisław Jaśkowski in 1948, give up explosion: from P and ¬P they do not let an arbitrary Q follow. A contradiction then stays local instead of making everything provable, which suits reasoning from records that conflict.

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.

RULEPrinciple of explosion

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.

Pitfall: Treating P → Q and P → ¬Q as contradictory

Taking P → Q and P → ¬Q to contradict each other overlooks the rows with P = F, where both are T. Together they are equivalent to ¬P, not to a contradiction; the formula here is always F only because it also asserts P.

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.

Pitfall: Keeping the connective in De Morgan's law

Rewriting ¬P ∧ ¬Q as ¬(P ∧ Q) turns the formula into (P ∨ Q) ∧ ¬(P ∧ Q), which is T at P = T, Q = F, so it is not a contradiction. De Morgan's law flips the connective: ¬P ∧ ¬Q ≡ ¬(P ∨ Q).

Reducing a formula to the pattern X ∧ ¬X is a quick way to recognise a contradiction without writing out the whole truth table. Each step of the reduction has to be an equivalence, so that the new formula agrees with the old one in every row. Only then does the verdict for X ∧ ¬X carry over to the formula the reduction started from.

Reach X ∧ ¬X through equivalences alone, and the contradiction is proved without writing a single row.

The section's formula in its two forms. Compare which one shows the contradiction at a glance.

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Contradictions

Reducing to X ∧ ¬X

Reach X ∧ ¬X through equivalences alone, and the contradiction is proved without writing a single row.

Before

Disjunction with both parts denied

(P \lor Q) \land \neg P \land \neg Q

At least one of P or Q, yet P false and Q false

De Morgan's law

¬P ∧ ¬Q ≡ ¬(P ∨ Q)

After

Pattern X ∧ ¬X

(P \lor Q) \land \neg(P \lor Q)

With X = P ∨ Q: the basic contradiction pattern

Keeping the connective gives ¬(P ∧ Q) instead, and (P ∨ Q) ∧ ¬(P ∧ Q) is true at P = T, Q = F, so it is not a contradiction. De Morgan's law turns ∧ into ∨.

What changed

The two separate denials ¬P and ¬Q merged into one negation of the whole disjunction.

What stayed

The truth value in every row: each step is an equivalence.

(P ∨ Q) ∧ ¬P ∧ ¬Q reduced to X ∧ ¬X·/logic/truth-tables/contradictionsLearn Math Class

Because De Morgan's law is an equivalence, the two formulas agree in every row of the truth table, so the verdict for X ∧ ¬X carries over: the original formula is a contradiction too.

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).

Fact: Arrow chains skip the truth table

Formulas built only from arrows like P → Q, bare letters like P and denials like ¬R, a kind of Horn formula, need no 2ⁿ rows. Mark true whatever the arrows force from the bare letters: if a denied letter gets marked, the formula is a contradiction; if not, the marking itself is a satisfying row.

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.

One contradiction inside a conjunction sinks the whole conjunction, whatever else is conjoined beside it.

(¬(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).

Pitfall: Negating an implication into an implication

Writing ¬(P → Q) as P → ¬Q fails at P = F, Q = T: there ¬(P → Q) is F but P → ¬Q is T. The negation is a conjunction, ¬(P → Q) ≡ 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 ∧ ⊥ = ⊥, where ⊥ is the symbol for the proposition that is false outright — the mirror of ⊤, and the reason anything conjoined with it collapses. The mark is set out in contradiction notation. The conflict between Q and ¬Q makes the formula unsatisfiable.

The section's reduction, one equivalence at a time. Watch Q meet ¬Q.

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Contradictions

Reducing to ⊥

The first conjunct requires Q to be false, the second requires Q to be true. The reduction makes the conflict explicit.

Negated implicationBefore

\neg(P \to Q) \land (P \land Q)

↓Negate the implication— P → Q ≡ ¬P ∨ Q, so ¬(P → Q) ≡ P ∧ ¬Q

Two conjunctions

(P \land \neg Q) \land (P \land Q)

↓Regroup— P twice is just P; put ¬Q next to Q

Q against ¬Q

P \land (\neg Q \land Q)

↓A statement and its negation— ¬Q ∧ Q is false outright: ⊥

Conjoined with ⊥

P \land \bot

↓Anything ∧ ⊥ collapses— P ∧ ⊥ = ⊥

False outrightAfter

\bot

Writing ¬(P → Q) as P → ¬Q fails at P = F, Q = T: there ¬(P → Q) is F but P → ¬Q is T. The negation of an implication is a conjunction, P ∧ ¬Q.

(¬(P → Q)) ∧ (P ∧ Q) reduced to ⊥·/logic/truth-tables/contradictionsLearn Math Class

Every step is an equivalence, so the original formula is false in every row: it is unsatisfiable. The P that survives the regrouping makes no difference once it is conjoined with ⊥.

Common Pitfalls

Four slips account for many wrong answers when formulas are tested for contradiction. Each is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

Calling a contingent formula a contradiction

A formula is found to be false in one or two rows of its truth table, and it is called a contradiction. A contradiction is false in every row; a formula that is false in some rows and true in others is contingent.

Wrong working: P → Q is F at P = T, Q = F, so P → Q is a contradiction.

Right working: for (P, Q) equal to (T, T), (T, F), (F, T), (F, F), P → Q takes T, F, T, T. Three rows are T, so the formula is satisfiable and not a contradiction. Because of its F row it is not a tautology either: it is contingent.

Why it happens: the F row is the memorable one, and the three ways a formula can behave, always T, always F or mixed, get collapsed into two. "Not a tautology" is heard as "a contradiction".

How to check: read the whole final column. A contradiction has no T at all; a single T makes the formula satisfiable.

Treating P → Q and P → ¬Q as contradictory

Two implications with the same antecedent and opposite consequents look as if they cannot both hold, so their conjunction is taken to be a contradiction. They clash only when the antecedent is true.

Wrong working: P → Q says P gives Q, P → ¬Q says P gives ¬Q, and Q and ¬Q cannot both be true, so (P → Q) ∧ (P → ¬Q) is always F.

Right working: for (P, Q) equal to (T, T), (T, F), (F, T), (F, F), P → Q takes T, F, T, T and P → ¬Q takes F, T, T, T, so the conjunction takes F, F, T, T. It is true in both rows with P = F and is equivalent to ¬P. The formula in section 4 is a contradiction only because it adds the third conjunct P, which is F in exactly those rows.

Why it happens: in everyday speech, "if it rains, the ground is wet" and "if it rains, the ground is not wet" sound incompatible, because the listener pictures rain. On a dry day both are vacuously true.

How to check: test the rows where the shared antecedent is false. If the conjunction is T there, it is satisfiable; the two implications together only say that the antecedent is false.

Keeping the connective in De Morgan's law

In the reduction of (P ∨ Q) ∧ ¬P ∧ ¬Q to the pattern X ∧ ¬X, the part ¬P ∧ ¬Q is rewritten by pulling the negation outside, but the ∧ is kept. The rewritten formula is no longer equivalent to the original.

Wrong working: ¬P ∧ ¬Q ≡ ¬(P ∧ Q), so the formula becomes (P ∨ Q) ∧ ¬(P ∧ Q).

Right working: at P = T, Q = F, ¬P ∧ ¬Q is F ∧ T, which is F, while ¬(P ∧ Q) is ¬F, which is T, so the rewrite is not an equivalence. The wrong formula (P ∨ Q) ∧ ¬(P ∧ Q) is T in that row: it is the exclusive or, not a contradiction. De Morgan's law changes the connective: ¬P ∧ ¬Q ≡ ¬(P ∨ Q), and the formula becomes (P ∨ Q) ∧ ¬(P ∨ Q), which is F in every row.

Why it happens: the negation is moved out like a common factor taken out of a bracket, with the operation between the terms left as it was. In words, "neither P nor Q" gets confused with "not both P and Q".

How to check: test a row where P and Q differ. "Neither P nor Q" is F there, while "not both" is T, so a rewrite that turns one into the other changes the value.

Negating an implication into an implication

To negate "if P, then Q", the negation is moved onto the consequent and the arrow is kept, giving P → ¬Q. That is another implication, not the negation.

Wrong working: ¬(P → Q) ≡ P → ¬Q, so the formula in section 12 would become (P → ¬Q) ∧ (P ∧ Q).

Right working: at P = F, Q = T, P → Q is T, so ¬(P → Q) is F, but P → ¬Q is F → F, which is T. With material implication and De Morgan's law, ¬(P → Q) ≡ ¬(¬P ∨ Q) ≡ ¬¬P ∧ ¬Q ≡ P ∧ ¬Q, which is T only at P = T, Q = F.

Why it happens: "it is not the case that if P then Q" sounds like a claim about what happens when P holds, so the "if" is kept. The negation of a conditional describes the single row where the conditional fails.

How to check: the negation must be T exactly where P → Q is F, which is only the row P = T, Q = F. P → ¬Q is T in both rows with P = F, so it fails the test.

Contradictions FAQ

How do you identify a contradiction in a truth table?

A formula is a contradiction if the final column of its truth table contains only false values — every row, without exception. One false row among true ones only makes the formula contingent, not contradictory. Build the table for any candidate formula in the truth table generator and inspect the final column.Read more →

What is the difference between a contradiction and a tautology?

A tautology is true under every truth assignment; a contradiction is false under every truth assignment. In a truth table, a tautology's final column contains only T values and a contradiction's only F values. The two are exact opposites: negating a tautology yields a contradiction, and negating a contradiction yields a tautology.Read more →

Can inconsistent premises produce a contradiction?

Yes. When premises are mutually inconsistent — such as P → Q, P → ¬Q, and P asserted together — their conjunction is a contradiction: no truth assignment satisfies them all. This gives a practical consistency test: conjoin the premises and check the truth table. If every row is false, the premise set is inconsistent and at least two of them conflict.Read more →

What does it mean for a formula to be unsatisfiable?

An unsatisfiable formula has no truth assignment that makes it true — every row of its truth table is false. Unsatisfiable and contradiction name the same property. A satisfiable formula, by contrast, is true under at least one assignment; a formula that is true under all of them is a tautology.Read more →

What is the principle of explosion?

The principle of explosion (ex falso quodlibet) states that from a contradiction, anything can be derived: once both P and ¬P are accepted, every statement whatever becomes provable. This is why contradictions are fatal to a formal system — a single one makes the system trivial, since it can no longer distinguish true claims from false ones.Read more →