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Contradiction


Definition
Contradiction Notation
Contradictions as Logic Laws
Contradictions that are not laws
Contradiction vs Tautology
Common Pitfalls
Contradiction FAQ



Introduction

Contradictions form a cornerstone concept in propositional logic, representing logical impossibility. A contradiction is a formula that evaluates to false under all possible interpretations of its variables. The simplest example is P ∧ ¬P, which states that a proposition and its negation are simultaneously true – an impossible scenario.

Unlike ordinary propositions that may be true or false depending on circumstances, contradictions are guaranteed to be false, providing absolute certainty in logical analysis. This property makes contradictions invaluable in proof techniques, particularly in reductio ad absurdum (proof by contradiction), where we demonstrate that assuming the opposite of what we want to prove leads to a logical impossibility.

This page explores contradictions in detail, examining their properties, relationship to tautologies, and applications in logical reasoning and formal proofs.


Definition

Some statements cannot be true, however the world turns out.
"The light is on and the light is not on" needs no checking: whatever the light is doing, the sentence fails. It is like a contract whose second clause cancels the first — it defeats itself, and no facts can rescue it. Logic needs a name for formulas of this kind.
The definition
A contradiction is a formula that is false under every assignment of truth values to its variables.
Example
                                        P∧¬PP∧¬P
(The law of non-contradiction: "P and not P" is always false.)
So P∧¬PP \land \neg P is a contradiction: it is false whether PP is true or false.
Being a contradiction is a property of a formula's shape, not of its content.
A contradiction is a proposition that is always false, regardless of the truth values of its components. In propositional logic, a contradiction is a formula that evaluates to false under all possible truth assignments.
AB
P ∧ ¬P covers no case

Let circle A stand for the cases where P is true. P ∧ ¬P asks for cases inside A and outside A at once, and there are none, so nothing is shaded: the formula is false under every assignment, which is what makes it a contradiction. Check other formulas on the Venn truth-table explorer.

Its negation, ¬(P ∧ ¬P), shades everything: the negation of a contradiction is a tautology.
Truth tables show it row by row: P ∧ ¬P in two rows, a longer formula in four.
To decide whether a formula is a contradiction, build its full truth table: a formula with nn variables has 2n2^n rows, one for each assignment of truth values. The verdict is read from the column under the main connective. P∧¬PP \land \neg P has a single variable and two rows, and its column reads F, F.
P¬PP ∧ ¬P
TFF
FTF
With two variables there are four rows to check. In (P∨Q)∧¬P∧¬Q(P \lor Q) \land \neg P \land \neg Q, the first part needs at least one of PP, QQ true and the second needs both false, so every row fails.
PQP ∨ Q¬P ∧ ¬Q(P ∨ Q) ∧ ¬P ∧ ¬Q
TTTFF
TFTFF
FTTFF
FFFTF
The definition promised false in every row — so one false row proves nothing.
Both tables kept the promise: the last column is F all the way down. A formula that is F in some rows and T in others is not a contradiction, however many F's it shows.
Pitfall: Judging from a single row
Calling P∧QP \land Q a contradiction because it is F at P=TP = T, Q=FQ = F ignores the row P=TP = T, Q=TQ = T, where it is T. A contradiction is F in every row; P∧QP \land Q is a contingency.
Scope: contradictions are one of three kinds of formula, and most formulas are none of the extremes.
A formula is a tautology (T in every row), a contradiction (F in every row) or a contingency (some of each). Negation swaps the first two, turning any contradiction into a tautology and back. Some contradictions are negated laws of logic and some are not, as the section on contradictions that are not laws shows.
The rest of the page covers the notation, contradictions as negated laws, other contradictions, and how contradictions mirror tautologies.

Contradiction Notation

Notation

Contradiction Notation

The mirror of the tautology marks: a constant for falsehood, a turnstile carrying a negation, and the equivalence spelling. Everything here is the tautology notation reflected — same grammar, opposite pole.
⊤\top, ⊢\vdash and ⊨\vDash are owned by the tautology page; the connectives and letters by propositional logic notation; ≡\equiv by equivalence notation.
⊥\bot
bottom; falsum; the false constant
The false constant — ⊤\top turned upside down, and the inversion is the whole idea: ⊥\bot denotes falsehood itself, a formula false under every assignment, as Definition above establishes with P∧¬PP \wedge \neg P.
CasesInside formulas it absorbs and annihilates in mirror image: P∧⊥P \wedge \bot is ⊥\bot, P∨⊥P \vee \bot is PP — the laws reflect the ⊤\top rules exactly. Proof by contradiction is the practical payoff: derive ⊥\bot and the assumption falls.
Also writtenFF or F\mathbf{F} in truth tables and elementary texts — the tables on this page use it; some systems write 00, borrowing Boolean algebra's values.
Same glyph elsewhereThe identical glyph is the perpendicular sign ⊥\perp of geometry and linear algebra, where it relates two objects rather than naming a constant — same shape, different arity.
⊢¬P\vdash \neg P
the negation of P is provable
The turnstile applied to a negation: ⊢¬P\vdash \neg P certifies that ¬P\neg P is derivable — which is the proof-side way of declaring PP contradictory, since a formula false in every case has a provable negation.
CasesThe mirror is exact — ⊢P\vdash P marks a tautology, ⊢¬P\vdash \neg P a contradiction; negation is what carries a claim from one pole to the other.
Do not confuse⊬P\nvdash P. "Not provable" is a much weaker statement than "the negation is provable" — the crossed turnstile denies a derivation exists; ⊢¬P\vdash \neg P supplies one for the opposite formula.
P≡⊥P \equiv \bot
P is equivalent to the false constant
The equivalence spelling: rather than call PP a contradiction, assert that it agrees with ⊥\bot in every case — the same construction the tautology page makes with ⊤\top.
CasesThe laws of Contradictions as Logic Laws below are written this way, in the FF dialect: P∧¬P≡FP \wedge \neg P \equiv F; a simplification chain ending at ⊥\bot has proved a contradiction.
Do not confuseA claim that PP is false right now. Like its twin, ≡\equiv delivers a verdict over all cases, not a truth assignment in one — the sentence "PP happens to be false" is a different, weaker statement.
DownloadPNG2×PDF
Notation · Cheat sheet
Propositional Logic Notation at a Glance
The marks from the notation cards across the propositional-logic pages: how each is read and what it means. Every group links to the full card on its page.
Syntax
P,Q,RP, Q, R · P1,P2P_1, P_2
proposition letters
Capital letters name atomic propositions; subscripts extend the supply.
¬  ∧  ∨  →  ↔\neg \; \wedge \; \vee \; \to \; \leftrightarrow
not; and; or; implies; if and only if
The five connectives: negation, conjunction, disjunction, implication, biconditional.
φ,ψ\varphi, \psi
phi and psi
Placeholders for whole formulas in the formation rules.
(φ∧ψ)(\varphi \wedge \psi)
parenthesised conjunction
Every binary connective brings its own parentheses; no precedence at this level.
Implication
p→qp \to q
if p then q; p implies q
The conditional connective: antecedent pp, consequent qq.
p⇒qp \Rightarrow q
p implies q, in the meta sense
The double shaft claims the implication holds in every case; older texts write p⊃qp \supset q.
p→qp \to q in words
the English phrasings of one arrow
"if pp, then qq"; "pp is sufficient for qq"; "qq is necessary for pp"; "pp only if qq".
Equivalences
P≡QP \equiv Q
P is logically equivalent to Q
A claim about formulas: PP and QQ agree in every case.
P⇔QP \Leftrightarrow Q
P if and only if Q, in the meta sense
The double-shafted spelling of the same claim.
P=QP = Q
P equals Q
Boolean algebra’s spelling, where formulas denote values in {0,1}\{0, 1\}.
Tautology
⊤\top
top; verum; the true constant
Truth itself: a formula true under every assignment.
⊢P\vdash P
turnstile P; P is provable
The single turnstile: a derivation of PP exists in the proof system.
⊨P\vDash P
double turnstile P; P is valid
The double turnstile: PP is true under every assignment.
P≡⊤P \equiv \top
P is equivalent to the true constant
The third way to say that PP is a tautology.
ContradictionThis page
⊥\bot
bottom; falsum; the false constant
Falsehood itself: a formula false under every assignment.
⊢¬P\vdash \neg P
the negation of P is provable
The proof-side way to declare PP a contradiction.
P≡⊥P \equiv \bot
P is equivalent to the false constant
The equivalence spelling of "contradiction".
Same marks, different job
→\to vs ⇒\RightarrowA connective inside a formula; a claim about formulas.
↔\leftrightarrow vs ≡\equivThe biconditional connective; logical equivalence in the metalanguage.
⊢\vdash vs ⊨\vDashProvable by the rules; true in every case.
⊃\supsetImplication in older logic texts; superset in set theory.
⊥\botThe false constant here; perpendicular in geometry.
Logic symbols tablePropositional logic
Propositional Logic Notation at a Glance·/logic/propositional-logicLearn Math Class

Contradictions as Logic Laws

As discussed in previous section, a contradiction is a proposition that always evaluates to false. Some contradictions are so fundamental that they represent core principles of propositional logic.

## Negations of Logical Laws as Contradictions


COROLLARY
When we negate certain fundamental laws of propositional logic, we obtain expressions that are always false, making them contradictions.

Examples:

Negation of the Law of Excluded Middle:
¬(P∨¬P)≡F¬(P∨¬P)≡F
This states that a proposition is neither true nor false, which is always false, making it a contradiction.

Negation of the Law of Non-Contradiction:
P∧¬P≡FP∧¬P≡F
This states that a proposition is both true and false simultaneously, which is always false.

## Contradictory Forms of Equivalences
We can also derive contradictions by asserting the negation of logical equivalences:

Negation of Commutative Laws:
¬((P∨Q)↔(Q∨P))≡F¬((P∨Q)↔(Q∨P))≡F
¬((P∧Q)↔(Q∧P))≡F¬((P∧Q)↔(Q∧P))≡F

Negation of De Morgan's Laws:
¬(¬(P∨Q)↔(¬P∧¬Q))≡F¬(¬(P∨Q)↔(¬P∧¬Q))≡F
¬(¬(P∧Q)↔(¬P∨¬Q))≡F¬(¬(P∧Q)↔(¬P∨¬Q))≡F

Pitfall: Pushing the negation in without flipping the connective
Rewriting ¬(P∨¬P)\neg(P \lor \neg P) as ¬P∨¬¬P\neg P \lor \neg\neg P gives ¬P∨P\neg P \lor P, which is T in both rows, a tautology. De Morgan's law turns the ∨\lor into ∧\land: ¬P∧P\neg P \land P, which is F in both rows.
In each of these formulas the negation sign stands in front of a parenthesis, so its scope is the whole law inside it. There are two ways to confirm that the result is a contradiction. One is to evaluate the law first and then negate its entire column. The other is to move the negation inward step by step, with De Morgan's laws and double negation, until it reaches the single letters. Both routes must end with F in every row.

These contradictions are the logical foundation of proof by contradiction methods, where assuming the negation of the statement to be proved leads to a contradiction, thereby validating the original statement.

Visit corresponding page to learn more about propositional logic laws.

Use this tool to evaluate truth tables.

Negate any law of logic and the result is a formula that no assignment can make true.

Contradictions that are not laws

While some contradictions are negations of fundamental logical laws, not all contradictions represent the negation of logical laws. A law in logic is a fundamental principle that defines how logical operations behave, often used in formal proofs and reasoning systems.

Fact: Twenty-four entries, one formula
P∧¬PP \land \neg P and Q∧¬QQ \land \neg Q share no letter, yet they are equivalent: in all four rows of (P,Q)(P, Q) both columns read F. Up to equivalence, all 24 entries of the table below are one and the same formula, ⊥\bot.
However, some contradictions are simply well-formed logical statements that always evaluate to false without being the direct negation of a fundamental principle.

These non-law contradictions may still be useful in proofs, particularly in proof by contradiction methods (reductio ad absurdum), but they do not represent the negation of core logical rules like De Morgan's Laws or the Law of Excluded Middle. Instead, they are often the result of specific logical constructions or transformations that yield contradictory results.

nameexpressionexplanation
Simple Contradiction
P ∧ ¬P
A proposition and its negation cannot both be true
Multiple Contradiction
(P ∧ Q) ∧ ¬(P ∧ Q)
A compound proposition and its negation cannot both be true
Implication Contradiction
(P → Q) ∧ (P ∧ ¬Q)
Cannot have P implying Q while P is true and Q is false
Biconditional Contradiction
(P ↔ Q) ∧ (P ∧ ¬Q)
Cannot have P equivalent to Q while P is true and Q is false
Disjunction Contradiction
¬(P ∨ Q) ∧ (P ∨ Q)
A disjunction and its negation cannot both be true
Exclusive Disjunction Contradiction
(P ⊕ Q) ∧ (P ↔ Q)
P and Q cannot be both different and the same
Triple Contradiction
P ∧ ¬P ∧ Q
Adding propositions to a contradiction still yields a contradiction
Negated Tautology
¬(P → P)
The negation of a self-implication tautology
Material Implication Contradiction
(P → Q) ∧ ¬(¬P ∨ Q)
Contradicts the material implication equivalence
Syllogism Contradiction
((P → Q) ∧ (Q → R)) ∧ (P ∧ ¬R)
Contradicts the transitive property of implication
Distributive Law Contradiction
¬((P ∧ (Q ∨ R)) ↔ ((P ∧ Q) ∨ (P ∧ R)))
Negation of the distributive property of conjunction over disjunction
Absorption Law Contradiction
¬((P ∨ (P ∧ Q)) ↔ P)
Negation of the absorption property
Double Negation Contradiction
¬(¬¬P ↔ P)
Negation of the double negation equivalence
Contraposition Contradiction
¬((P → Q) ↔ (¬Q → ¬P))
Negation of the contrapositive equivalence
Material Equivalence Contradiction
¬((P ↔ Q) ↔ ((P → Q) ∧ (Q → P)))
Negation of the definition of the biconditional
Exportation Law Contradiction
¬(((P ∧ Q) → R) ↔ (P → (Q → R)))
Negation of the exportation equivalence
Self-Contradiction with Implication
(P → Q) ∧ (P → ¬Q) ∧ P
P implies contradictory outcomes while P is true
Conjunction-Disjunction Contradiction
(P ∧ Q) ∧ ¬(P ∨ Q)
Cannot have a conjunction be true while its disjunction is false
Tautology-Contradiction Implication
((P ∨ ¬P) → (Q ∧ ¬Q))
A tautology cannot imply a contradiction
Vacuous Truth Contradiction
¬P ∧ ¬(P → Q)
Contradicts the principle of vacuous truth for implication
Biconditional Chain Contradiction
(P ↔ Q) ∧ (Q ↔ R) ∧ ¬(P ↔ R)
Contradicts the transitivity of the biconditional
Disjunctive Syllogism Contradiction
(P ∨ Q) ∧ ¬P ∧ ¬Q
Contradicts the disjunctive syllogism principle
Hypothetical Syllogism Contradiction
(P → Q) ∧ (Q → R) ∧ (P ∧ ¬R)
Contradicts the hypothetical syllogism principle
Exclusive Or Contradiction
(P ⊕ Q) ∧ ¬(P ∨ Q)
Exclusive OR requires at least one proposition to be true
Pitfall: Joining a contradiction with or
Treating (P∧¬P)∨Q(P \land \neg P) \lor Q as a contradiction, like the Triple Contradiction, fails at Q=TQ = T, where the formula is T. Only ∧\land keeps a contradiction false; (P∧¬P)∨Q(P \land \neg P) \lor Q is equivalent to QQ.
Several entries of the table build a larger contradiction from a smaller one, as the Triple Contradiction attaches QQ to P∧¬PP \land \neg P. The connective that does the attaching decides the outcome. A conjunction is true only in the rows where every part is true, so a part that is false in every row keeps the whole conjunction false in every row, whatever the other parts are.

Use this tool to generate truth tables dynamically and evaluate these contradictions.

Contradiction vs Tautology

Fact: From a contradiction, anything follows
(P∧¬P)→Q(P \land \neg P) \to Q is a tautology whatever QQ says, even "the moon is made of cheese": its left side is never true, so no row can make the implication false. This is the principle of explosion, ex falso quodlibet.
In propositional logic, tautologies and contradictions represent opposite ends of logical certainty.
While tautology is a logical formula that always evaluates to true, no matter what truth values are assigned to its variables, a contradiction, in contrast, is always false regardless of its variables' values.
These concepts are perfect mirrors of each other - applying negation to a tautology produces a contradiction, and negating a contradiction creates a tautology.
This relationship is not just a curiosity but forms the foundation of logical reasoning. Tautologies tell us what must be true in all possible worlds, while contradictions show us what cannot be true under any circumstances. Together, they establish the boundaries of logical possibility and impossibility, providing the framework for all logical deduction in propositional logic.
negation (¬) negation (¬) Tautology Contradiction
Pitfall: Negating a formula that is not a tautology
Taking ¬(P→Q)\neg(P \to Q) for a contradiction because P→QP \to Q is not a tautology fails at P=TP = T, Q=FQ = F, where ¬(P→Q)\neg(P \to Q) is T. A contingency negates to a contingency: ¬(P→Q)\neg(P \to Q) is T in one row and F in the other three.
These two related and opposite concepts exist in perfect duality through negation as shown in the diagram – the negation of any tautology produces a contradiction, and negating any contradiction creates a tautology. This relationship reveals a fundamental symmetry in logical reasoning.

This duality plays a crucial role in logical analysis. Tautologies provide the foundation for valid arguments and proofs, as they represent statements that must necessarily be true. Contradictions enable powerful proof techniques like reductio ad absurdum, where we disprove statements by showing they lead to contradictions. Together, they establish the boundary conditions of logical reasoning – what must be true and what cannot be true – providing the fixed points around which all logical deduction revolves in propositional logic.

Common Pitfalls

Four slips account for many wrong answers about contradictions. 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.
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1Judging from a single row
A formula is evaluated in one assignment, comes out false there, and is declared a contradiction. A contradiction must be false in every assignment; a formula that is false in some rows and true in others is a contingency.
Start
At P=TP = T, Q=FQ = F the formula P∧QP \land Q is T∧FT \land F, which is F.
Wrong turn
Error here
So P∧QP \land Q is called a contradiction.
Right path
For (P,Q)(P, Q) equal to (T,T)(T, T), (T,F)(T, F), (F,T)(F, T), (F,F)(F, F), the formula P∧QP \land Q takes the values T, F, F, F. The first row is T, so P∧QP \land Q is satisfiable: it is a contingency, not a contradiction.
Why it happensAn ordinary false statement and a contradiction both come out F, and the checking stops at the first F. The word "every" in the definition is easy to skip.
How to checkLook for one row where the formula is T. If such a row exists, the formula is not a contradiction.
2Pushing the negation in without flipping the connective
To simplify a negated law, the negation is moved inside the parentheses and applied to each part, but the connective between the parts is kept. De Morgan's laws require the connective to change as well: ∨\lor becomes ∧\land, and ∧\land becomes ∨\lor.
Wrong turn
Error here
¬(P∨¬P)\neg(P \lor \neg P) becomes ¬P∨¬¬P\neg P \lor \neg\neg P, which is ¬P∨P\neg P \lor P. At P=TP = T this is F∨TF \lor T, which is T, and at P=FP = F it is T∨FT \lor F, which is T. The negated law of excluded middle comes out as a tautology.
Right path
¬(P∨¬P)≡¬P∧¬¬P≡¬P∧P\neg(P \lor \neg P) \equiv \neg P \land \neg\neg P \equiv \neg P \land P. At P=TP = T this is F∧TF \land T, which is F, and at P=FP = F it is T∧FT \land F, which is F. The formula is a contradiction, as the section states.
Why it happensNegating each part looks like the whole job, much as a minus sign is multiplied into a bracket in algebra. The change of connective has no counterpart there, so it is forgotten.
How to checkA negation must have the opposite value of the original in every row. At P=TP = T, P∨¬PP \lor \neg P is T, so its negation must be F there; ¬P∨P\neg P \lor P is T at P=TP = T, so it cannot be the negation.
3Joining a contradiction with or
The rule that a contradiction stays a contradiction when more letters are attached is applied to every connective. It holds for ∧\land only. A disjunction is true as soon as one part is true, so the attached part can make the whole formula true.
Wrong turn
Error here
P∧¬PP \land \neg P is a contradiction, and the Triple Contradiction P∧¬P∧QP \land \neg P \land Q shows that attaching QQ keeps it one. So (P∧¬P)∨Q(P \land \neg P) \lor Q is called a contradiction too.
Right path
P∧¬PP \land \neg P is F in every row, so (P∧¬P)∨Q(P \land \neg P) \lor Q has the same value as F∨QF \lor Q, which is the value of QQ. At Q=TQ = T the formula is T, so it is a contingency, equivalent to QQ.
Why it happensThe table shows the case with ∧\land, and the pattern is carried over without checking which connective does the attaching. The identity law P∨⊥≡PP \lor \bot \equiv P and the domination law P∧⊥≡⊥P \land \bot \equiv \bot are easy to mix up.
How to checkReplace the contradictory part by F and simplify. F∧QF \land Q is F in every row; F∨QF \lor Q is just QQ.
4Negating a formula that is not a tautology
The fact that negating a tautology gives a contradiction is stretched to every formula that is not a tautology. It applies to tautologies only: a formula that is neither a tautology nor a contradiction is a contingency, and so is its negation.
Start
P→QP \to Q is F at P=TP = T, Q=FQ = F, so it is not a tautology.
Wrong turn
Error here
Therefore ¬(P→Q)\neg(P \to Q) is called a contradiction.
Right path
For (P,Q)(P, Q) equal to (T,T)(T, T), (T,F)(T, F), (F,T)(F, T), (F,F)(F, F), the formula P→QP \to Q takes the values T, F, T, T, and ¬(P→Q)\neg(P \to Q) takes the values F, T, F, F. The row P=TP = T, Q=FQ = F makes the negation T, so it is a contingency, equivalent to P∧¬QP \land \neg Q.
Why it happensTautologies and contradictions are learned as a pair, and every formula gets sorted into one of the two. The third group, the contingencies, drops out of view.
How to checkNegation flips every entry of the column. A contradiction needs a column of F only, so the original column must be T only; if it holds even one F, the negation holds a T.
Common pitfalls with contradictions·/logic/propositional-logic/semantics/contradictionLearn Math Class
Each slip calls a formula always false on too little evidence, or keeps it false after a change that frees it. A contradiction is F in every row, so one F row proves nothing; moving a negation inside turns ∨\lor into ∧\land; joining a contradiction to QQ with ∨\lor leaves just QQ; and the negation of a formula that is not a tautology still has a T somewhere. Looking for a single T row, or replacing the contradictory part by F, catches each slip.

Contradiction FAQ

What does the symbol ⊥ mean in logic?

+
⊥ (read 'bottom' or 'falsum') is the constant for falsehood: a formula false under every assignment, the upside-down twin of ⊤. Truth tables often write it F or 0. Inside formulas it absorbs conjunctions and vanishes from disjunctions: P ∧ ⊥ is ⊥, while P ∨ ⊥ is P. The same glyph also serves as the perpendicularity sign in geometry — one shape, two unrelated jobs.Read more →

How does proof by contradiction work?

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Proof by contradiction (reductio ad absurdum) establishes a statement by assuming its negation and deriving something impossible. Assume ¬P, reason until a contradiction such as Q ∧ ¬Q appears, and conclude that the assumption was untenable — so P holds. The method works because a contradiction is false under every assignment: any assumption that forces one cannot be true.Read more →

Is every false statement a contradiction?

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No. An ordinary false statement is false under the particular circumstances at hand but could be true under others — its truth table contains both values. A contradiction is false under every possible assignment: no circumstances whatever make it true. 'It is raining' may happen to be false; 'it is raining and it is not raining' cannot be true at all. Test any formula in the truth table generator.Read more →

Are all contradictions equivalent to each other?

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Yes. Two formulas are equivalent when their columns agree in every row, and every contradiction has a column of F only. So P∧¬PP \land \neg P and Q∧¬QQ \land \neg Q are equivalent even though they share no letter: in all four rows of (P,Q)(P, Q) both read F. Up to equivalence there is only one contradiction, the constant ⊥\bot.Read more →

Does joining a contradiction with another statement keep it a contradiction?

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Only when the join is AND. P∧¬P∧QP \land \neg P \land Q is a contradiction, because a conjunction with a part that is false in every row is false in every row. With OR it fails: (P∧¬P)∨Q(P \land \neg P) \lor Q has the value of F∨QF \lor Q, which is just QQ, so it is T whenever QQ is T. The formula is a contingency, equivalent to QQ.Read more →