Contradiction

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 and Notation

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.

Example:

P∧¬P

(The law of non-contradiction: "P and not P" is always false.)

The notation commonly used for contradiction in logic is:

\bot

(The bottom symbol, also called "falsum" or "false constant").

Alternatively, a contradiction can be denoted explicitly as:

\vdash ¬P

which means "the negation of P is provable" when P is a tautology.

In some texts, contradictions are also expressed using equivalence sign like this:

P \equiv \bot

to explicitly state that a proposition

( P )

is always false (equivalent to false).

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
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

This states that it's false that a proposition can be neither true nor false, making it a contradiction.

Law of Non-Contradiction (direct form):

P∧¬P≡F

This states that a proposition cannot be 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

Negation of De Morgan's Laws:

¬(¬(P∨Q)≡(¬P∧¬Q))≡F

¬(¬(P∧Q)≡(¬P∨¬Q))≡F

These contradictions are the logical foundation of proof by contradiction methods, where assuming the negation of a true statement 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.

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.

However, some contradictions are simply valid 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.

name	expression	explanation
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) ∧ ¬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

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

Contradiction vs Tautology

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.

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.

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