##  [Negation](/negation-0) 

 Definition

A unary logical connective that yields the opposite truth value of its operand formula; typically written ¬A and defined so that ¬A is true exactly when A is false (in classical propositional logic).

 

 

 

 

 

 





## Principle

Principle

Operates as a truth-function mapping: given the truth value of A, the truth value of ¬A is determined by a fixed rule (in classical logic the Boolean complement). The behaviour of negation can vary across logical systems (classical, intuitionistic, paraconsistent, modal).

 

 

 

 

 





## Demonstration

Demonstration

If p stands for 'It is raining' and v(p)=true, then ¬p represents 'It is not raining' and v(¬p)=false under classical valuation. Double negation: in classical logic ¬¬A is equivalent to A, so ¬¬p is true when p is true.

 

 

 

 

## Misapplication

Misapplication

Assuming classical double-negation elimination or De Morgan equivalences in non-classical systems where they fail (for example treating ¬¬A as equivalent to A in intuitionistic logic), or reading ¬ as simply 'absence of evidence' rather than explicit denial.

 

 

 

 

 





## Consequence

Consequence

Negation allows expression of denial and the construction of contrapositives and contradictions; its formal properties underpin proof techniques like proof by contradiction in classical systems (though this depends on the logic's acceptance of excluded middle and double negation rules).

 

 

 

 

## Reversal

Reversal

The reversal is affirmation (removing negation yields the positive formula); treating negation as affirmation or as a mere syntactic marker without semantic consequence inverts its role and collapses distinctions between truth and falsity.

 

 

 

 

 





## Boundary

Boundary

Refers to the object-language unary connective; excludes modal, epistemic, or paraconsistent operators that are named 'negation' but have distinct semantic clauses, and excludes meta-linguistic negations like 'it is not provable that...'.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension between classical negation (truth-functional complement), constructive/intuitionistic negation (related to provability of contradiction), and paraconsistent negation (which tolerates some contradictions without explosion).

 

 

 

 

 





## Synthesis

Synthesis

Negation is the unary connective that in a given logic maps a formula to its denial under the logic's semantics; its exact inferential and semantic properties depend on whether the logic is classical, constructive, or paraconsistent.