# Negative formula

A *negative formula* is a formula *N* such that (thus a algebra for the monad ). As a consequence *N* and are equivalent.

A formula *N* is negative if and only if is positive.

## Negative connectives

A connective *c* of arity *n* is *negative* if for any negative formulas *N*_{1},...,*N*_{n}, is negative.

**Proposition** (Negative connectives)

, , , , and are negative connectives.

*Proof.*
This is equivalent to the fact that , , , , and are positive connectives.

More generally, is negative for any formula *A*.

The notion of negative connective is related with but different from the notion of synchronous connective.

## Generalized structural rules

Negative formulas admit generalized right structural rules corresponding to a structure of -monoid: and . The following rule is derivable:

*Proof.*
This is equivalent to the generalized left structural rules for positive formulas.

Negative formulas are also acceptable in the context of the promotion rule. The following rule is derivable:

*Proof.*
This is equivalent to the possibility of having positive formulas in the left-hand side context of the promotion rule.