# Positive formula

(New page: A ''positive formula'' is a formula <math>P</math> such that <math>P\limp\oc P</math> (thus a coalgebra for the comonad <math>\oc</math>). A...) |
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− | A ''positive formula'' is a formula <math>P</math> such that <math>P\limp\oc P</math> (thus a [[Wikipedia:F-coalgebra|coalgebra]] for the [[Wikipedia:Comonad|comonad]] <math>\oc</math>). As a consequence <math>P</math> and <math>\oc P</math> are [[Sequent calculus#Equivalences and definability|equivalent]]. |
+ | A ''positive formula'' is a formula <math>P</math> such that <math>P\limp\oc P</math> (thus a [[Wikipedia:F-coalgebra|coalgebra]] for the [[Wikipedia:Comonad|comonad]] <math>\oc</math>). As a consequence <math>P</math> and <math>\oc P</math> are [[Sequent calculus#Equivalences|equivalent]]. |

== Positive connectives == |
== Positive connectives == |

## Revision as of 17:53, 12 March 2009

A *positive formula* is a formula *P* such that (thus a coalgebra for the comonad ). As a consequence *P* and are equivalent.

## Positive connectives

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

**Proposition** (Positive connectives)

, , , , and are positive connectives.

*Proof.*

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

The notion of positive connective is related with but different from the notion of asynchronous connective.

## Generalized structural rules

Positive formulas admit generalized left structural rules corresponding to a structure of -comonoid: and . The following rule is derivable:

*Proof.*

Positive formulas are also acceptable in the left-hand side context of the promotion rule. The following rule is derivable:

*Proof.*