As part of a bigger project I am researching a propositional logic, without a negation. And I would like to know, whether this already exists, to avoid double work and have proper references.

In this propositional logic, every theorem consists of exactly one implication $\rightarrow$ of two boolean expressions that consists of propositional variables, $\wedge$, $\vee$, $true$, $false$.

For instance an expression in normal proposition logic:

$$(a \wedge \neg b) \rightarrow (c \wedge \neg d)$$

Must be expressed in this system in two theorems:

$$a \rightarrow (b \vee c)$$ $$(a \wedge d) \rightarrow b$$

Every propositional expression can be transformed in multiple theorems of this system. That can easily be seen by turning an expression in CNF. Although, this form is more relaxed than CNF.

Although it doesn't have a not, $\neg a$ can be expressed as $a \rightarrow false$

So, the idea is that any sub-expression on the left is negative charged and on the right side positive. This allows of easy deep inference. If:

$$\alpha \rightarrow \beta$$ and $$\gamma \rightarrow \delta$$ and $\beta$ is a sub-expression of $\gamma$ then $\beta$ can be replaced by $\alpha$: $$ \gamma_{\beta/\alpha} \rightarrow \delta $$

I have been searching on the internet, but couldn't find anything in this direction. Does anyone has seen something like this, or something where I should look?

Note, due to the absence of negation, you don't have the discussion of excluded middle.