Classical logic has the theorem ($p\wedge\lnot p)\rightarrow q$, which I will call EFQ ("ex falso quodlibet"). Constructive logic often has the principle built in, in the form of an axiom $\bot\rightarrow q$ which one can use to prove EFQ via $(p\wedge\lnot p)\rightarrow \bot$.
Suppose one takes some usual system of constructive logic and deletes $\bot\rightarrow q$. Then although $(p\wedge\lnot p)\rightarrow \bot$ is still provable, there is no longer any way to eliminate $\bot$, so one can't get anything further from a contradiction. Or so I believe; did I miss anything? Can one still deduce EFQ? If one deletes the $\bot$-elimination rule as I suggested, does the resulting logic depend on which equivalent formalization one starts with?
This system is no longer complete with respect to the usual model (subsets of $\Bbb R$), but perhaps one can adjust the model a little bit and get a slightly different model with respect to which this logic is complete and consistent.
Is there anything obviously wrong with this logic that I've overlooked? Is it discussed anywhere? (I don't remember reading about it in Priest's book In Contradiction, but I read it some years ago and may have forgotten.)