What you describe is not a Boolean algebra, for several reasons:
Boolean algebra has three operations, $+,\cdot,-$ and two constants $0,1$. You only specified addition and constants.
Addition is commutative, so $a+0=0+a$, unlike how you stated $0+1\neq 1+0$.
The $0$ element is a neutral element with respect to addition, that is $0+a=a$, against in contradiction to $1\neq 1+0\neq 0+1\neq 1$.
You could consider the Boolean algebra defined on $\{0,1,a,b\}$ such that:
- $0,1$ are the usual constants;
- $-a=b$ (and so $-b=a$);
- $a\cdot b=0$ and $a+b=1$.
One can consider this as $\mathcal P(\{a,b\})$ defined with the usual intersection, union and complement. In which case $0=\varnothing$ and $1=\{a,b\}$. Furthermore, every Boolean algebra which has exactly four elements is isomorphic to this one.
This algebra is complete, so it is very convenient to define Boolean-valued logic on this algebra.
This is somewhat similar the ideas met in usual logic, only now there is a new way to define how true is a sentence. We can consider this as measuring how true is something. It does not correspond very well to any real life intuition that I can provide, so my advice is to stick with formal proofs until you get the hang of the idea.
The truth value of a sentence in this logic would be defined, as usually, by induction. We denote it as $||\varphi||$, for convenience. (This is not limited to the Boolean algebra discussed above, of course.)
- $||\varphi_1\lor\varphi_2||=||\varphi_1||+||\varphi_2||$;
- $||\varphi_1\land\varphi_2||=||\varphi_1||\cdot||\varphi_2||$;
- $||\lnot\varphi||=-||\varphi||$;
For adding the definition for quantifier we need to use the completeness.
- $\displaystyle||\exists x\varphi||=\sum ||\varphi[t]||$, that is the supremum (union if we think about the power set representation) of all possible values for placing something into $\varphi$ and evaluating the sentence.
- $\displaystyle||\forall x\varphi||=\prod ||\varphi[t]||$, that is the infimum (intersection if we think about power set) of possible values as before.
To see how this corresponds to the usual two-valued logic, $\exists x\varphi$ is true if and only if there exists some element which witnesses $\varphi$ - that is for some element in the model $||\varphi[x]||=1$, so the supremum is indeed $1$; and similarly for $\forall x\varphi$ we take the least element.
This approach is less familiar than fuzzy logic, but has a very important use in set theory when constructing forcing extensions.
I am not sure it answers your question, however it shows how to create a Boolean-valued model. Suppose you declare that the function which returns truth values for assignments into formulae is non-deterministic, or probabilistic, then you can have yourself a nice probabilistic logic, or fuzzy logic, and so on.