double interpretation of Boolean algebra It bothers me that there seems to be only a flawed double interpretation of Boolean algebra in terms of classes and of propositions. One can say
A∪B indicates  union or inclusive disjunction
A∩B indicates   intersection or conjunction
A' indicates complement or negation
but when one comes to 
A⊂B
one encounters a difficulty: if we say it indicates   subclass or implication we have a grammatical problem. In the other pairs above , the class interpretation yields another class ( e.g. A, B and A∪B are all classes) while the propositional interpretation makes them all propositions. However, in the case of A⊂B  , both interpretations are sentences. This creates a problem in that , e.g., 
A⊂(B ⊂C)
is grammatical interpreted propositionally but ungrammatical when interpreted in class theory.  I tried fixing it by using 
A' ∪ B
for
A⊂B
but it doesn’t work (check it with a Venn diagram). Can we fix this?
 A: As "$A\subseteq B$" is a statement about sets and not a set in itself, I prefer to view its logical analogue as "$A\vdash B$", which is a statement about formulas and not a formula itself. 
Therefore, in the context of Boolean algebra, "$A\subseteq B$" seem to be more properly translated as "entails" than as "implies".

The proper way to view $\to$, or logical implication, is then by $A\to B\equiv \lnot A\lor B$. Hence $\to$ is not a primitive notion in a Boolean algebra, but a defined one. This makes $A'\cup B$ the corresponding set theoretic analogue of $A\to B$.
A: One immediate difference is that $\subset$ is used to make a claim about sets, while $\cap$ and $\cup$ are operators that work on sets. Indeed, where $A \cap B$ is another set (i.e. $\cap$ is function that takes in two sets and outputs a new set), $A \subset B$ is not a set ... but a claim.  So that's why $\subset$ does not have a logical operator counterpart. 
Indeed, when you say:

I tried fixing it by using 
A' ∪ B
for
A⊂B

that makes absolutely no sense: $A ' \cup B$ is a set, but $A \subset B$ is not a set in the first place.
