Q:

What logic is modelled by generalized boolean algebra?

While it is well known that classical propositional logic is modelled by booleaan algebras, I have never heard of the logic modelled by generalized boolean algebras (GBA) defined by M.Stone (author of Stone's reprezentation theorem). Does anybody know anything about this logic or any attempts to find it? It would be nice to know at least what could be a good analog of implication in GBA. Since Stone axiomatized the GBAs in terms of equations, this is an equational logic, which I am not treating as proper logic - it is more algebra.But how this logic can be formulated in more regular manner by axioms in the form of implications rather than identities?

A:

Propositional logic captures the partial order of a Boolean algebra, i.e., logical entailment $p_1, \ldots, p_n \vdash q$ corresponds to $p_1 \land \cdots \land p_n \leq q$. To obtain a logic for a generalised Boolean algebra we should express its laws in terms of partial order. If the laws are written as adjunctions, the rules of inference will be clearly visible.
A GBA has finite binary meets $\land$ and joins $\lor$, a least element $\bot$, and relative complements $\setminus$. There will be no surprises in the rules for $\land$, $\lor$ and $\bot$.
Conjunction is the easiest. Meet is characterised as
$$r \leq p \land q \quad\text{iff}\quad \text{$r \leq p$ and $r \leq q$}.$$
Reading this from left to right gives us the elimination rules (where I write $\Gamma$ to indicate an arbitrary number of hypotheses $p_1, \ldots, p_n$)
$$\frac{\Gamma \vdash p \land q}{\Gamma \vdash p}
\quad\text{and}\quad
\frac{\Gamma \vdash p \land q}{\Gamma \vdash q}
$$
while the other direction gives the introduction rule
$$\frac{\Gamma \vdash p \quad \Gamma \vdash q}{\Gamma \vdash p \land q}.$$
Notice how $\Gamma$ took the role of $r$.
A naive conversion of the characterisation of $\bot$, namely
$$\bot \leq p,$$
would give us the rule $\bot \vdash p$. We actually want $\Gamma, \bot \vdash p$, but this is ok because $\bot$ is just as well characterised by
$$r \land \bot \leq p.$$
This is a small point which becomes very important when we think about disjunctions. Again, a naive conversion of
$$p \lor q \leq r \quad\text{iff}\quad \text{$p \leq r$ and $q \leq r$}$$
would give us something without $\Gamma$. What we really need is the characterisation
$$s \land (p \lor q) \leq r \quad\text{iff}\quad \text{$s \land p \leq r$ and $s \land q \leq r$}.$$
But does this really characterize joins? Yes, thanks to the distributivity law! And so by writing $\Gamma$ instead of $s$ we get the laws
$$\frac{\Gamma, p \lor q \vdash r}{\Gamma, p \vdash r}
\quad\text{and}\quad
\frac{\Gamma, p \lor q \vdash r}{\Gamma, q \vdash r}$$
and
$$\frac{\Gamma, p \vdash r \quad \Gamma, q \vdash r}{\Gamma, p \lor q \vdash r}.$$
You may find these rules for $\lor$ a bit odd, but they are equivalent to whatever variant you are used to.
The laws for disjunction baked in just enough distributivity to make the distributivity law provable from the rules stated so far. So we need not worry about distributivity.
The interesting connective is the relative complement. If we secretly think of $p \setminus q$ as "$p$ and not $q$", then it would seem that the relative complement is to be characterised in terms of its lower bounds because it is like a conjunction. Indeed, we have
$$r \leq p \setminus q
\quad\text{iff}\quad
\text{$r \leq p$ and $r \land q \leq \bot$}$$
which suggests the rules
$$\frac{\Gamma \vdash p \quad \Gamma, q \vdash \bot}{\Gamma \vdash p \setminus q}$$
and
$$\frac{\Gamma \vdash p \setminus q}{\Gamma \vdash p}
\quad\text{and}\quad
\frac{\Gamma \vdash p \setminus q}{\Gamma, q \vdash \bot}$$
These seem perfectly reasonable to me.
The point here is not what precise rules I derived, but how I derived them in a principled way:

logical entailment corresponds to the partial order
logical operations correspond to the operations
logical rules corespond to adjunctions that characterize the operations

By the way, there is of course no truth $\top$ in this calculus. If we add it, we get the usual classical propositional calculus, just like a GBA with a top element is a Boolean algebra.

A:

The question is not quite well posed. In algebraic logic, logics are not defined by algebras, but by logical matrices: these are pairs $\langle A,D\rangle$, where $D$ is a subset of $A$, termed the set of designated values. The logic of a class $K$ of matrices is then the consequence relation $\models_K$ such that for any formula $\phi$ and a set of formulas $\Gamma$, $\Gamma\models_K\phi$ holds iff for every $\langle A,D\rangle\in K$ and every homomorphism $v$ from the algebra of formulas to $A$: if $v(\Gamma)\subseteq D$, then $v(\phi)\in D$. In the most important situations, $D$ is equationally definable in $A$: $D=\{x\in A:A\models E(x)\}$ for a set of equations $E$, which reduces matrices back to pure algebraic language. However, there may be many different choices of $E$, hence it is not sufficient to specify just a class of algebras.
For example, the matrices for classical logic are $\langle A,\{1\}\rangle$, where $A$ is a Boolean algebra. Here, $\{1\}$ is definable by $E(x)=\{x\approx1\}$.
See R. Jansana’s SEP article Propositional consequence relations and algebraic logic for a comprehensive introduction to algebraic propositional logic.
Since you didn’t specify which sets of designated values to take in generalized Boolean algebras (and the most obvious choice doesn’t work as GBA do not need to have a top), the question does not necessarily admit a unique answer. Let me give some specific examples. For the following, I assume GBA formulated in the signature $\{\land,\lor,0,-\}$, where $x-y$ is the relative complement of $y$ in $[0,x]$:

Let $K$ be the class of matrices $\langle A,D\rangle$, where $A$ is a GBA, and $D$ is a nonempty filter in $A$. Then the logic of $K$ is the $\{\land,\lor,0,-\}$-fragment of classical logic.
Let $K$ be the class of matrices $\langle A,\{0\}\rangle$, where $A$ is a GBA. Then the logic of $K$ is a notational variant of the positive fragment (i.e., $\{\lor,\land,1,\to\}$) of classical logic, where the connectives have been renamed to their duals as indicated by the order in which I have written them.
Let $K$ be the class of matrices $\langle A,\varnothing\rangle$, where $A$ is a GBA. Then the logic of $K$ is the maximal logic with no theorems, i.e., $\Gamma\models_K\phi$ iff $\Gamma\ne\varnothing$.
Let $K$ be the class of matrices $\langle A,A\rangle$, where $A$ is a GBA. Then the logic of $K$ is the inconsistent logic, i.e., $\Gamma\models_K\phi$ for every $\Gamma,\phi$.

Choice #2 is better behaved than the other three as the matrices in question are (equationally definable and) reduced, and in particular, the logic obtained is (finitely, strongly, and regularly) algebraizable, with GBA being its equivalent semantics. (See Jansana’s article for the basic definitions.) While in principle the condition of algebraizability still does not lead to a unique logic from a given class of algebras, it means that the upside-down positive fragment of classical logic corresponds to GBA in as good a sense as the full classical logic corresponds to BA.
[EDIT 2: Let me qualify the previous sentence. It’s true that in general, more than one logic can be algebraizable wrt the same variety of algebras. However, in the case of GBA, there are not that many possible choices for $E(x)$, and it is in fact easy to check that positive classical logic is the unique logic algebraized by GBA (and the translations given below are also unique up to equivalence). Thus, the question does have a well-defined unique answer after all.]
EDIT: I will spell out explicitly what algebraizability of the positive fragment means, and how it provides a logic modelled by GBA. Consider the propositional logic $\vdash$ defined by the following Hilbert calculus:
$$\begin{gather}
(\phi-\psi)-\phi\\\\
((\chi-\phi)-(\psi-\phi))-((\chi-\psi)-\phi)\\\\
\phi-(\phi-(\psi-\phi))\\\\
0\\\\
\phi-(\phi\lor\psi)\\\\
\psi-(\phi\lor\psi)\\\\
((\phi\lor\psi)-\phi)-\psi\\\\
(\phi\land\psi)-\phi\\\\
(\phi\land\psi)-\psi\\\\
((\chi-(\phi\land\psi))-(\chi-\psi))-(\chi-\phi)\\\\
\phi,\psi-\phi\vdash\psi
\end{gather}$$
Let $\let\sd\vartriangle\phi\sd\psi:=(\phi-\psi)\lor(\psi-\phi)$, and let $\models$ denote validity in GBA. Then we have:

$\phi_1\approx\psi_1,\dots,\phi_n\approx\psi_n\models\phi\approx\psi$ iff $\phi_1\sd\psi_1,\dots,\phi_n\sd\psi_n\vdash\phi\sd\psi$
$\phi_1,\dots,\phi_n\vdash\psi$ iff $\phi_1\approx0,\dots,\phi_n\approx0\models\psi\approx0$
$\phi\approx\psi\models(\phi\sd\psi)\approx0$, $(\phi\sd\psi)\approx0\models\phi\approx\psi$
$\phi\vdash\phi\sd0$, $\phi\sd0\vdash\phi$

Thus, the mappings $\phi\approx\psi\mapsto\phi\sd\psi$ and $\phi\mapsto\phi\approx0$ provide a bi-interpretation of the quasiequational theory of GBA with the logic given by $\vdash$. (This is what distinguishes this case from other choices of matrices based on GBA, such as #1,3,4 above. These choices give logics modeled in GBA’s, but they do not have matching translations of algebra into logic, hence passing from GBA to such logics is losing structure and information.)

