Suppose we have a logic for counterfactuals as with David Lewis. I here use $\Rrightarrow$ for the counterfactual conditional. So suppose we have:


Rules:


(1) If $A$ and $A\rightarrow B$ are theorems, then $B$ is a theorem.


(2) If $(B\_1\wedge ...)\rightarrow C$ is a theorem, then so is $((A\Rrightarrow 
 B\_1)\wedge ...)\rightarrow (A\Rrightarrow C)$


Axioms:


(1) All truth functional tautologies


(2) $A\Rrightarrow A$


(3) $((A\Rrightarrow B)\wedge(B\Rrightarrow A))\rightarrow ((A\Rrightarrow C)\leftrightarrow (B\Rrightarrow C))$


(4) $(((A\vee B)\Rrightarrow A)\vee ((A \vee B)\Rrightarrow B))\vee (((A\vee B)\Rrightarrow C)\leftrightarrow((A\Rrightarrow C)\wedge(B\Rrightarrow C))$


(5) $(A\Rrightarrow B)\rightarrow(A\rightarrow B)$


(6) $(A\wedge B)\rightarrow(A\Rrightarrow B)$


Given Lewis's semantics so that $\alpha\Rrightarrow\beta$ holds iff $\beta$ holds in all closest possible worlds where $\alpha$ holds, we may define the modal operator for necessity


$\mathbf{Definition}$


$\Box \alpha :=\lnot\alpha\Rrightarrow\alpha$.


$\mathbf{Question}$


How do I most elegantly get modal logics in the hierarchy up to $S5$ on the basis of axiomatic principles for $\Rrightarrow$ while presupposing the Definition. 


$\mathbf{Initial \ example}$:


Given the Definition and the instance of axiom (5) that $(\lnot\alpha\Rrightarrow\alpha)\rightarrow(\lnot\alpha\rightarrow\alpha)$, we immediately get the $T$-$axiom$: $\Box\alpha\rightarrow\alpha$.



Axioms for modal logics based upon counterfactuals
In ["Completeness and decidability..."](http://www.andrewmbailey.com/dkl/Completeness_and_Decidability.pdf), Lewis shows the system C1 to be decidable and complete with respect to the semantics of corresponding canonical $\alpha$-models (81-4). It seems that K holds in all relevant filtrations of standard $\alpha$-models, so there should exist a syntactic proof of K. I don't see an obvious reason why a comparatively elegant syntactic proof, i.e. one that doesn't effectively enumerate all relevant cases, should be possible though.


In ["Counterfactuals and Comparative Possibility"](http://www.andrewmbailey.com/dkl/Counterfactuals_Comparative.pdf), Lewis uses the equivalent system VC, and proposes axiom U ($\Box A \rightarrow \Box\Box A$ and $\Diamond A \rightarrow \Box\Diamond A$) corresponding to a frame condition of (local) uniformity (442-4). Assuming K is valid, C implies T (and N), and U corresponds to 4+5, so VCU entails S5. However, U is independent of VC, since it corresponds to dropping accessibility relations from VC canonical models (445).


So, even with a syntactic proof for K in C1, you can only prove systems K and T. It is very simple though to prove K, 5, etc. by including an axiom for S5 semantics, e.g. as used by Donald Nute, MOD: $(\lnot A \Rrightarrow A) \rightarrow (B \Rrightarrow A)$.


