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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 $A$A and $A\rightarrow B$A→B are theorems, then $B$B is a theorem.

(2) If $(B_1\wedge (B1...)\rightarrow C$→C is a theorem, then so is $((A\Rrightarrow B_1A⇛B1)\wedge ...)\rightarrow (A\Rrightarrow CA⇛C)$

Axioms:

(1) All truth functional tautologies

(2) $A\Rrightarrow A$A⇛A

(3) $((A\Rrightarrow BA⇛B)\wedge(B\Rrightarrow AB⇛A))\rightarrow ((A\Rrightarrow CA⇛C)\leftrightarrow (B\Rrightarrow CB⇛C))$

(4) $(((A\vee BA∨B)\Rrightarrow A⇛A)\vee ((A \vee BA∨B)\Rrightarrow B⇛B))\vee (((A\vee BA∨B)\Rrightarrow C⇛C)\leftrightarrow((A\Rrightarrow CA⇛C)\wedge(B\Rrightarrow CB⇛C))$

(5) $(A\Rrightarrow BA⇛B)\rightarrow(A\rightarrow BA→B)$

(6) $(A\wedge BA∧B)\rightarrow(A\Rrightarrow BA⇛B)$

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

$\mathbf{Definition}$Definition

$\Box \alpha □α:=\lnot\alpha\Rrightarrow\alpha$=¬α⇛α.

$\mathbf{Question}$Question

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

$\mathbf{Initial \ example}$Initial example:

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

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 $T$-$axiom$: $\Box\alpha\rightarrow\alpha$.

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

Rules:

(1) If A and A→B are theorems, then B is a theorem.

(2) If (B1...)→C is a theorem, then so is ((A⇛B1)...)(A⇛C)

Axioms:

(1) All truth functional tautologies

(2) A⇛A

(3) ((A⇛B)(B⇛A))((A⇛C)(B⇛C))

(4) (((A∨B)⇛A)((A∨B)⇛B))(((A∨B)⇛C)((A⇛C)(B⇛C))

(5) (A⇛B)(A→B)

(6) (A∧B)(A⇛B)

Given Lewis's semantics so that α⇛β holds iff β holds in all closest possible worlds where α holds, we may define the modal operator for necessity

Definition

□α:=¬α⇛α.

Question

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

Initial example:

Given the Definition and the instance of axiom (5) that (¬α⇛α)(¬α→α), we immediately get the T-axiom: □α→α.

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Axioms for modal logics based upon counterfactuals

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 $T$-$axiom$: $\Box\alpha\rightarrow\alpha$.