An extension of the disjunction property in modal logic

Question

A normal modal propositional logic $\Delta$ has the disjunction property if and only if

For any formulas $A_1,\dotsc,A_n$, if $\Box A_1 \vee \dotsb\vee \Box A_n \in \Delta$ then $A_k\in \Delta$ for some $k$ with $1\leq k\leq n$.

Let us say that $\Delta$ has the extended disjunction property if and only if

For any formulas $A_1,\dotsc,A_n$ and non-modal formula $A_0$, if $A_0 \vee \Box A_1 \vee \dotsb \vee \Box A_n \in \Delta$ then $A_k\in \Delta$ for some $k$ with $0\leq k\leq n$.

Does there exist a normal modal propositional logic with the disjunction property but not the extended disjunction property?

I came across the second property in some unpublished notes of Kit Fine's from the '70s, but I do not know of much discussion of it elsewhere.

Answer 1

The extended disjunction property is equivalent to plain disjunction property for all normal modal logics $\Delta$.

Assume that $\Delta$ contains $$A_0\lor\bigvee_{i=1}^n\Box A_i,$$ where $A_0$ is box-free.

Let $P$ denote the set of propositional variables occurring in the formula. For each $a\colon P\to\{0,1\}$, let $\sigma_a$ denote the substitution such that $\sigma_a(p)=p^{a(p)}$ for all $p\in P$, where $p^1=p$, $p^0=\neg p$.

Since $\Delta$ is closed under substitution and classical reasoning, $\Delta$ contains $$\bigwedge_a\Bigl(\sigma_a(A_0)\lor\bigvee_{i=1}^n\Box\sigma_a(A_i)\Bigr).\tag{$*$}\label{star}$$

If $A_0$ is a classical tautology, then $A_0\in\Delta$ and we are done. Otherwise, \eqref{star} implies $$\bigvee_a\bigvee_{i=1}^n\Box\sigma_a(A_i)\tag{$**$}\label{starstar}$$ by classical propositional reasoning: for any assignment $e$, there is $a$ such that $e(\sigma_a(A_0))=0$, thus the rest of the disjuction holds under $\sigma_a$.

Applying the disjuction property to \eqref{starstar}, $\sigma_a(A_i)\in\Delta$ for some $i=1,\dotsc,n$ and $a\colon P\to\{0,1\}$. Applying closure under substitution once more, $\Delta$ contains $\sigma_a(\sigma_a(A_i))$, which is equivalent to $A_i$ over K.

The argument above applies not just to normal modal logics, but also to quasinormal logics, and even more generally, to all logics $\Delta$ that are closed under modus ponens and box-free substitutions, and that contain the tautologies of E. Here, E is the smallest logic that includes classical logic, and is closed under modus ponens and under the rule $A\leftrightarrow B\mathrel/\Box(A\leftrightarrow B)$.