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I came across the second property in some unpublished notes of Kit Fine's from the '70s, but I do not know of much discussion of it elsewhere.



An extension of the disjunction property in modal logic
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 $A\leftrightarrow B\mathrel/\Box(A\leftrightarrow B)$.


