# Modal logic as a quotient Boolean-valued logic

I never really studied modal logic, but to my better understanding this is similar to classical logic adding two modal operations:

1. $\square P\$ meaning necessarily $P$,
2. $\lozenge P\$ meaning possibly $P$.

Now if we consider Boolean-valued logic, in which we take a complete Boolean algebra and let the truth values be elements of the Boolean algebra where the evaluation uses the Boolean algebra, so:

• $\|\psi\land\varphi\| = \|\psi\|\cdot\|\varphi\|$
• $\|\psi\lor\varphi\|=\|\psi\|+\|\varphi\|$
• $\|\lnot\psi\| = -\|\psi\|$
• $\|\exists x\varphi(x)\| = \sum\|\varphi(y)\|$

If we take an ultrafilter on the Boolean algebra, we return to the usual two-valued logic by letting $\|\varphi\|\in\mathcal U$ being true, and false otherwise.

However if only take $\mathcal U$ to be a filter, can we think of it as a "necessary" predicate on $B$? that is $\|\square\varphi\|=1\iff\|\varphi\|\in\mathcal U$, and $\|\lozenge\varphi\|=1\iff\|\lnot\varphi\|\notin\mathcal U$ (both values are $0$ otherwise).

If the answer is indeed yes, is this a complete characterization of all modal logics, that is every quotient of a Boolean-valued logic is modal logic, and vice versa?

• I find this a little dubious for the following reason: in this scheme, $\| \Box \varphi \|$ and $\| \Diamond \varphi \|$ can only ever be the maximum or minimum elements, so, if this scheme were complete for modal logic, $\Box \Box \varphi$ would be logically equivalent to $\Box \varphi$. But there are modal logics which aren't even monotonic! (Take the simplest possible modal logic which has no axioms other than the usual ones for classical logic.) Perhaps you would like to read up more about Kripke semantics. – Zhen Lin Jan 5 '12 at 2:27
• As Zhen Lin has pointed out, there are many different modal logics, usually they are built on two main axioms: $\square (A \Rightarrow B) \Rightarrow (\square A \Rightarrow \square B)$ and $\square A \Rightarrow A$. I am not an expert on modal logic but I really liked this article in SEP: plato.stanford.edu/entries/logic-modal – Daniil Jan 5 '12 at 8:45