An informal definition of a logical truth is a sentence that's true in virtue of its form alone: $\phi$ is logically true iff all substitutions of $\phi$ that leave its logical vocabulary alone are true.

We might try to formulate a version of this idea in modal logic. Let $\mathcal{L}$ be a modal language, and let $v: \mathcal{L} \to \{0,1\}$ be a Boolean valuation (i.e. $v(A\wedge B) = min(v(A),v(B))$ and $v(\neg A)=1-v(A)$). Say that $v$ is proper if it additionally satisfies the constraint:

• $v(\Box A)=1$ if and only if $v(iA)=1$ for every substitution $i$

here we represent a substitution, $i$, as a function from letters to arbitrary sentences and write $iA$ for the result of applying that substitution to $A$. (If $S$ is a restricted set of substitutions, say that $v$ is an $S$-valuation if the corresponding biconditional with $i$ restricted to $S$ holds instead.)

This conception of modality validates some interesting principles: for example if $v(\Box A)=1$ then $v(iA)=1$ for every substitution $i$. In particular, for any given $j$, $v(i(jA)) =1$ for every $i$, since $i\circ j$ is also a substitution. So $v(\Box jA)=1$ for every $j$, and so $v(\Box\Box A)=1$. It follows that the S4 principle is true in every proper valuation. Indeed, one can show that every theorem of S4M is true in every proper valuation. (M is the McKinsey axiom, $\Box \Diamond A \to \Diamond \Box A$, and can be seen to be validated by considering substitutions that map letters to $\top$ and $\bot$.)

Note, however, that it's not obvious that there are any proper valuations. The bulleted claim is a constraint, not a definition, as it involves circularity. For example $v(\Box p) = 1$ iff $v(ip)$ is true for every $i$, and the circularity arises in cases where $ip =\Box p$. But the circularity is not vicious in this case, e.g. $v(\Box p)=0$ since $v(ip)=0$ when $ip= \bot$. I conjecture that the constraint is never vicious, and can always be satisfied. So I was wondering:

• Are there any proper valuations?

I have some thoughts about constructing a valuation, but they haven't delivered anything so far. One useful fact to note is that if there is a proper valuation $v$, we can construct a Kripke model by letting $W$ be the set of substitutions, letting $i R (j\circ i)$ for all $i,j$, and letting $i \Vdash p$ iff $v(ip)=1$. Conversely, if there's a Kripke model on this frame satisfying $i \Vdash p$ iff $id\Vdash ip$ then we can construct a proper valuation by letting $v(\phi)=1$ iff $id\Vdash \phi$, where $id$ is the identity substitution. So this gives us another way of thinking about the problem. (There is also a topological reformulation of the problem, but I think that's enough for now.)

(Background: McKinsey talks about related notions of necessity here and investigates their logic. He constructs what I've called an $S$-valuation for a very restricted $S$. However, he doesn't seem to raise or recognize the issue with the unrestricted notion.)