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.)

Second-order propositional logichas essentially the same circularity, and quite unproblematic semantics. Rather, the issue in your setup is the (lack of) treatment of bound variables: the semantics of $\Box \varphi$ quantifies over propositional variables in $\varphi$, but syntactically, $\Box$ doesn’t bind them. E.g. any proper valuation must have $v(\lnot \Box A) = \top$ (so one might think $\lnot \Box A$ was “logically true”) but also $v(\Box \lnot \Box A) = \bot$. $\endgroup$ – Peter LeFanu Lumsdaine Jan 16 '18 at 13:11and only if$\phi(A/p)$ is true for every sentence $A$. $\endgroup$ – Andrew Bacon Jan 16 '18 at 19:25