I believe I learned that Carnap's state description semantics for propositional modal logic suffered from validating $\lozenge p$ for all atomic variables p. Re-reading Nino Cochiarelli's primary semantics for quantified modal logic in Handbook of Philosophical Logic 1st ed. Vol. 2 we have, on page 312 conditions (2) and (6):

(2) $\mathfrak M, A\vDash P^{n}(x_1,...,s_n)$ iff $<A(x_{1}),...,A(x_{n})>\in R(P^{n})$

(6) $\mathfrak M, A\vDash \square \phi$ iff for all R', if $<D,R'>$ is an L-model, then $<D,R'>,A\vDash \phi$

Let F a primitive dyadic (or any other arity with accompanying change in stating my point) relation sign of the language considered. We then, by interdefinability of modal operators have, it seems to me, $\mathfrak M, A\vDash \lozenge Fx_{1}x_{2}$ as surely there is an L-model $<D,R'>$ such that $<A(x_{1},A(x_{2})>\in R'(F)$.

Is my consideration correct? Would Carnap or other logical atomists welcome such a result as this?

I believe I learned that Carnap's state description semantics for propositional modal logic suffered from validating $\lozenge p$ for all atomic variables p. Re-reading Nino Cochiarelli's primary semantics for quantified modal logic in Handbook of Philosophical Logic 1st ed. Vol. 2 we have, on page 312 conditions (2) and (6):

(2) $\mathfrak M, A\vDash P^{n}(x_1,...,s_n)$ iff $<A(x_{1}),...,A(x_{n})>\in R(P^{n})$

(6) $\mathfrak M, A\vDash \square \phi$ iff for all R', if $<D,R'>$ is an L-model, then $<D,R'>,A\vDash \phi$

Let F a primitive dyadic (or any other arity with accompanying change in stating my point) relation sign of the language considered. We then, by interdefinability of modal operators have, it seems to me, $\mathfrak M, A\vDash \lozenge Fx_{1}x_{2}$ as surely there is an L-model $<D,R'>$ such that $<A(x_{1}),A(x_{2})>\in R'(F)$.

Is my consideration correct? Would Carnap or other logical atomists welcome such a result as this?