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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?

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    $\begingroup$ The book referred to here is: "Handbook of Philosophical Logic", Vol. II "Extensions of Classical Logic", eds. Dov Gabbay, Franz Guenthner; §II.6, "Philosophical Perspectives on Quantification in Tense and Modal Logic", Nino B Cocchiarella; ISBN-13: 9789027716040; ISBN: 9027716048: books.google.co.uk/books/about/…. $\endgroup$
    – Rhubbarb
    Commented Mar 25, 2015 at 22:07

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My consideration is correct, and Carnap indeed welcomed this kind of approach in the case of modal predicate logic. There is a nice discussion on Carnap's modal semantics which sort out these matters in the following article by Max Cresswell: http://www.iep.utm.edu/cmlogic/#H4

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