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I've been working through James Freeman's "Algebraic Semantics for Modal Predicate Logic" (1976), where he introduces modal cylindric algebras -- essentially (special) cylindric algebras augmented with a modal operator. One of the axioms he gives, however, has me a bit puzzled. The axiom is $$ c_k (*c_k x) = *c_k x,$$ where the asterisk is the algebraic equivalent of a modal diamond. What has me confused is the import this has for quantified modal logic (QML).

I believe the translation of this axiom in QML is $$ \exists x \Diamond \exists x \phi \leftrightarrow \Diamond \exists x \phi.$$ In the right-to-left direction this seems to be just an instance of the following axiom/theorem of first-order logic: $$ \phi \rightarrow \exists x \phi.$$ So it would seem that the left-to-right direction must be what holds modal interest and ensures some desirable connection between the modal operators and the quantifiers.

The left-to-right direction is $$ \exists x \Diamond \exists x \phi \rightarrow \Diamond \exists x \phi,$$ which is also equivalent to $$ \Box \forall x \phi \rightarrow \forall x \Box \forall x \phi.$$ But then this looks like it's just the axiom of first-order logic sometimes called "vacuous quantification": $$ \phi \rightarrow \forall x \phi.$$

Is there something of modal significance to the above axiom for modal cylindric algebras, or is it just an axiom establishing conditions on quantifiers/cylindrifications in which the modal operator appears mysteriously? Or did I blunder somewhere?

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  • $\begingroup$ Isn't this just (as you say) a way of encoding the principle of vacuous quantification in a modal cylindric-algebraic context? $\endgroup$ – anygivenpoint Feb 28 at 9:36
  • $\begingroup$ (That sounds more aggressive than intended - I mean that I can't see what distinctively modal significance this has either.) $\endgroup$ – anygivenpoint Feb 28 at 9:37

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