# Help with modal logic countermodel

I am trying to understand a modal logic countermodel, illustrating that the QK theorem, $$\Box (\phi(x) \wedge \forall x \phi (x)) \rightarrow \Box \forall x \phi (x),$$ fails in the alternate semantics for modal logic given by David Lewis's counterpart theory. The countermodel is the following:

It comes from Oliver Kutz's Kripke-Typ Semantiken für die modale Prädikatenlogik , p. 31. I don't understand German, but based on what I could glean from context and Google Translate it doesn't seem like the diagram is explained at all.

Here's what I do understand:

• The large circles represent the two worlds, $$w_1$$ and $$w_2$$.

• The smaller dotted circles, which I would prefer to be loops, indicate that $$C (u_1, u_1)$$ and $$C (u_2, u_2)$$ (where $$C$$ is the "counterpart relation").

• At the bottom of each circle is an indication that $$\phi/\lnot \phi$$ hold of $$u_1/u_2$$ in their respective worlds.

What I don't get are the dotted arcs. Based on context I believe the quantifiers below the arcs are quantifiers over worlds. This makes sense because the original presentation of counterpart theory utilized a translation procedure to link statements of quantified modal logic to statements in the two-sorted (worlds and individuals) framework of counterpart theory. So, e.g., the formula $$\Box \phi (x)$$ gets translated as $$\forall v \forall y [W (v) \wedge I (y, v) \wedge C (y, x) \rightarrow \phi^v (y)],$$ which is read as "for every world $$v$$ if $$y$$ inhabits $$v$$ and $$y$$ is a counterpart of $$x$$ then $$\phi$$ is true of $$y$$ in $$v$$". Similarly, a statement like $$\Diamond \phi (x)$$ gets translated as $$\exists v \exists y (W (v) \wedge I (y, v) \wedge C (y, x) \wedge \phi^v (b)).$$

As a result there are various completion I could imagine to the quantification over worlds. It will be true that there is some world where $$\exists x \lnot \phi (x)$$ is true, namely $$w_2$$, and so $$\Diamond \exists x \lnot \phi (x)$$ is true in $$w_1$$ (and therefore $$\lnot \Box \forall x \phi (x)$$ is also true in $$w_1$$, falsifying the consequent of the conditional).

Additionally, both $$\phi (x)$$ and $$\forall x \phi (x)$$ are true in $$w_1$$ since the only counterpart of $$u_1$$ is itself it also follows that $$\Box (\phi (x) \wedge \forall x \phi (x))$$ is true in $$w_1$$. From this and the preceding we have that the conditional QK theorem is false at $$w_1$$ in this counterpart theory model.

So obviously there is some quantification over worlds in the box and diamond statements, but I can't quite figure out what sort of a relationship the dotted arcs are attempting to indicate.

Any help would be appreciated.

Note that in the last case [(ii) following from the assumption $$\mathcal{M} \vDash \neg \phi^{w_2}$$], no conditions on the existence of counterparts are imposed, since the formula $$\neg \Box \forall x \phi(x)$$ contains no free variables. Indeed, this is the source of of the failure of box distribution. For note that due to the double quantification $$\forall v \forall y \ldots$$ over worlds and objects, subformulas of a formula $$\psi$$ of the form $$\Diamond \chi(\overline{z})$$, depending on which free variables $$\overline{z}$$ they contain, are evaluated in multiple possible worlds. In particular, a sentence of the form $$\Box \phi$$ is evaluated in all possible worlds, without restrictions in terms of an accessibility relation on the set of possible worlds taking effect. [...]
I therefore believe that $$\forall v \ldots$$ is the translation of the formula (i), and $$\exists v \ldots$$ is a translation of the formula (ii), and the dotted arcs indicate that the formulas are evaluated in all possible worlds $$w_1, w_2$$ disregarding constraints on accessibility, since they contain no free variables.