Show $K_\rho$ is a proper extension of $K_\eta$

Question

Show $K_\rho$ is a proper extension of $K_\eta$:

$K_\rho$ is an extension of $K$ where $R$ is reflexive. And $K_\eta$ is an extension of $K$ where $R$ is extendable.

So, I reasoned as follows:

$K_\rho$ is a proper extension of $K$ because it can prove $(\Box p \supset p)$ and $K$ cannot.

And $K_\eta$ is a proper extension of $K$ because it can prove $\Box p \supset\Diamond p$ and $K$ cannot.

Now, $K_\rho$ can prove $\Box p \supset\Diamond p$ but $K_\eta$ cannot prove $\Box p \supset p$.

So, $K_\rho$ can prove more inferences than $K_\eta$.

And so here finally is my question: I am asked to supply an example of such an inference. To do this, could I just use the inference from $\Box p$ to $ p$?

Also, what does this tell me about the set of interpretations that make the sets of inferences of either $K_\eta$ and $K_\rho$ true?

For this last question, is it correct that if there are more restrictions on the $R$ relationship there will be more inferences it can prove but, conversely, fewer interpretations that model those inferences, compared to another logic with fewer restrictions on $R$?

And if so, I don't quite follow how $K_\rho$ is thought to have more restrictions on its $R$ than $K_\eta$. I feel like reflexivity is more generalized than extendability. But perhaps I'm just getting confused.

Thanks for any and all help!

BTW this is all from Graham Priests's Non-Classical Logic Book, pg 37, question 3.10.7a.

Answer 1

Concerning your reasoning: You still have to proof that every deduction in $D$ is a deduction in $T$. For this it suffices to show that there is a $T$-deduction of $\Box\varphi \rightarrow \Diamond \varphi$ from the empty set. If you you use a Hilbert-system this is trivial, if you have proved that $\varphi \rightarrow \Diamond \varphi$ is a $T$-theorem. This is trivial as well, as this proof sketch shows:

1. $\Box \neg \varphi \rightarrow \neg \varphi$,$\space$ T-axiom

2. $\Diamond \neg \varphi \rightarrow \Box \neg \varphi, $ Duality-axiom

3. $\varphi \rightarrow \Diamond \varphi, \space$ PC, Duality, 1,2

Concerning your main question: Yes, that works; there are very simple counter models based on serial frames.

Concerning your question on the relation between restrictions and validities: As a rule of thumb, this is correct. However, the 'number' of models of a certain class is not so easy to determine. Note that any ordinal with its well-ordering is both a $T$- as well as a $D$-frame. So, the proper classes of $T$- and $D$-frames each include the class of all ordinals and I'm not enough into higher infinities even to guess or whether these classes have the same 'size' or not.

Concerning your supposed confusion: It's simply a mathematical fact that every reflexive relation is serial but not every serial relation is reflexive (take the relation of being an immediate successor on the natural numbers). So the class of reflexive relations is properly included in the class of serial relations. If talk of generalization helps here, I'm not sure.