Solutions to Chapter 44 Semantics for ML

We have presented all of the counter-interpretations diagrammatically. If you would prefer to write them out explicitly, then that would be fine too!

A. Present counter-interpretations to the following:

1.

$\neg P\vDash_{\mathbf{K}}\neg\Diamond P$
2.

$\Box(P\vee Q)\vDash_{\mathbf{K}}\Box P\vee\Box Q$
3.

$\vDash_{\mathbf{K}}\neg\Box(A\wedge\neg A)$
4.

$\Box A\vDash_{\mathbf{K}}A$

B. Present counter-interpretations to the following:

1.

$\Box(M\rightarrow O),\Diamond M\vDash_{\mathbf{T}}O$
2.

$\Box A\vDash_{\mathbf{T}}\Box\Box A$

C. Present counter-interpretations to the following:

1.

$\Diamond A\vDash_{\mathbf{S4}}\Box\Diamond A$
2.

$\Diamond A,\Box(\Diamond A\rightarrow B)\vDash_{\mathbf{S4}}\Box B$