# Help with modal logic exercise

1. Jun 9, 2009

### hatsoff

I'm teaching myself modal logic, and I'm curious about the following exercise, taken from this book...

1. The problem statement, all variables and given/known data

Prove $$T:\; \Box A \rightarrow A$$ using the following rules:

2. Relevant equations

Given the structure $$M=<W,R,P>$$ in which $$W$$ and $$P$$ are as they are in a model, and $$R$$ is an equivalence relation on $$W$$, we have:

(1) $$\vDash_{\alpha}^M\mathbb{P}_n$$ iff $$\alpha\in P_n$$, for $$n=0,1,2,...$$

(2) $$\vDash_{\alpha}^M\mathbb\top$$

(3) Not $$\vDash_{\alpha}^M\bot$$

(4) $$\vDash_{\alpha}^M\neg A$$ iff not $$\vDash_{\alpha}^MA$$

(5) $$\vDash_{\alpha}^MA\wedge B$$ iff both $$\vDash_{\alpha}^MA$$ and $$\vDash_{\alpha}^MB$$.

(6) $$\vDash_{\alpha}^MA\vee B$$ iff either $$\vDash_{\alpha}^MA$$ or $$\vDash_{\alpha}^MB$$, or both.

(7) $$\vDash_{\alpha}^MA\rightarrow B$$ iff if $$\vDash_{\alpha}^MA$$ then $$\vDash_{\alpha}^MB$$.

(8) $$\vDash_{\alpha}^MA\leftrightarrow B$$ iff $$\vDash_{\alpha}^MA$$ if and only if $$\vDash_{\alpha}^MB$$

(9') $$\vDash_{\alpha}^M\Box A$$ iff for every $$\beta\in M$$ such that $$\alpha R \beta$$, $$\vDash_{\alpha}^MA$$.

(10') $$\vDash_{\alpha}^M\Diamond A$$ iff for some $$\beta\in M$$ such that $$\alpha R \beta$$, $$\vDash_{\alpha}^MA$$.

3. The attempt at a solution

I'm unsure of the proper notation to begin this. I would think that we would begin by declaring a variable to represent the actual world. Let's use $$\alpha\in W$$. Then for $$M=<W,R,P>$$ as described above:

$$\Box A\rightarrow\vDash_{\alpha}^M\Box A$$

Correct?

And if this is the case, then by (9') and the reflexivity of $$R$$ (all equivalence relations are reflexive), we can say:

$$\vDash_{\alpha}^M\Box A$$ with $$\alpha\in M$$ and $$\alpha R \alpha \;\rightarrow\; \vDash_{\alpha}^MA$$

And since $$\alpha$$ is the actual world, we have $$\vDash_{\alpha}^MA\rightarrow A$$. Correct?

So, assuming all this is proper, then we give the following proof...

Choose $$\alpha\in W$$ with $$\alpha$$ the actual world and $$M=<W,R,P>$$ as described above. Then:

$$\Box A\rightarrow \vDash_{\alpha}^M\Box A$$ with $$\alpha\in M$$ and $$\alpha R \alpha \;\rightarrow\; \vDash_{\alpha}^MA\rightarrow A$$

Does that look okay?