problems

modal propositional logic 

1. Draw a possible world model that shows that the relevant formula is not true in all worlds of all models of the specified sort.

   1. \(\Diamond p \to \Box p\) / euclidean
   2. \(\Box (\Box p \to p)\) / symmetric
   3. \(\Diamond(p \to q) \to (\Diamond p \to \Diamond q)\) / reflexive
   4. \(\Box \Box p \to \Box p\) / transitive
   5. \(\Diamond p \to \Box \Diamond p\) / reflexive and transitive

   
   

2. Justify the claim that no formula of the form \(\Box \Diamond \varphi\) is valid in all frames.

   Hint. You should be able to find a model in which all such formulas are in fact false at some world.

3. Call a relation \(R\) is serial on \(W\) if, and only if, for all \(u \in W\), there is some \(v\in W\) such that \(uRv\). On the other hand, a relation \(R\) is functional on \(W\) if, and only if, for all \(w, u, v \in W\), if \(wRu\) and \(wRv\), then \(u = v\). Justify the claims below:

   1. \(\Box p \to \Diamond p\) defines the class of serial frames.

   2. \(\Diamond p \to \Box p\) defines the class of functional frames.

4. Find a modal formula to define the class of identity frames, that is, fames of the form \((W, R)\), where \(R = \{(u,u): u \in W\}\).

5. Consider the rule of inference: \(\Box \varphi / \varphi\) Justify the claim that it preserves validity with respect to the class of all frames.

   Hint. Take the contrapositive. That is, you can argue that if \(\varphi\) is not valid in all frames, then neither is \(\Box \varphi\).