what am I doing wrong?

Hi my initial solution the first part of exercise 5.3.1 (d) shows that the sentence is globally true. The solution shows that it is not globally true. Here is my solution - where does my argument fail?

M can globally fail <>(a -> b) -> (<>a -> <>b) only if there exists some x in G such that M satisfies <>(a -> b) at x and fails to satisfy (<>a -> <>b) at x. And M can fail to satisfy (<>a -> <>b) at x only by satisfying <>a at x but not satisfying <>b at x. And M fails to satisfy <>b at x only if it fails to satisfy ¬[]¬b at x. i.e. it satisfies []¬b at x. And M satisfies []¬b only if for all y in G with (x,y) in R such that M does not satisfy b at y.

This cannot be because <>(a -> b) is satisfied at x. i.e. ¬[]¬(a -> b) is satisfied at x. i.e. []¬(a -> b) fails to be satisfied by M at x and since y is accessible from x; ¬(a -> b) fails to be satisfied at y. i.e. a -> b is satisfied at y. Similarly, since <>a is satisfied at x, a is satisfied at y. Therefore b must be satisfied by y.

so the sentence must be globally true.

I know this to be wrong - I'm just not sure why.

Your solution is not right in that in the first paragraph you generalise, not showing a particular M and immediately in the second paragraph you give specific values for M.y does not satisfy b and also the part in your second paragraph where you say "since y is accessible from x; ¬(a -> b) fails to be satisfied at y." is not right.

What you should take note of is as follows:

Given some sentence S of a Modal Language KL ALL M= (G, R, V) and this is allowed.

There are only 2 possibilities:

1. S is globally true in all M (which means in every world w of G for every M). To prove this (when this is the case) you say
Let M= (G, R, V) be any modal model (which means we do not assume anything about M) and w be any element of G. We show that S is true in w.... you can give an example.

2. The other possibility is that S is not globally true i.e. there exist an M= (G, R, V) and a w which is an element of G such that S is false in w. To prove this, you give M= (G, R, V) and w explicitly and prove that S is false in w.

There are a variety of examples given in your study guide on page 276-277, which gives you a clear picture. Study them.

Lecturer COS 361-F

thank you. I have, in fact, done every exercise in the study guide. but I still make mistakes and sometimes I can't see where I went wrong. feedback like this is really helpful.