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From: George Levy <GLevy.domain.name.hidden>

Date: Sun, 15 Apr 2001 15:48:35 -0700

Reflexive, Transitive and Symmetric applies only to the relation R that define

accessibility. So:

Reflexive:

W ---->|

<------|

Transitive:

W1 ------> W2 --------> W3

Symmetric

W1 <-----> W2

And the Goedel-like formula

<>p --> -[]<>p

means: if p is true in at least one world accessed from w, then it is false that

in any world x accessed from w, that p is true in at least one world accessed from

x:

|---->w121 p false

|---->w12: p true ---->|---->w122 p false

|---->w13: p false |---->w123 p false

w1--|---->w14: p false

|---->w15: p false |----->w161 p false

|---->w16 p false---->|----->w162 p false

|----->w163 p false

I have a little bit of trouble with reading -[]<>p from left to right....Is it - (

[] ( <>p ) )?

Also does the frame of reference changes when you talk about [] i.e., any world x?

In other words you start with world w but then, as you express [] all (any)

world(s) x accessible from w do you have to change the frame to x when you pursue

the reasoning to <>?

If what I assumed is correct, then the Goedel-like formula makes sense.

Now reinterpreting [] and <> to mean provable and consistent we get for

<>p --> -[]<>p

if p is consistent in at least one world accessed from w, then it is false that in

any world x accessed from w, that p is provable in at least one world accessed

from x:

Now going to your LASE

p-> []<>p

if p is true in w, then in any world x accessed from w, it is possible to access

one world where p is true.

This statement seems to be correct only when R is symmetric or if there is a

transitive loop back to w.

I assume the following:

A terminal world is defined as a world from which no other world is accessible.

---->w --|

A transitory world is one from which at least one other world is

accessible w---->w1

An ideal world is one with no access to a terminal

world w---->w1

A realist world is one with access to at least one terminal

world. w---->w1 --|

Your theorem

5. (W,R) respects []p -> <>p iff (W,R) is ideal

means

if (if p is true in all worlds accessible from w, then p is true in at least one

world accessible from w.) is true then w does not lead to a terminal world.

In a terminal world nothing is possible, everything is false so p cannot be true.

The premise doesn't even hold. Since an ideal world does not lead to a terminal

world, then from an ideal world (W,R) respects []p -> <>p is always true.

6. (W,R) respects <>p -> -[]<>p iff (W,R) is realist

means

if (if there is at least one world accessible from w where p is true, then it is

false that from any world x accessible from w, it is possible to access at least

one world in which p is true) is true, then w leads to at least one terminal

world.

hmmm.. All this reasoning could be quickly shown with one drawing......When I read

this statement I suffer from a mental stack overflow. :-)

George

Received on Sun Apr 15 2001 - 15:52:28 PDT

Date: Sun, 15 Apr 2001 15:48:35 -0700

Reflexive, Transitive and Symmetric applies only to the relation R that define

accessibility. So:

Reflexive:

W ---->|

<------|

Transitive:

W1 ------> W2 --------> W3

Symmetric

W1 <-----> W2

And the Goedel-like formula

<>p --> -[]<>p

means: if p is true in at least one world accessed from w, then it is false that

in any world x accessed from w, that p is true in at least one world accessed from

x:

|---->w121 p false

|---->w12: p true ---->|---->w122 p false

|---->w13: p false |---->w123 p false

w1--|---->w14: p false

|---->w15: p false |----->w161 p false

|---->w16 p false---->|----->w162 p false

|----->w163 p false

I have a little bit of trouble with reading -[]<>p from left to right....Is it - (

[] ( <>p ) )?

Also does the frame of reference changes when you talk about [] i.e., any world x?

In other words you start with world w but then, as you express [] all (any)

world(s) x accessible from w do you have to change the frame to x when you pursue

the reasoning to <>?

If what I assumed is correct, then the Goedel-like formula makes sense.

Now reinterpreting [] and <> to mean provable and consistent we get for

<>p --> -[]<>p

if p is consistent in at least one world accessed from w, then it is false that in

any world x accessed from w, that p is provable in at least one world accessed

from x:

Now going to your LASE

p-> []<>p

if p is true in w, then in any world x accessed from w, it is possible to access

one world where p is true.

This statement seems to be correct only when R is symmetric or if there is a

transitive loop back to w.

I assume the following:

A terminal world is defined as a world from which no other world is accessible.

---->w --|

A transitory world is one from which at least one other world is

accessible w---->w1

An ideal world is one with no access to a terminal

world w---->w1

A realist world is one with access to at least one terminal

world. w---->w1 --|

Your theorem

5. (W,R) respects []p -> <>p iff (W,R) is ideal

means

if (if p is true in all worlds accessible from w, then p is true in at least one

world accessible from w.) is true then w does not lead to a terminal world.

In a terminal world nothing is possible, everything is false so p cannot be true.

The premise doesn't even hold. Since an ideal world does not lead to a terminal

world, then from an ideal world (W,R) respects []p -> <>p is always true.

6. (W,R) respects <>p -> -[]<>p iff (W,R) is realist

means

if (if there is at least one world accessible from w where p is true, then it is

false that from any world x accessible from w, it is possible to access at least

one world in which p is true) is true, then w leads to at least one terminal

world.

hmmm.. All this reasoning could be quickly shown with one drawing......When I read

this statement I suffer from a mental stack overflow. :-)

George

Received on Sun Apr 15 2001 - 15:52:28 PDT

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