How prove or refute $diamond Box A$ → A characterizes symmetry
0
$begingroup$
Can some of you nice people help me and show me how to prove
$diamond Box A$ → A characterizes symmetry.
I really appreciate it
Bests
modal-logic
edited Dec 9 '18 at 0:15
Rob Arthan
29.3k42966
asked Dec 9 '18 at 0:12
NormanNorman
257
$endgroup$
|
show 3 more comments
0
$begingroup$
Can some of you nice people help me and show me how to prove
$diamond Box A$ → A characterizes symmetry.
I really appreciate it
Bests
modal-logic
edited Dec 9 '18 at 0:15
Rob Arthan
29.3k42966
asked Dec 9 '18 at 0:12
NormanNorman
257
$endgroup$
$begingroup$
What have you tried? Are you sure you have got the statement of the problem right?
$endgroup$
– Rob Arthan
Dec 9 '18 at 0:22
$begingroup$
Can you spell out what does it mean to 'characterize symmetry' here? One direction of the statement is straightforward..
$endgroup$
– Berci
Dec 9 '18 at 1:23
$begingroup$
@RobArthan, all i know is we have to define a Frame F = <W,R> and some world like s,t in W and if sRt then tRs and prove in both direction. but don't know how
$endgroup$
– Norman
Dec 9 '18 at 11:48
$begingroup$
@Berci i think this means that in this modal system if we define 2 world w,t in W and have a frame F<W,R> if sRt then tRs. but i don't know how should i prove it.
$endgroup$
– Norman
Dec 9 '18 at 11:51
$begingroup$
Ok. One statement is that if $R$ is symmetric in a frame, then all Kripke models on this frame satisfy the given formula. (This is the straightforward direction.) The other statement to prove is: if all Kripke models on a fixed frame satisfy the given formula, then the frame is symmetric. Can you prove any of these statements?
$endgroup$
– Berci
Dec 9 '18 at 12:06
|
show 3 more comments
0
0
0
$begingroup$
Can some of you nice people help me and show me how to prove
$diamond Box A$ → A characterizes symmetry.
I really appreciate it
Bests
modal-logic
edited Dec 9 '18 at 0:15
Rob Arthan
29.3k42966
asked Dec 9 '18 at 0:12
NormanNorman
257
$endgroup$
Can some of you nice people help me and show me how to prove
$diamond Box A$ → A characterizes symmetry.
I really appreciate it
Bests
modal-logic
modal-logic
edited Dec 9 '18 at 0:15
Rob Arthan
29.3k42966
asked Dec 9 '18 at 0:12
NormanNorman
257
edited Dec 9 '18 at 0:15
Rob Arthan
29.3k42966
asked Dec 9 '18 at 0:12
NormanNorman
257
edited Dec 9 '18 at 0:15
Rob Arthan
29.3k42966
edited Dec 9 '18 at 0:15
Rob Arthan
29.3k42966
edited Dec 9 '18 at 0:15
Rob Arthan
29.3k42966
29.3k42966
asked Dec 9 '18 at 0:12
NormanNorman
257
asked Dec 9 '18 at 0:12
NormanNorman
257
asked Dec 9 '18 at 0:12
NormanNorman
257
257
$begingroup$
What have you tried? Are you sure you have got the statement of the problem right?
$endgroup$
– Rob Arthan
Dec 9 '18 at 0:22
$begingroup$
Can you spell out what does it mean to 'characterize symmetry' here? One direction of the statement is straightforward..
$endgroup$
– Berci
Dec 9 '18 at 1:23
$begingroup$
@RobArthan, all i know is we have to define a Frame F = <W,R> and some world like s,t in W and if sRt then tRs and prove in both direction. but don't know how
$endgroup$
– Norman
Dec 9 '18 at 11:48
$begingroup$
@Berci i think this means that in this modal system if we define 2 world w,t in W and have a frame F<W,R> if sRt then tRs. but i don't know how should i prove it.
$endgroup$
– Norman
Dec 9 '18 at 11:51
$begingroup$
Ok. One statement is that if $R$ is symmetric in a frame, then all Kripke models on this frame satisfy the given formula. (This is the straightforward direction.) The other statement to prove is: if all Kripke models on a fixed frame satisfy the given formula, then the frame is symmetric. Can you prove any of these statements?
$endgroup$
– Berci
Dec 9 '18 at 12:06
|
show 3 more comments
$begingroup$
What have you tried? Are you sure you have got the statement of the problem right?
$endgroup$
– Rob Arthan
Dec 9 '18 at 0:22
$begingroup$
Can you spell out what does it mean to 'characterize symmetry' here? One direction of the statement is straightforward..
$endgroup$
– Berci
Dec 9 '18 at 1:23
$begingroup$
@RobArthan, all i know is we have to define a Frame F = <W,R> and some world like s,t in W and if sRt then tRs and prove in both direction. but don't know how
$endgroup$
– Norman
Dec 9 '18 at 11:48
$begingroup$
@Berci i think this means that in this modal system if we define 2 world w,t in W and have a frame F<W,R> if sRt then tRs. but i don't know how should i prove it.
$endgroup$
– Norman
Dec 9 '18 at 11:51
$begingroup$
Ok. One statement is that if $R$ is symmetric in a frame, then all Kripke models on this frame satisfy the given formula. (This is the straightforward direction.) The other statement to prove is: if all Kripke models on a fixed frame satisfy the given formula, then the frame is symmetric. Can you prove any of these statements?
$endgroup$
– Berci
Dec 9 '18 at 12:06
$begingroup$
What have you tried? Are you sure you have got the statement of the problem right?
$endgroup$
– Rob Arthan
Dec 9 '18 at 0:22
$begingroup$
What have you tried? Are you sure you have got the statement of the problem right?
$endgroup$
– Rob Arthan
Dec 9 '18 at 0:22
$begingroup$
Can you spell out what does it mean to 'characterize symmetry' here? One direction of the statement is straightforward..
$endgroup$
– Berci
Dec 9 '18 at 1:23
$begingroup$
Can you spell out what does it mean to 'characterize symmetry' here? One direction of the statement is straightforward..
$endgroup$
– Berci
Dec 9 '18 at 1:23
$begingroup$
@RobArthan, all i know is we have to define a Frame F = <W,R> and some world like s,t in W and if sRt then tRs and prove in both direction. but don't know how
$endgroup$
– Norman
Dec 9 '18 at 11:48
$begingroup$
@RobArthan, all i know is we have to define a Frame F = <W,R> and some world like s,t in W and if sRt then tRs and prove in both direction. but don't know how
$endgroup$
– Norman
Dec 9 '18 at 11:48
$begingroup$
@Berci i think this means that in this modal system if we define 2 world w,t in W and have a frame F<W,R> if sRt then tRs. but i don't know how should i prove it.
$endgroup$
– Norman
Dec 9 '18 at 11:51
$begingroup$
@Berci i think this means that in this modal system if we define 2 world w,t in W and have a frame F<W,R> if sRt then tRs. but i don't know how should i prove it.
$endgroup$
– Norman
Dec 9 '18 at 11:51
$begingroup$
Ok. One statement is that if $R$ is symmetric in a frame, then all Kripke models on this frame satisfy the given formula. (This is the straightforward direction.) The other statement to prove is: if all Kripke models on a fixed frame satisfy the given formula, then the frame is symmetric. Can you prove any of these statements?
$endgroup$
– Berci
Dec 9 '18 at 12:06
$begingroup$
Ok. One statement is that if $R$ is symmetric in a frame, then all Kripke models on this frame satisfy the given formula. (This is the straightforward direction.) The other statement to prove is: if all Kripke models on a fixed frame satisfy the given formula, then the frame is symmetric. Can you prove any of these statements?
$endgroup$
– Berci
Dec 9 '18 at 12:06
|
show 3 more comments
1 Answer
1
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0
$begingroup$
Hints:
- If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), vVdash diamondBox A$ and deduce that $A$ is valid at $v$.
- If $(W,R)$ is not symmetric, there are $v, win W$ such that $vRw$ but not $wRv$, then define a valuation $p$ (of $A$) that makes $vVdash diamondBox A$ but $A$ is false at $v$.
answered Dec 9 '18 at 18:17
BerciBerci
60.8k23673
$endgroup$
$begingroup$
I don't think part 2 is right. See my comment on the question.
$endgroup$
– Rob Arthan
Dec 9 '18 at 22:03
add a comment |
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1 Answer
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1 Answer
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0
$begingroup$
Hints:
- If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), vVdash diamondBox A$ and deduce that $A$ is valid at $v$.
- If $(W,R)$ is not symmetric, there are $v, win W$ such that $vRw$ but not $wRv$, then define a valuation $p$ (of $A$) that makes $vVdash diamondBox A$ but $A$ is false at $v$.
answered Dec 9 '18 at 18:17
BerciBerci
60.8k23673
$endgroup$
$begingroup$
I don't think part 2 is right. See my comment on the question.
$endgroup$
– Rob Arthan
Dec 9 '18 at 22:03
add a comment |
0
$begingroup$
Hints:
- If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), vVdash diamondBox A$ and deduce that $A$ is valid at $v$.
- If $(W,R)$ is not symmetric, there are $v, win W$ such that $vRw$ but not $wRv$, then define a valuation $p$ (of $A$) that makes $vVdash diamondBox A$ but $A$ is false at $v$.
answered Dec 9 '18 at 18:17
BerciBerci
60.8k23673
$endgroup$
$begingroup$
I don't think part 2 is right. See my comment on the question.
$endgroup$
– Rob Arthan
Dec 9 '18 at 22:03
add a comment |
0
0
0
$begingroup$
Hints:
- If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), vVdash diamondBox A$ and deduce that $A$ is valid at $v$.
- If $(W,R)$ is not symmetric, there are $v, win W$ such that $vRw$ but not $wRv$, then define a valuation $p$ (of $A$) that makes $vVdash diamondBox A$ but $A$ is false at $v$.
answered Dec 9 '18 at 18:17
BerciBerci
60.8k23673
$endgroup$
Hints:
- If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), vVdash diamondBox A$ and deduce that $A$ is valid at $v$.
- If $(W,R)$ is not symmetric, there are $v, win W$ such that $vRw$ but not $wRv$, then define a valuation $p$ (of $A$) that makes $vVdash diamondBox A$ but $A$ is false at $v$.
answered Dec 9 '18 at 18:17
BerciBerci
60.8k23673
answered Dec 9 '18 at 18:17
BerciBerci
60.8k23673
answered Dec 9 '18 at 18:17
BerciBerci
60.8k23673
answered Dec 9 '18 at 18:17
BerciBerci
60.8k23673
60.8k23673
$begingroup$
I don't think part 2 is right. See my comment on the question.
$endgroup$
– Rob Arthan
Dec 9 '18 at 22:03
add a comment |
$begingroup$
I don't think part 2 is right. See my comment on the question.
$endgroup$
– Rob Arthan
Dec 9 '18 at 22:03
$begingroup$
I don't think part 2 is right. See my comment on the question.
$endgroup$
– Rob Arthan
Dec 9 '18 at 22:03
$begingroup$
I don't think part 2 is right. See my comment on the question.
$endgroup$
– Rob Arthan
Dec 9 '18 at 22:03
add a comment |
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$begingroup$
What have you tried? Are you sure you have got the statement of the problem right?
$endgroup$
– Rob Arthan
Dec 9 '18 at 0:22
$begingroup$
Can you spell out what does it mean to 'characterize symmetry' here? One direction of the statement is straightforward..
$endgroup$
– Berci
Dec 9 '18 at 1:23
$begingroup$
@RobArthan, all i know is we have to define a Frame F = <W,R> and some world like s,t in W and if sRt then tRs and prove in both direction. but don't know how
$endgroup$
– Norman
Dec 9 '18 at 11:48
$begingroup$
@Berci i think this means that in this modal system if we define 2 world w,t in W and have a frame F<W,R> if sRt then tRs. but i don't know how should i prove it.
$endgroup$
– Norman
Dec 9 '18 at 11:51
$begingroup$
Ok. One statement is that if $R$ is symmetric in a frame, then all Kripke models on this frame satisfy the given formula. (This is the straightforward direction.) The other statement to prove is: if all Kripke models on a fixed frame satisfy the given formula, then the frame is symmetric. Can you prove any of these statements?
$endgroup$
– Berci
Dec 9 '18 at 12:06