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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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
















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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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














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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 9 '18 at 0:15









Rob Arthan

29.3k42966




29.3k42966










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


















  • $begingroup$
    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










1 Answer
1






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oldest

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0












$begingroup$

Hints:




  1. If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), vVdash diamondBox A$ and deduce that $A$ is valid at $v$.

  2. 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 $v$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I don't think part 2 is right. See my comment on the question.
    $endgroup$
    – Rob Arthan
    Dec 9 '18 at 22:03











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0












$begingroup$

Hints:




  1. If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), vVdash diamondBox A$ and deduce that $A$ is valid at $v$.

  2. 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 $v$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I don't think part 2 is right. See my comment on the question.
    $endgroup$
    – Rob Arthan
    Dec 9 '18 at 22:03
















0












$begingroup$

Hints:




  1. If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), vVdash diamondBox A$ and deduce that $A$ is valid at $v$.

  2. 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 $v$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I don't think part 2 is right. See my comment on the question.
    $endgroup$
    – Rob Arthan
    Dec 9 '18 at 22:03














0












0








0





$begingroup$

Hints:




  1. If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), vVdash diamondBox A$ and deduce that $A$ is valid at $v$.

  2. 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 $v$.






share|cite|improve this answer









$endgroup$



Hints:




  1. If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), vVdash diamondBox A$ and deduce that $A$ is valid at $v$.

  2. 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 $v$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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


















  • $begingroup$
    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


















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