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

29.1k42966




29.1k42966










asked Dec 9 '18 at 0:12









NormanNorman

127




127












  • $begingroup$
    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






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











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3031839%2fhow-prove-or-refute-diamond-box-a-%25e2%2586%2592-a-characterizes-symmetry%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























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

60.1k23672




60.1k23672












  • $begingroup$
    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


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3031839%2fhow-prove-or-refute-diamond-box-a-%25e2%2586%2592-a-characterizes-symmetry%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Berounka

Image
Berounka.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Caractéristiques Longueur 139  km Bassin 8 861  km 2 Bassin collecteur Elbe Débit moyen ? Cours Source Confluence de la Mže et de la Radbuza · Localisation Pilsen · Altitude 305  m · Coordonnées 49° 45′ 13″ N, 13° 23′ 24″ E Confluence Vltava · Localisation Prague · Altitude 187  m · Coordonnées 49° 59′ 43″ N, 14° 24′ 04″ E Géographie Pays traversés   République tchèque modifier   La Berounka est une rivière de la République tchèque. Elle porte le nom de Mže depuis sa source en Allemagne près de la frontière jusqu'à sa confluence avec la Radbuza à Plzeň. Elle continue ensuite sous le nom de Berounka jusqu'à sa confluence avec la Vltava, près de Prague. La rivière est une destination prisée pour les amateurs de canoë, qui appréc...
Read more

Sphinx de Gizeh

Image
Pour les articles homonymes, voir Sphinx. Sphinx de Gizeh.mw-parser-output .entete.egypte-antique{background-image:url("//upload.wikimedia.org/wikipedia/commons/c/cc/Picto_infobox_ancient_Egypt.png")} Site d'Égypte antique Le visage du sphinx de Gizeh. Noms en arabe أبو الهول , Abou al-Hôl Localisation Région Basse-Égypte Coordonnées 29° 58′ 31″ nord, 31° 08′ 16″ est Géolocalisation sur la carte : Égypte Sphinx de Gizeh modifier   Le sphinx de Gizeh est la statue thérianthrope qui se dresse devant les grandes pyramides du plateau de Gizeh, en Basse-Égypte. Sculpture monumentale monolithique la plus grande du monde [ 1 ] avec 73,5 mètres de longueur, 14 mètres de largeur et 20,22 mètres de hauteur, elle représente un sphinx couchant. Réalisée vers -2500, elle serait attribuée à Djédefrê, l'un des pharaons de la IV e  dynastie, représentant son père le pharaon Khéops. Sommaire 1 Étymologie 2 Desc...
Read more

Different font size/position of beamer's navigation symbols template's content depending on regular/plain...

Image
3 Let's suppose I'd like to divert the navigation symbols template (e.g. in order replace them by the frame numbers, see this bug report ;) I would be facing troubles regarding the font size/position, depending on regular/plain frames, as shown by the following MCE. How could I get rid of these troubles? documentclass{beamer} usetheme{PaloAlto} setbeamertemplate{navigation symbols}{insertframenumber/inserttotalframenumber} begin{document} begin{frame} frametitle{A regular frame} end{frame} begin{frame}[plain] frametitle{A plain frame} end{frame} end{document} beamer header-footer share | improve this question asked Dec 10 '18 a...
Read more