$Gammamodelsphi$ if and only if $Gamma,negphimodelspsilandnegpsi$ The 2019 Stack Overflow Developer Survey Results Are InShow that $(phi rightarrow psi), (phi rightarrow neg psi) vdash neg phi$Is it correct that If $mathcal A $ is a model of $Gamma $, and if $Gamma models psi$ then $mathcal A models psi $?Show that $Gamma cup neg phi$ is satisfiable if and only if $Gammanot models phi$How to show that if $neg b = a land d$ then $a land neg b = neg b$ and $b land neg a = neg a$If $models neg phi$, then $models phi^circ$, where $phi^circ$ is the “semi-dual” of $phi$Prove that a theory $Gamma$ is consistent if and only if there is a structure $M$ so that $M$ $models$ $Gamma$.not always $A models phi$ or $A models neg phi$ exampleIf $Gamma$ is consistent and $Gammanotvdashphi$, then $Gammacupnegphi$ is also consistent. Why?Is there any way to simplify $(Aland B land C) lor (neg A land neg B land neg C)$?Show that $vdash Gamma cup psi$ implies $vdash Gamma cup psi'$ where $psi'$ is $psi$ with one of its bound variables renamed.
Carnot-Caratheodory metric
Should I use my personal or workplace e-mail when registering to external websites for work purpose?
What tool would a Roman-age civilization have to grind silver and other metals into dust?
Inline version of a function returns different value then non-inline version
What does "rabbited" mean/imply in this sentence?
Does it makes sense to buy a new cycle to learn riding?
Why is Grand Jury testimony secret?
Could JWST stay at L2 "forever"?
Is it possible for the two major parties in the UK to form a coalition with each other instead of a much smaller party?
What is this 4-propeller plane?
Which Sci-Fi work first showed weapon of galactic-scale mass destruction?
What is the use of option -o in the useradd command?
"What time...?" or "At what time...?" - what is more grammatically correct?
Inflated grade on resume at previous job, might former employer tell new employer?
If a poisoned arrow's piercing damage is reduced to 0, do you still get poisoned?
Any good smartcontract for "business calendar" oracles?
Does duplicating a spell with Wish count as casting that spell?
Springs with some finite mass
What are the motivations for publishing new editions of an existing textbook, beyond new discoveries in a field?
aging parents with no investments
Output the Arecibo Message
JSON.serialize: is it possible to suppress null values of a map?
It's possible to achieve negative score?
Deadlock Graph and Interpretation, solution to avoid
$Gammamodelsphi$ if and only if $Gamma,negphimodelspsilandnegpsi$
The 2019 Stack Overflow Developer Survey Results Are InShow that $(phi rightarrow psi), (phi rightarrow neg psi) vdash neg phi$Is it correct that If $mathcal A $ is a model of $Gamma $, and if $Gamma models psi$ then $mathcal A models psi $?Show that $Gamma cup neg phi$ is satisfiable if and only if $Gammanot models phi$How to show that if $neg b = a land d$ then $a land neg b = neg b$ and $b land neg a = neg a$If $models neg phi$, then $models phi^circ$, where $phi^circ$ is the “semi-dual” of $phi$Prove that a theory $Gamma$ is consistent if and only if there is a structure $M$ so that $M$ $models$ $Gamma$.not always $A models phi$ or $A models neg phi$ exampleIf $Gamma$ is consistent and $Gammanotvdashphi$, then $Gammacupnegphi$ is also consistent. Why?Is there any way to simplify $(Aland B land C) lor (neg A land neg B land neg C)$?Show that $vdash Gamma cup psi$ implies $vdash Gamma cup psi'$ where $psi'$ is $psi$ with one of its bound variables renamed.
$begingroup$
Let $Gammacupphi,psisubseteq L epsilon$ then $Gammamodelspsi$ if and only if $Gamma,(negphi)models(psiland(negpsi))$. I don't seem to understand how the reverse implication goes. Can anyone help me out ? Thanks.
logic first-order-logic
edited Mar 30 at 11:09
blub
3,299929
asked Mar 30 at 10:50
Pedro SantosPedro Santos
16810
$endgroup$
add a comment |
$begingroup$
Let $Gammacupphi,psisubseteq L epsilon$ then $Gammamodelspsi$ if and only if $Gamma,(negphi)models(psiland(negpsi))$. I don't seem to understand how the reverse implication goes. Can anyone help me out ? Thanks.
logic first-order-logic
edited Mar 30 at 11:09
blub
3,299929
asked Mar 30 at 10:50
Pedro SantosPedro Santos
16810
$endgroup$
add a comment |
$begingroup$
Let $Gammacupphi,psisubseteq L epsilon$ then $Gammamodelspsi$ if and only if $Gamma,(negphi)models(psiland(negpsi))$. I don't seem to understand how the reverse implication goes. Can anyone help me out ? Thanks.
logic first-order-logic
edited Mar 30 at 11:09
blub
3,299929
asked Mar 30 at 10:50
Pedro SantosPedro Santos
16810
$endgroup$
Let $Gammacupphi,psisubseteq L epsilon$ then $Gammamodelspsi$ if and only if $Gamma,(negphi)models(psiland(negpsi))$. I don't seem to understand how the reverse implication goes. Can anyone help me out ? Thanks.
logic first-order-logic
logic first-order-logic
edited Mar 30 at 11:09
blub
3,299929
asked Mar 30 at 10:50
Pedro SantosPedro Santos
16810
edited Mar 30 at 11:09
blub
3,299929
asked Mar 30 at 10:50
Pedro SantosPedro Santos
16810
edited Mar 30 at 11:09
blub
3,299929
edited Mar 30 at 11:09
blub
3,299929
edited Mar 30 at 11:09
blub
3,299929
3,299929
asked Mar 30 at 10:50
Pedro SantosPedro Santos
16810
asked Mar 30 at 10:50
Pedro SantosPedro Santos
16810
asked Mar 30 at 10:50
Pedro SantosPedro Santos
16810
16810
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Written like this, it makes no sense. I assume you wanted to write
$$Gammamodelsphitext iff Gamma,negphimodelspsilandnegpsi$$
To prove this, it is helpful to note that for $Deltacuppsisubseteqmathcal L_FO$, $Deltanotmodelspsilandnegpsi$ iff $Delta$ is satisfiable, as then there is an interpretation $mathcal I$ s.t. $mathcal ImodelsDelta$, and naturally $mathcal Inotmodelspsilandnegpsi$.
Now on to proving the equivalence. Let $Gammacupphi,psisubseteqmathcal L_FO$.
From left to right, assume $Gammamodelsphi$, i.e. for every interpretation $mathcal I$: $mathcal ImodelsGamma$ implies $mathcal Imodelsphi$. Thus, no interpretation $mathcal I$ models $Gamma,negphi$ and thus for every interpretation $mathcal I$: $mathcal ImodelsGamma,negphi$ implies $mathcal Imodelspsilandnegpsi$.
From right to left, assume $Gammanotmodelsphi$, i.e. there is an interpretation $mathcal I$ s.t. $mathcal ImodelsGamma$ but $mathcal Inotmodelsphi$. The latter implies $mathcal Imodelsnegphi$. Thus $mathcal ImodelsGamma,negphi$, i.e. $Gamma,negphi$ is satisfiable and thus $Gamma,negphinotmodelspsilandnegpsi$.
answered Mar 30 at 10:57
blubblub
3,299929
$endgroup$
$begingroup$
Ah yes thats what i meant , yes Thank you my friend !
$endgroup$
– Pedro Santos
Mar 30 at 11:00
add a comment |
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
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3168146%2fgamma-models-phi-if-and-only-if-gamma-neg-phi-models-psi-land-neg-psi%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
$begingroup$
Written like this, it makes no sense. I assume you wanted to write
$$Gammamodelsphitext iff Gamma,negphimodelspsilandnegpsi$$
To prove this, it is helpful to note that for $Deltacuppsisubseteqmathcal L_FO$, $Deltanotmodelspsilandnegpsi$ iff $Delta$ is satisfiable, as then there is an interpretation $mathcal I$ s.t. $mathcal ImodelsDelta$, and naturally $mathcal Inotmodelspsilandnegpsi$.
Now on to proving the equivalence. Let $Gammacupphi,psisubseteqmathcal L_FO$.
From left to right, assume $Gammamodelsphi$, i.e. for every interpretation $mathcal I$: $mathcal ImodelsGamma$ implies $mathcal Imodelsphi$. Thus, no interpretation $mathcal I$ models $Gamma,negphi$ and thus for every interpretation $mathcal I$: $mathcal ImodelsGamma,negphi$ implies $mathcal Imodelspsilandnegpsi$.
From right to left, assume $Gammanotmodelsphi$, i.e. there is an interpretation $mathcal I$ s.t. $mathcal ImodelsGamma$ but $mathcal Inotmodelsphi$. The latter implies $mathcal Imodelsnegphi$. Thus $mathcal ImodelsGamma,negphi$, i.e. $Gamma,negphi$ is satisfiable and thus $Gamma,negphinotmodelspsilandnegpsi$.
answered Mar 30 at 10:57
blubblub
3,299929
$endgroup$
$begingroup$
Ah yes thats what i meant , yes Thank you my friend !
$endgroup$
– Pedro Santos
Mar 30 at 11:00
add a comment |
$begingroup$
Written like this, it makes no sense. I assume you wanted to write
$$Gammamodelsphitext iff Gamma,negphimodelspsilandnegpsi$$
To prove this, it is helpful to note that for $Deltacuppsisubseteqmathcal L_FO$, $Deltanotmodelspsilandnegpsi$ iff $Delta$ is satisfiable, as then there is an interpretation $mathcal I$ s.t. $mathcal ImodelsDelta$, and naturally $mathcal Inotmodelspsilandnegpsi$.
Now on to proving the equivalence. Let $Gammacupphi,psisubseteqmathcal L_FO$.
From left to right, assume $Gammamodelsphi$, i.e. for every interpretation $mathcal I$: $mathcal ImodelsGamma$ implies $mathcal Imodelsphi$. Thus, no interpretation $mathcal I$ models $Gamma,negphi$ and thus for every interpretation $mathcal I$: $mathcal ImodelsGamma,negphi$ implies $mathcal Imodelspsilandnegpsi$.
From right to left, assume $Gammanotmodelsphi$, i.e. there is an interpretation $mathcal I$ s.t. $mathcal ImodelsGamma$ but $mathcal Inotmodelsphi$. The latter implies $mathcal Imodelsnegphi$. Thus $mathcal ImodelsGamma,negphi$, i.e. $Gamma,negphi$ is satisfiable and thus $Gamma,negphinotmodelspsilandnegpsi$.
answered Mar 30 at 10:57
blubblub
3,299929
$endgroup$
$begingroup$
Ah yes thats what i meant , yes Thank you my friend !
$endgroup$
– Pedro Santos
Mar 30 at 11:00
add a comment |
$begingroup$
Written like this, it makes no sense. I assume you wanted to write
$$Gammamodelsphitext iff Gamma,negphimodelspsilandnegpsi$$
To prove this, it is helpful to note that for $Deltacuppsisubseteqmathcal L_FO$, $Deltanotmodelspsilandnegpsi$ iff $Delta$ is satisfiable, as then there is an interpretation $mathcal I$ s.t. $mathcal ImodelsDelta$, and naturally $mathcal Inotmodelspsilandnegpsi$.
Now on to proving the equivalence. Let $Gammacupphi,psisubseteqmathcal L_FO$.
From left to right, assume $Gammamodelsphi$, i.e. for every interpretation $mathcal I$: $mathcal ImodelsGamma$ implies $mathcal Imodelsphi$. Thus, no interpretation $mathcal I$ models $Gamma,negphi$ and thus for every interpretation $mathcal I$: $mathcal ImodelsGamma,negphi$ implies $mathcal Imodelspsilandnegpsi$.
From right to left, assume $Gammanotmodelsphi$, i.e. there is an interpretation $mathcal I$ s.t. $mathcal ImodelsGamma$ but $mathcal Inotmodelsphi$. The latter implies $mathcal Imodelsnegphi$. Thus $mathcal ImodelsGamma,negphi$, i.e. $Gamma,negphi$ is satisfiable and thus $Gamma,negphinotmodelspsilandnegpsi$.
answered Mar 30 at 10:57
blubblub
3,299929
$endgroup$
Written like this, it makes no sense. I assume you wanted to write
$$Gammamodelsphitext iff Gamma,negphimodelspsilandnegpsi$$
To prove this, it is helpful to note that for $Deltacuppsisubseteqmathcal L_FO$, $Deltanotmodelspsilandnegpsi$ iff $Delta$ is satisfiable, as then there is an interpretation $mathcal I$ s.t. $mathcal ImodelsDelta$, and naturally $mathcal Inotmodelspsilandnegpsi$.
Now on to proving the equivalence. Let $Gammacupphi,psisubseteqmathcal L_FO$.
From left to right, assume $Gammamodelsphi$, i.e. for every interpretation $mathcal I$: $mathcal ImodelsGamma$ implies $mathcal Imodelsphi$. Thus, no interpretation $mathcal I$ models $Gamma,negphi$ and thus for every interpretation $mathcal I$: $mathcal ImodelsGamma,negphi$ implies $mathcal Imodelspsilandnegpsi$.
From right to left, assume $Gammanotmodelsphi$, i.e. there is an interpretation $mathcal I$ s.t. $mathcal ImodelsGamma$ but $mathcal Inotmodelsphi$. The latter implies $mathcal Imodelsnegphi$. Thus $mathcal ImodelsGamma,negphi$, i.e. $Gamma,negphi$ is satisfiable and thus $Gamma,negphinotmodelspsilandnegpsi$.
answered Mar 30 at 10:57
blubblub
3,299929
edited Mar 30 at 11:02
answered Mar 30 at 10:57
blubblub
3,299929
answered Mar 30 at 10:57
blubblub
3,299929
answered Mar 30 at 10:57
blubblub
3,299929
3,299929
$begingroup$
Ah yes thats what i meant , yes Thank you my friend !
$endgroup$
– Pedro Santos
Mar 30 at 11:00
add a comment |
$begingroup$
Ah yes thats what i meant , yes Thank you my friend !
$endgroup$
– Pedro Santos
Mar 30 at 11:00
$begingroup$
Ah yes thats what i meant , yes Thank you my friend !
$endgroup$
– Pedro Santos
Mar 30 at 11:00
$begingroup$
Ah yes thats what i meant , yes Thank you my friend !
$endgroup$
– Pedro Santos
Mar 30 at 11:00
add a comment |
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.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3168146%2fgamma-models-phi-if-and-only-if-gamma-neg-phi-models-psi-land-neg-psi%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
