English statement to logic












1












$begingroup$


If anyone cheats, everyone suffers.



$S1:forall x(cheat(x) rightarrow forall y, suffer(y))$



$S2:forall x forall y(cheat(x) rightarrow suffer(y))$



I thought Both S1 and S2 are wrong because they imply that if everyone cheats then everyone suffers, but the original sentence will be true even if one person from the domain of person cheats.



Please let me know what I am missing.



I need to know why I failed to recognize these two statements.






predicate-logic







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    Well, the statements says about existence of a cheater. So, it should be rather $exists x left( text{cheats} left( x right) right) rightarrow forall y left( text{suffer} left( y right) right)$.
    $endgroup$
    – Aniruddha Deshmukh
    Dec 10 '18 at 6:55












  • $begingroup$
    Latex didn't render properly in your comment Aniruddha
    $endgroup$
    – user3767495
    Dec 10 '18 at 6:56










  • $begingroup$
    @Shaun-Any Reference for your statement?
    $endgroup$
    – user3767495
    Dec 10 '18 at 6:59










  • $begingroup$
    I do not really think so. The statement (in English), means that if someone cheats, then everyone suffers. It should not mean "If everyone cheats, then everyone suffers".
    $endgroup$
    – Aniruddha Deshmukh
    Dec 10 '18 at 7:00






  • 1




    $begingroup$
    $forall x:(cheat (x) to M)$ means for every person, if that person cheats then M. If everyone cheats then M would be $(forall x(cheat(x)))to M$. You have to see what statement is being said about "forall x". In S1 it's $cheat(x)to M$ which means it is to for everybody if they cheat then M.
    $endgroup$
    – fleablood
    Dec 10 '18 at 7:37
















1












$begingroup$


If anyone cheats, everyone suffers.



$S1:forall x(cheat(x) rightarrow forall y, suffer(y))$



$S2:forall x forall y(cheat(x) rightarrow suffer(y))$



I thought Both S1 and S2 are wrong because they imply that if everyone cheats then everyone suffers, but the original sentence will be true even if one person from the domain of person cheats.



Please let me know what I am missing.



I need to know why I failed to recognize these two statements.






predicate-logic







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    Well, the statements says about existence of a cheater. So, it should be rather $exists x left( text{cheats} left( x right) right) rightarrow forall y left( text{suffer} left( y right) right)$.
    $endgroup$
    – Aniruddha Deshmukh
    Dec 10 '18 at 6:55












  • $begingroup$
    Latex didn't render properly in your comment Aniruddha
    $endgroup$
    – user3767495
    Dec 10 '18 at 6:56










  • $begingroup$
    @Shaun-Any Reference for your statement?
    $endgroup$
    – user3767495
    Dec 10 '18 at 6:59










  • $begingroup$
    I do not really think so. The statement (in English), means that if someone cheats, then everyone suffers. It should not mean "If everyone cheats, then everyone suffers".
    $endgroup$
    – Aniruddha Deshmukh
    Dec 10 '18 at 7:00






  • 1




    $begingroup$
    $forall x:(cheat (x) to M)$ means for every person, if that person cheats then M. If everyone cheats then M would be $(forall x(cheat(x)))to M$. You have to see what statement is being said about "forall x". In S1 it's $cheat(x)to M$ which means it is to for everybody if they cheat then M.
    $endgroup$
    – fleablood
    Dec 10 '18 at 7:37














1












1








1





$begingroup$


If anyone cheats, everyone suffers.



$S1:forall x(cheat(x) rightarrow forall y, suffer(y))$



$S2:forall x forall y(cheat(x) rightarrow suffer(y))$



I thought Both S1 and S2 are wrong because they imply that if everyone cheats then everyone suffers, but the original sentence will be true even if one person from the domain of person cheats.



Please let me know what I am missing.



I need to know why I failed to recognize these two statements.






predicate-logic







share|cite|improve this question









$endgroup$




If anyone cheats, everyone suffers.



$S1:forall x(cheat(x) rightarrow forall y, suffer(y))$



$S2:forall x forall y(cheat(x) rightarrow suffer(y))$



I thought Both S1 and S2 are wrong because they imply that if everyone cheats then everyone suffers, but the original sentence will be true even if one person from the domain of person cheats.



Please let me know what I am missing.



I need to know why I failed to recognize these two statements.






predicate-logic




predicate-logic






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 10 '18 at 6:52









user3767495user3767495

3988




3988








  • 3




    $begingroup$
    Well, the statements says about existence of a cheater. So, it should be rather $exists x left( text{cheats} left( x right) right) rightarrow forall y left( text{suffer} left( y right) right)$.
    $endgroup$
    – Aniruddha Deshmukh
    Dec 10 '18 at 6:55












  • $begingroup$
    Latex didn't render properly in your comment Aniruddha
    $endgroup$
    – user3767495
    Dec 10 '18 at 6:56










  • $begingroup$
    @Shaun-Any Reference for your statement?
    $endgroup$
    – user3767495
    Dec 10 '18 at 6:59










  • $begingroup$
    I do not really think so. The statement (in English), means that if someone cheats, then everyone suffers. It should not mean "If everyone cheats, then everyone suffers".
    $endgroup$
    – Aniruddha Deshmukh
    Dec 10 '18 at 7:00






  • 1




    $begingroup$
    $forall x:(cheat (x) to M)$ means for every person, if that person cheats then M. If everyone cheats then M would be $(forall x(cheat(x)))to M$. You have to see what statement is being said about "forall x". In S1 it's $cheat(x)to M$ which means it is to for everybody if they cheat then M.
    $endgroup$
    – fleablood
    Dec 10 '18 at 7:37














  • 3




    $begingroup$
    Well, the statements says about existence of a cheater. So, it should be rather $exists x left( text{cheats} left( x right) right) rightarrow forall y left( text{suffer} left( y right) right)$.
    $endgroup$
    – Aniruddha Deshmukh
    Dec 10 '18 at 6:55












  • $begingroup$
    Latex didn't render properly in your comment Aniruddha
    $endgroup$
    – user3767495
    Dec 10 '18 at 6:56










  • $begingroup$
    @Shaun-Any Reference for your statement?
    $endgroup$
    – user3767495
    Dec 10 '18 at 6:59










  • $begingroup$
    I do not really think so. The statement (in English), means that if someone cheats, then everyone suffers. It should not mean "If everyone cheats, then everyone suffers".
    $endgroup$
    – Aniruddha Deshmukh
    Dec 10 '18 at 7:00






  • 1




    $begingroup$
    $forall x:(cheat (x) to M)$ means for every person, if that person cheats then M. If everyone cheats then M would be $(forall x(cheat(x)))to M$. You have to see what statement is being said about "forall x". In S1 it's $cheat(x)to M$ which means it is to for everybody if they cheat then M.
    $endgroup$
    – fleablood
    Dec 10 '18 at 7:37








3




3




$begingroup$
Well, the statements says about existence of a cheater. So, it should be rather $exists x left( text{cheats} left( x right) right) rightarrow forall y left( text{suffer} left( y right) right)$.
$endgroup$
– Aniruddha Deshmukh
Dec 10 '18 at 6:55






$begingroup$
Well, the statements says about existence of a cheater. So, it should be rather $exists x left( text{cheats} left( x right) right) rightarrow forall y left( text{suffer} left( y right) right)$.
$endgroup$
– Aniruddha Deshmukh
Dec 10 '18 at 6:55














$begingroup$
Latex didn't render properly in your comment Aniruddha
$endgroup$
– user3767495
Dec 10 '18 at 6:56




$begingroup$
Latex didn't render properly in your comment Aniruddha
$endgroup$
– user3767495
Dec 10 '18 at 6:56












$begingroup$
@Shaun-Any Reference for your statement?
$endgroup$
– user3767495
Dec 10 '18 at 6:59




$begingroup$
@Shaun-Any Reference for your statement?
$endgroup$
– user3767495
Dec 10 '18 at 6:59












$begingroup$
I do not really think so. The statement (in English), means that if someone cheats, then everyone suffers. It should not mean "If everyone cheats, then everyone suffers".
$endgroup$
– Aniruddha Deshmukh
Dec 10 '18 at 7:00




$begingroup$
I do not really think so. The statement (in English), means that if someone cheats, then everyone suffers. It should not mean "If everyone cheats, then everyone suffers".
$endgroup$
– Aniruddha Deshmukh
Dec 10 '18 at 7:00




1




1




$begingroup$
$forall x:(cheat (x) to M)$ means for every person, if that person cheats then M. If everyone cheats then M would be $(forall x(cheat(x)))to M$. You have to see what statement is being said about "forall x". In S1 it's $cheat(x)to M$ which means it is to for everybody if they cheat then M.
$endgroup$
– fleablood
Dec 10 '18 at 7:37




$begingroup$
$forall x:(cheat (x) to M)$ means for every person, if that person cheats then M. If everyone cheats then M would be $(forall x(cheat(x)))to M$. You have to see what statement is being said about "forall x". In S1 it's $cheat(x)to M$ which means it is to for everybody if they cheat then M.
$endgroup$
– fleablood
Dec 10 '18 at 7:37










1 Answer
1






active

oldest

votes


















0












$begingroup$

You aren't applying the "forall condition to the right clause".



Consider the Four sentences. 1)"For everyone, it is true if they cheat then everyone will suffer" and 2)"If everyone cheats, everyone will suffer" 3) "If there exists someone who cheats, then everyone suffers" and 4) "There exists someone who if he cheats everyone will suffer".



What's the difference between the sentences? For 1) and 4) the quantifiers "for everyone" and "there is someone" refer to the entire sentence "if he cheats then everyone suffers". For 2) and 3) the quantifiers only only refer to the conditional clause and if it is true for that person of for everyone, then the result.



So



1) $color{blue}{forall x(}color{orange}{cheats(x)to forall y(suffer(y))}color{blue}{)}$



means "for everyone, if he cheats everyone suffers" = "Hey, margaret! If you cheat we all suffer. Tom, if you cheat, we all suffer. Everybody! If any of you cheat we all suffer" = "if anyone cheats then everyone suffers".



This one you want.



2)$color{blue}{(forall x(cheats(x))}color{orange}{to forall y(suffer(y)))}$



That means "If everybody, every single person cheats, then we all suffer" = "if everybody cheats, then everyone suffers".



This one you don't



3)$color{blue}{(exists x(cheats(x))}color{orange}{to forall y(suffer(y)))}$



This means "if there is someone who cheats, then everyone suffers" or "if someone cheats, then everyone suffer" or "if anyone cheats, then everyone suffers".



You want this one too.



4)$color{blue}{exists x(}color{orange}{cheats(x)to forall y(suffer(y))}color{blue}{)}$



This means "there exists someone who if he cheats, then everyone cheats". Or "You see that person over there? If that person cheats the everyone suffers but if someone else cheats nothing will happen.



We don't want that one either.



We can isolate the qualifiers for the $y$ as well.



S2) $color{blue}{forall x}color{orange}{forall y}color{green}{(cheat(color{blue}x )to suffer(color{orange}y))}$



means "for every person x and for every person y, if $x$ cheats, however they are, then $y$ suffers whoever they are."= "if anyone cheats, then everyone suffers".



This one is one you want too.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This makes sense.Awesome!!. Thanks fleablood.
    $endgroup$
    – user3767495
    Dec 11 '18 at 4:14











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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

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0












$begingroup$

You aren't applying the "forall condition to the right clause".



Consider the Four sentences. 1)"For everyone, it is true if they cheat then everyone will suffer" and 2)"If everyone cheats, everyone will suffer" 3) "If there exists someone who cheats, then everyone suffers" and 4) "There exists someone who if he cheats everyone will suffer".



What's the difference between the sentences? For 1) and 4) the quantifiers "for everyone" and "there is someone" refer to the entire sentence "if he cheats then everyone suffers". For 2) and 3) the quantifiers only only refer to the conditional clause and if it is true for that person of for everyone, then the result.



So



1) $color{blue}{forall x(}color{orange}{cheats(x)to forall y(suffer(y))}color{blue}{)}$



means "for everyone, if he cheats everyone suffers" = "Hey, margaret! If you cheat we all suffer. Tom, if you cheat, we all suffer. Everybody! If any of you cheat we all suffer" = "if anyone cheats then everyone suffers".



This one you want.



2)$color{blue}{(forall x(cheats(x))}color{orange}{to forall y(suffer(y)))}$



That means "If everybody, every single person cheats, then we all suffer" = "if everybody cheats, then everyone suffers".



This one you don't



3)$color{blue}{(exists x(cheats(x))}color{orange}{to forall y(suffer(y)))}$



This means "if there is someone who cheats, then everyone suffers" or "if someone cheats, then everyone suffer" or "if anyone cheats, then everyone suffers".



You want this one too.



4)$color{blue}{exists x(}color{orange}{cheats(x)to forall y(suffer(y))}color{blue}{)}$



This means "there exists someone who if he cheats, then everyone cheats". Or "You see that person over there? If that person cheats the everyone suffers but if someone else cheats nothing will happen.



We don't want that one either.



We can isolate the qualifiers for the $y$ as well.



S2) $color{blue}{forall x}color{orange}{forall y}color{green}{(cheat(color{blue}x )to suffer(color{orange}y))}$



means "for every person x and for every person y, if $x$ cheats, however they are, then $y$ suffers whoever they are."= "if anyone cheats, then everyone suffers".



This one is one you want too.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This makes sense.Awesome!!. Thanks fleablood.
    $endgroup$
    – user3767495
    Dec 11 '18 at 4:14
















0












$begingroup$

You aren't applying the "forall condition to the right clause".



Consider the Four sentences. 1)"For everyone, it is true if they cheat then everyone will suffer" and 2)"If everyone cheats, everyone will suffer" 3) "If there exists someone who cheats, then everyone suffers" and 4) "There exists someone who if he cheats everyone will suffer".



What's the difference between the sentences? For 1) and 4) the quantifiers "for everyone" and "there is someone" refer to the entire sentence "if he cheats then everyone suffers". For 2) and 3) the quantifiers only only refer to the conditional clause and if it is true for that person of for everyone, then the result.



So



1) $color{blue}{forall x(}color{orange}{cheats(x)to forall y(suffer(y))}color{blue}{)}$



means "for everyone, if he cheats everyone suffers" = "Hey, margaret! If you cheat we all suffer. Tom, if you cheat, we all suffer. Everybody! If any of you cheat we all suffer" = "if anyone cheats then everyone suffers".



This one you want.



2)$color{blue}{(forall x(cheats(x))}color{orange}{to forall y(suffer(y)))}$



That means "If everybody, every single person cheats, then we all suffer" = "if everybody cheats, then everyone suffers".



This one you don't



3)$color{blue}{(exists x(cheats(x))}color{orange}{to forall y(suffer(y)))}$



This means "if there is someone who cheats, then everyone suffers" or "if someone cheats, then everyone suffer" or "if anyone cheats, then everyone suffers".



You want this one too.



4)$color{blue}{exists x(}color{orange}{cheats(x)to forall y(suffer(y))}color{blue}{)}$



This means "there exists someone who if he cheats, then everyone cheats". Or "You see that person over there? If that person cheats the everyone suffers but if someone else cheats nothing will happen.



We don't want that one either.



We can isolate the qualifiers for the $y$ as well.



S2) $color{blue}{forall x}color{orange}{forall y}color{green}{(cheat(color{blue}x )to suffer(color{orange}y))}$



means "for every person x and for every person y, if $x$ cheats, however they are, then $y$ suffers whoever they are."= "if anyone cheats, then everyone suffers".



This one is one you want too.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This makes sense.Awesome!!. Thanks fleablood.
    $endgroup$
    – user3767495
    Dec 11 '18 at 4:14














0












0








0





$begingroup$

You aren't applying the "forall condition to the right clause".



Consider the Four sentences. 1)"For everyone, it is true if they cheat then everyone will suffer" and 2)"If everyone cheats, everyone will suffer" 3) "If there exists someone who cheats, then everyone suffers" and 4) "There exists someone who if he cheats everyone will suffer".



What's the difference between the sentences? For 1) and 4) the quantifiers "for everyone" and "there is someone" refer to the entire sentence "if he cheats then everyone suffers". For 2) and 3) the quantifiers only only refer to the conditional clause and if it is true for that person of for everyone, then the result.



So



1) $color{blue}{forall x(}color{orange}{cheats(x)to forall y(suffer(y))}color{blue}{)}$



means "for everyone, if he cheats everyone suffers" = "Hey, margaret! If you cheat we all suffer. Tom, if you cheat, we all suffer. Everybody! If any of you cheat we all suffer" = "if anyone cheats then everyone suffers".



This one you want.



2)$color{blue}{(forall x(cheats(x))}color{orange}{to forall y(suffer(y)))}$



That means "If everybody, every single person cheats, then we all suffer" = "if everybody cheats, then everyone suffers".



This one you don't



3)$color{blue}{(exists x(cheats(x))}color{orange}{to forall y(suffer(y)))}$



This means "if there is someone who cheats, then everyone suffers" or "if someone cheats, then everyone suffer" or "if anyone cheats, then everyone suffers".



You want this one too.



4)$color{blue}{exists x(}color{orange}{cheats(x)to forall y(suffer(y))}color{blue}{)}$



This means "there exists someone who if he cheats, then everyone cheats". Or "You see that person over there? If that person cheats the everyone suffers but if someone else cheats nothing will happen.



We don't want that one either.



We can isolate the qualifiers for the $y$ as well.



S2) $color{blue}{forall x}color{orange}{forall y}color{green}{(cheat(color{blue}x )to suffer(color{orange}y))}$



means "for every person x and for every person y, if $x$ cheats, however they are, then $y$ suffers whoever they are."= "if anyone cheats, then everyone suffers".



This one is one you want too.






share|cite|improve this answer









$endgroup$



You aren't applying the "forall condition to the right clause".



Consider the Four sentences. 1)"For everyone, it is true if they cheat then everyone will suffer" and 2)"If everyone cheats, everyone will suffer" 3) "If there exists someone who cheats, then everyone suffers" and 4) "There exists someone who if he cheats everyone will suffer".



What's the difference between the sentences? For 1) and 4) the quantifiers "for everyone" and "there is someone" refer to the entire sentence "if he cheats then everyone suffers". For 2) and 3) the quantifiers only only refer to the conditional clause and if it is true for that person of for everyone, then the result.



So



1) $color{blue}{forall x(}color{orange}{cheats(x)to forall y(suffer(y))}color{blue}{)}$



means "for everyone, if he cheats everyone suffers" = "Hey, margaret! If you cheat we all suffer. Tom, if you cheat, we all suffer. Everybody! If any of you cheat we all suffer" = "if anyone cheats then everyone suffers".



This one you want.



2)$color{blue}{(forall x(cheats(x))}color{orange}{to forall y(suffer(y)))}$



That means "If everybody, every single person cheats, then we all suffer" = "if everybody cheats, then everyone suffers".



This one you don't



3)$color{blue}{(exists x(cheats(x))}color{orange}{to forall y(suffer(y)))}$



This means "if there is someone who cheats, then everyone suffers" or "if someone cheats, then everyone suffer" or "if anyone cheats, then everyone suffers".



You want this one too.



4)$color{blue}{exists x(}color{orange}{cheats(x)to forall y(suffer(y))}color{blue}{)}$



This means "there exists someone who if he cheats, then everyone cheats". Or "You see that person over there? If that person cheats the everyone suffers but if someone else cheats nothing will happen.



We don't want that one either.



We can isolate the qualifiers for the $y$ as well.



S2) $color{blue}{forall x}color{orange}{forall y}color{green}{(cheat(color{blue}x )to suffer(color{orange}y))}$



means "for every person x and for every person y, if $x$ cheats, however they are, then $y$ suffers whoever they are."= "if anyone cheats, then everyone suffers".



This one is one you want too.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 10 '18 at 8:09









fleabloodfleablood

71.1k22686




71.1k22686












  • $begingroup$
    This makes sense.Awesome!!. Thanks fleablood.
    $endgroup$
    – user3767495
    Dec 11 '18 at 4:14


















  • $begingroup$
    This makes sense.Awesome!!. Thanks fleablood.
    $endgroup$
    – user3767495
    Dec 11 '18 at 4:14
















$begingroup$
This makes sense.Awesome!!. Thanks fleablood.
$endgroup$
– user3767495
Dec 11 '18 at 4:14




$begingroup$
This makes sense.Awesome!!. Thanks fleablood.
$endgroup$
– user3767495
Dec 11 '18 at 4:14


















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Puebla de Zaragoza

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Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
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Musa

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Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
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