# English statement to logic

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.

• Well, the statements says about existence of a cheater. So, it should be rather $\exists x \left( \text{cheats} \left( x \right) \right) \rightarrow \forall y \left( \text{suffer} \left( y \right) \right)$. – Aniruddha Deshmukh Dec 10 '18 at 6:55
• Latex didn't render properly in your comment Aniruddha – user3767495 Dec 10 '18 at 6:56
• @Shaun-Any Reference for your statement? – user3767495 Dec 10 '18 at 6:59
• 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". – Aniruddha Deshmukh Dec 10 '18 at 7:00
• $\forall x:(cheat (x) \to M)$ 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. – fleablood Dec 10 '18 at 7:37

## 1 Answer

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.

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