Some formulas can get pretty hairy: ∀** x** :(∃

**:(∀**

*y**: (*

**z****likes (**∧

*x*,*y*)**¬**

**likes (**))). The zeroth step is to take a breath, and read this in English: for every

*y*,*z*)*x*, there's some

*y*such that for every

**,**

*z**x*likes

**but**

*y***doesn't like**

*y***. Even so, how do we approach getting a handle on what this means? Given an interpretation, how do we know it's true?**

*z*
The top-down way would be to read this formula left-to-right. Is the whole formula true? Well, it's only true if, for every possible value of *x*, some smaller formula is true (namely, "
there exists a ** y** such that forall

*z*,

**likes (**and ¬

*x*,*y*)**likes**(

**). "). (This is a formula with**

*y*,*z**x*free, that is, it's a statement about

*x*.) And is that formula true? Well, precisely when we can find some

*y*such that ... (and so on). This direct approach is hard to keep inside your head all at once.

Most people prefer approaching the problems in a bottom-up manner (or if you prefer, right-to-left or inside-out): First consider at the small inner bits alone, figure out what they mean, and only then figure out how they relate.

- What does the innermost formula
**likes**(*x*,*y*) ∧¬**likes**(*y*,*z*) mean, in English? That's not so bad:*x*likes*y*, and*y*dislikes*z*. A statement about three people called*x*,*y*,*z*. - Working outward, what does ∀
*z*: (**likes**(*x*,*y*) ∧¬**likes**(*y*,*z*)) mean? Ah, not so bad either:*x*likes*y*, and*y*dislikes everybody.^{6} - Keep on going: ∃
*y*:(∀*z*: (**likes**(*x*,*y*) ∧¬**likes**(*y*,*z*))) becomes "*x*likes some misanthrope". - Now it's clear: ∀
*x*:(∃*y*:(∀*z*: (**likes**(*x*,*y*) ∧¬**likes**(*y*,*z*)))) is just "everybody likes some misanthrope".

Phew!

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