# Predicate logic translation

Given:

$$Sx: x$$ is a sith

$$Kxy: x$$ kills $$y$$

$$Dx: x$$ succumbs to the dark side.

Translate: Not everyone who kills a Sith succumbs to the dark side.

Not (everyone who kills a sith succumbs to the dark side)

Give the implicit people names:

Not (everyone, $a$, who kills a sith, $b$, succumbs to the dark side)

Not is a '$\neg$':

$\neg($ for everyone$, a,$ who (kills a sith $b$) implies $a$ succumbs to the dark side)

Saying (kills a sith $b$) can be translated as $\exists b(b$ is a sith and $a$ kills $b)$

$\neg($ for everyone, $a$, $\exists b(b$ is a sith and $a$ kills $b)$ implies $a$ succumbs to the dark side)

quantify over $a$ to account for 'everyone':

$\neg\forall a( \exists b(b$ is a sith and $a$ kills $b)$ implies $a$ succumbs to the dark side)

finish:

$\neg\forall a(\exists b(Sb\wedge Kab)\rightarrow Da)$

• Thank you so much! Then how can I translate into Predicate language with identity? (with "=") Commented May 2, 2015 at 6:36
• You wouldn't need identity for this statement anyway, so what I did works whether or not your language includes identity. First order logic without identity is strictly contained in first order logic with identity, so anything valid in the language without identity is valid in the language with identity (but not vice versa, ofcourse).
– jack
Commented May 2, 2015 at 7:00

$\sim (\forall x,y) ( Sy \wedge Kxy \wedge Dx )$

Replace "For all" with upside down A