# logic quantifiers truth

$$\forall x \forall y \forall z[P(x,y) \land P(y,z) \implies P(x,z)]$$

for the domain all people in the world, and
$$P(x,y) \iff$$ x likes y

how can i know that this is true or false?

i can change the equation to: $$\forall x \forall y \forall z[ \neg P(x,y) \lor \neg P(y,z) \lor P(x,z)]$$

i tried to draw truth tables

P(x,y) P(y,z) P(x,z) |$$\neg P(x,y)$$|$$\neg P(y,z)$$|P(x,z)|result

TTT |F F T T

TTF |F F F F

TFT |F F T T

TFF |F T F T

FTT |T F T T

FTF |T F F T

FFT |T T T T

FFF |T T F T

means is not true for all domain?

is the negation will be $$\exists x \exists y \exists z[P(x,y) \land P(y,z) \land \neg P(x,z)]$$

The proposition you gave $$\forall x \forall y \forall z[P(x,y) \land P(y,z) \implies P(x,z)]$$ Is the definition of transitivity. It is true in some context, but not in all context.
E.g. let $$x, y, z, \in \Bbb R$$ and $$P(x, y) \iff x. Then your proposition is true. $$x and $$y implies $$x.
The negative of this proposition is $$\exists x \exists y \exists z[P(x,y) \land P(y,z) \land \neg P(x,z)]$$