Formal logic language

Introduction

Formal logic plays an important role, especially so in set theory, but overall an important role. It is important to have a concrete understanding of this.

Example 1

$$\forall x\forall y\forall z((P(x,y)\wedge P(y,z))\rightarrow P(x,z))$$	$$\forall x\forall y\forall z(P(x,y)\wedge P(y,z)\rightarrow P(x,z))$$
for all $x$ for all $y$ for all $z$ if we have both $P(x,y)$ $P(y,z)$ then $P(x,z)$	for all $x$ for all $y$ for all $z$ we have both $P(x,y)$ if we have $P(y,z)$ then $P(x,z)$
Whenever we have ($P(x,y)$ and $P(y,z)$) we also have $P(x,z)$	$P(x,y)$ is true and whenever $P(y,z)$ is true then $P(x,z)$ is true
It is always true that if ($P(x,y)$ and $P(y,z)$) then $P(x,z)$	It is always true that $P(x,y))$ and (if $P(y,z)$ then $P(x,z)$)
$P(x,y)$ $P(y,z)$ $P(x,z)$ Uncertain True Uncertain True Uncertain True True True	$P(x,y)$ $P(y,z)$ $P(x,z)$ True (definition) Uncertain True (definition) True True