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

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