I've trying to solve this for a while.
Problem
Let $L$ be a first order language with the predicate symbol $p$. Let S be the extension obtained by adding to $K_L$ the open $wf. p(x, y)$. Is S consistent?
Definitions
In my bibliography there is not definition of openness, but closeness. It is: A $wf. A$ of $L$ is said to be closed if no variable occurs free in $A$.
$K$ is a consistent first order system.
The book I'm using is: Logic for mathematicians, Hamilton.
My thoughts
Well, I think that it is consistent because a model can be created for $S$.
My question
In reality, what it is really puzzling me is the open word. What is the difference of creating an extension by adding an open or a closed $wf.$?
Thanks in advance.