# Precedence of satisfiability operator

I'm just reading a textbook in mathematical logic, as following:

What is the precedence to consider the equality and satisfiability operators in the equation pinpointed in red?! In other words, which of following cases right:

(A \sat x) = y[s] or A \sat (x = y[s])

• Probably neither. I guess $s$ is an environment, i.e. a mapping from variables to members of the universe. In this case $\vDash$ is a ternary relation between a model, a formula, and an environment. The formula in the example is $x = y$. – Martin Berger Jan 12 '17 at 1:59
• But the book has been considered ⊨ as a binary operator. How could it be defined in a ternary semantic? – Roboticist Jan 12 '17 at 2:00
• I don't know the book, so how can I answer this. But to interpret a FOL formula, you need to have a denotation of the formula's free variables. That's not given by the standard notion of) model. That's the purpose of the environment, or "assignment function". – Martin Berger Jan 12 '17 at 2:03
• Sorry.. You're right... The book is "A Friendly Introduction to Mathematical Logic" by Leary and Kristiansen. – Roboticist Jan 12 '17 at 2:05
• Well, it's online here. Indeed, $\vDash$ is a ternary relation in this book. – Martin Berger Jan 12 '17 at 2:20