# Terminology First-Order Logic

A graph $$G$$ is said to be a model of a first-order sentence $$\varphi$$ if $$G$$ satisfies $$\varphi$$. Now let $$\varphi(x_1,...,x_r)$$ be a first order formula with free variables $$x_1,...,x_r$$. What is the standard terminology for a tuple of vertices $$(v_1,...,v_r)$$ of $$G$$ such that $$G$$ satisfies the formula $$\varphi(v_1,...,v_r)$$ obtained by substituting $$x_i$$ with $$v_i$$?

What I'm having trouble is to find the right terminology to connect $$G$$ to the assignment. The only thing I can think now is the following.

"$$(v_1,...,v_r)$$ is a satisfying assignment for $$\varphi$$ in $$G$$."

But maybe there is a shorter way.

I think that's often called a "variable assignment", since it assigns to each variable a value (a vertex in the graph, in your case).

If the graph is equipped with a set of such tuples, these might be considered to be hyperedges, i.e. edges connected to an arbitrary number of vertices (not necessarily two of them).

• Thanks. I'm fine with the fact that it is an assignment. What I'm having trouble is to find the right terminology to connect $G$ to the assignment. For instance, would the standard be to say that $(v_1,...,v_r)$ is a satisfying assignment for $\varphi$ in $G$? – verifying Mar 1 '19 at 20:28
• @verifying I would understand that. You could even say that $(v_1,\ldots,v_r)$ satisfies $\varphi$ for short. I can't guarantee that's the most standard way, but I find that very reasonable and clear. – chi Mar 1 '19 at 23:00