Definition. Structures and Variable Assignments (FOL) [0013]

A structure \(\mathcal {M}\) for a first-order language \(\mathcal {L}\) consists of:

• A non-empty domain \(D\), which is the set of objects that the variables can refer to.
• An interpretation function \(I\) that assigns meanings to the non-logical symbols in \(\mathcal {L}\):
• For each constant symbol \(c\) in \(\mathcal {L}\), \(I(c)\) is an element of \(D\).
• For each n-ary function symbol \(f\) in \(\mathcal {L}\), \(I(f)\) is a function from \(D^n\) to \(D\).
• For each n-ary predicate symbol \(P\) in \(\mathcal {L}\), \(I(P)\) is a subset of \(D^n\).

A variable assignment \(\sigma \) for a structure \(S\) is a function that assigns each variable to an element of the domain \(D\) of the structure. That is, for each variable \(x\), \(\sigma (x) \in D\).