Definition. Interpretation (FOL) [0012]

  • November 22, 2025

In first order logic a Interpretation; similar to the case of propositional logic; is a mapping which assigns meaning to the syntax of the language. In this instance it's a mapping from constants, function symbols and predicate symbols to specific objects, functions and relations in a given domain. We can define an interpretation as a sort of piece wise function as follows:

\[ \begin {align*} I &: \mathcal C \cup \mathcal F \cup \mathcal R \to D \cup (D^n \to D) \cup (D^n \to \{\top , \bot \}) \\ I(c) & = d \quad \forall c \in \mathcal C, d \in D \\ I(f) & = f_D: D^n \to D \quad \forall f \in \mathcal F \\ I(p) & = p_D: D^n \to \{\top , \bot \} \quad \forall p \in \mathcal R \\ \end {align*} \]

Where \(D\) (sometimes also denoted as \(U\)) is the domain of discourse which is a non-empty set of objects over which the quantifiers range. Intuitively the domain of discourse represents what we wish to talk about in our interpretation. We can break this down into 3 parts:

  • constants (\(c \in \mathcal C\)): are mapped to specific elements in the domain \(d \in D\). For example if \(c\) is the constant \("a"\) and \(D = \{1, 2, 3\}\), then \(I(a) = 1\) could be a valid mapping.
  • function symbols (\(f \in \mathcal F\)): are mapped to functions that take elements from the domain and return elements in the domain. For example if \(f\) is a unary function symbol \("f"\) and \(D = \{1, 2, 3\}\), then \(I(f) = f_D\) where \(f_D(1) = 2\), \(f_D(2) = 3\), and \(f_D(3) = 1\) could be a valid mapping.
  • relation symbols (\(p \in \mathcal R\)): are mapped to relations (or predicates) that take elements from the domain and return truth values \(\{\top , \bot \}\). For example if \(p\) is a binary relation symbol \("R"\) and \(D = \{1, 2, 3\}\), then \(I(R) = R_D\) where \(R_D(1, 2) = \top \), \(R_D(2, 3) = \bot \), and \(R_D(3, 1) = \top \) could be a valid mapping.