Interpretation (FOL)

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Definition

An interpretation is a mapping, $I:\mathscr{L}\rightarrow\mathbb{M}$ where $\mathscr{L}$ is a first order language and $\mathbb{M}$ is a domain. As we often identify a domain with its set, we may write $I:\mathscr{L}\rightarrow M$ instead. Recall a domain is a 3-tuple, $(M,\mathcal{F},\mathcal{R})$. An interpretation has the following properties:

1. For each constant symbol, $c$ in $\mathscr{L}$, $I(c)$ is an element of $M$
2. For each $n$-ary function symbol, $f$ in $\mathscr{L}$, $I(c)$ is an element of $\mathcal{F}$
3. For each $n$-ary predicate symbol, $P$ in $\mathscr{L}$, $I(P)$ is an element of $\mathcal{R}$

An interpretation is usually used as part of a structure (of a first order language, $\mathscr{L}$), $\mathbf{M}$, which is a 2-tuple: $\mathbf{M}:=(\mathbb{M},I)$ where $\mathbb{M}$ is a domain and $I$ an interpretation as defined above. When an interpretation is used as a part of a structure we adopt the following notation:

• $I(c)$ is written $c_\mathbf{M}$,
• $I(f)$ is written $f_\mathbf{M}$ and
• $I(P)$ is written $P_\mathbf{M}$

See next

• Structure
• Model