Interpretation (FOL)

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Created so I don't have to sift through notes or scour PDFs, it needs more references and fleshing out

Definition

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

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

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

• [ilmath]I(c)[/ilmath] is written [ilmath]c_\mathbf{M} [/ilmath],
• [ilmath]I(f)[/ilmath] is written [ilmath]f_\mathbf{M} [/ilmath] and
• [ilmath]I(P)[/ilmath] is written [ilmath]P_\mathbf{M} [/ilmath]

See next

• Structure
• Model