Difference between revisions of "Interpretation (FOL)"
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Latest revision as of 06:54, 10 September 2016
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Created so I don't have to sift through notes or scour PDFs, it needs more references and fleshing out
Contents
[hide]Definition
An interpretation is a mapping, I:L→M where \mathscr{L} is a first order language and \mathbb{M} is a domain[1]. 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
- For each n-ary function symbol, f in \mathscr{L} , I(c) is an element of \mathcal{F}
- 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}