Difference between revisions of "Interpretation (FOL)"

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==References==
 
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{{Definition|Formal Logic}}

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

Definition

An interpretation is a mapping, I:LM 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]:

  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

References

  1. Jump up to: 1.0 1.1 Mathematical Logic - Foundations for Information Science - Wei Li

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