Term interpretation

The term interpretation is a term used in mathematical logic, it is a special interpretation in the first order predicate logic .

If a set of expressions of a language is given, an interpretation of the language dependent on it should be constructed. This essentially uses the terms of the language. An interpretation is given by its universe (non-empty set), by an interpretation of the symbols in and a variable assignment. We start by defining the universe of interpretation. By ${\displaystyle \Phi }$ ${\displaystyle L_{I}^{S}}$ ${\displaystyle \Phi }$ ${\displaystyle S}$

${\displaystyle t_{1}\sim t_{2}\quad \Leftrightarrow \quad \Phi \vdash t_{1}\equiv t_{2}}$

an equivalence relation is defined on the set of all terms of the language. The set of equivalence classes is denoted by, the equivalence class of a term by . We use one interpretation as the universe ${\displaystyle T}$ ${\displaystyle T/\sim }$ ${\displaystyle T^{\Phi }\,}$ ${\displaystyle [t]_{\Phi }\,}$ ${\displaystyle T^{\Phi }\,}$ ${\displaystyle {\mathcal {T}}^{\Phi }}$

Next, give the interpretations of the constant, function, and relation symbols. Set for a constant symbol ${\displaystyle c}$

${\displaystyle c^{{\mathcal {T}}^{\Phi }}\quad :=\quad [c]_{\Phi }\in T^{\Phi }}$.

Define for an n-digit function symbol ${\displaystyle f}$

${\displaystyle f^{{\mathcal {T}}^{\Phi }}:(T^{\Phi })^{n}\rightarrow T^{\Phi },\quad f^{{\mathcal {T}}^{\Phi }}([t_{1}]_{\Phi },\ldots ,[t_{n}]_{\Phi }):=[ft_{1}\ldots t_{n}]_{\Phi }}$

and for an n-place relation symbol ${\displaystyle R}$

${\displaystyle R^{{\mathcal {T}}^{\Phi }}([t_{1}]_{\Phi },\ldots ,[t_{n}]_{\Phi })\quad :\Leftrightarrow \quad \Phi \vdash Rt_{1}\ldots t_{n}}$.

One can show that these determinations are well defined. Finally, a variable assignment must be specified; you just bet ${\displaystyle \beta ^{\Phi }}$

${\displaystyle \beta ^{\Phi }(v_{i})\,:=\,[v_{i}]_{\Phi }}$, where the variables are. ${\displaystyle v_{0},v_{1},v_{2},\ldots }$

Overall, this defines what is known as the term interpretation . ${\displaystyle {\mathcal {T}}^{\Phi }=(T^{\Phi },\beta ^{\Phi })}$

You can immediately see that from the above definitions

${\displaystyle T_{k}^{\Phi }:=\{[t]_{\Phi };\,t\in T,\mathrm {var} (t)\subset \{v_{0},\ldots v_{k-1}\}\}}$

Substructures are defined, where stands for the set of variables occurring in the term and the set of symbols must contain at least one constant symbol so that it is not empty. This gives you further interpretations if you define the assignment: ${\displaystyle \mathrm {var} (t)}$ ${\displaystyle t}$ ${\displaystyle k=0}$ ${\displaystyle c}$ ${\displaystyle T_{k}^{\Phi }}$ ${\displaystyle {\mathcal {T}}_{k}^{\Phi }=(T_{k}^{\Phi },\beta _{k}^{\Phi })}$

${\displaystyle \beta _{k}^{\Phi }(v_{i})={\begin{cases}\,[v_{i}]_{\Phi },&{\mbox{wenn }}i<k\\\,[v_{0}]_{\Phi },&{\mbox{wenn }}i\geq k>0\\\,[c]_{\Phi },&{\mbox{wenn }}i\geq k=0\end{cases}}}$

Term interpretations occur in Herbrand structures and Henkin's theorem .

Individual evidence

1. ^ Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas: Introduction to mathematical logic. Spektrum Akademischer Verlag, Heidelberg / Berlin / Oxford 1996, ISBN 3-8274-0130-5 , in particular Chapter V, § 1.
2. ^ Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas: Introduction to mathematical logic. Spectrum Academic Publishing House, Heidelberg / Berlin / Oxford 1996, ISBN 3-8274-0130-5 , especially Chapter XI, § 1.