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Free interpretation, quotient interpretation and substitution of a letter with a term for first-order languages. (English) Zbl 1276.03033
Summary: Fourth of a series of articles laying down the bases for classical first-order model theory. This paper supplies a toolkit of constructions to work with languages and interpretations, and results relating them. The free interpretation of a language, having as a universe the set of terms of the language itself, is defined.
The quotient of an interpreteation with respect to an equivalence relation is built, and shown to remain an interpretation when the relation respects it. Both the concepts of quotient and of respecting relation are defined in broadest terms, with respect to objects as general as possible.
Along with the trivial symbol substitution generally previously defined, the more complex substitution of a letter with a term is defined, basing right on the free interpretation just introduced, which is a novel approach, to the author’s knowledge. A first important result shown is that the quotient operation commute in some sense with term evaluation and reassignment functors. A second result proved is substitution lemma. This will be vital for proving satisfiability theorem and correctness of a certain sequent derivation rule. A third result supplied is that if two given languages coincide on the letters of a given FinSequence, their evaluation of it will also coincide. This too will be instrumental for proving correctness of another rule. Also, the Depth functor is shown to be invariant with respect to term substitution in a formula.

##### MSC:
 03C07 Basic properties of first-order languages and structures 03B35 Mechanization of proofs and logical operations 68T15 Theorem proving (deduction, resolution, etc.) (MSC2010)
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