Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
See all formats and pricing
More options …
Volume 19, Issue 3 (Jan 2011)


Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages

Marco Caminati
  • Mathematics Department "G. Castelnuovo", Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, Italy
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2012-04-26 | DOI: https://doi.org/10.2478/v10037-011-0028-z

Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages

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 defined in [11], 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, both introduced in [13] (theorem 3, theorem 15). A second result proved is substitution lemma (theorem 10, corresponding to III.8.3 of [15]). This will be vital for proving satisfiability theorem and correctness of a certain sequent derivation rule in [14]. 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 in [14] for proving correctness of another rule. Also, the Depth functor is shown to be invariant with respect to term substitution in a formula.

  • Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Google Scholar

  • Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Google Scholar

  • Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Google Scholar

  • Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Google Scholar

  • Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Google Scholar

  • Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.Google Scholar

  • Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Google Scholar

  • Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • Marco B. Caminati. Preliminaries to classical first order model theory. Formalized Mathematics, 19(3):155-167, 2011, doi: 10.2478/v10037-011-0025-2.CrossrefGoogle Scholar

  • Marco B. Caminati. Definition of first order language with arbitrary alphabet. Syntax of terms, atomic formulas and their subterms. Formalized Mathematics, 19(3):169-178, 2011, doi: 10.2478/v10037-011-0026-1.CrossrefGoogle Scholar

  • Marco B. Caminati. First order languages: Further syntax and semantics. Formalized Mathematics, 19(3):179-192, 2011, doi: 10.2478/v10037-011-0027-0.CrossrefGoogle Scholar

  • Marco B. Caminati. Sequent calculus, derivability, provability. Gödel's completeness theorem. Formalized Mathematics, 19(3):205-222, 2011, doi: 10.2478/v10037-011-0029-y.CrossrefGoogle Scholar

  • H. D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical logic. Springer, 1994.Google Scholar

  • Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.Google Scholar

  • Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Google Scholar

  • Krzysztof Retel. Properties of first and second order cutting of binary relations. Formalized Mathematics, 13(3):361-365, 2005.Google Scholar

  • Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Google Scholar

  • Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Google Scholar

  • Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • Edmund Woronowicz. Many-argument relations. Formalized Mathematics, 1(4):733-737, 1990.Google Scholar

  • Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Google Scholar

  • Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Google Scholar

  • Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990.Google Scholar

About the article

Published Online: 2012-04-26

Published in Print: 2011-01-01

Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-011-0028-z.

Export Citation

This content is open access.

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Marco Caminati
Formalized Mathematics, 2011, Volume 19, Number 3

Comments (0)

Please log in or register to comment.
Log in