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A characterization of Continuous Logic by using quantale-valued logics

David Reyes, Pedro H. Zambrano

Abstract

In this paper, we propose a generalization of Continuous Logic ([BBHU08]) where the distances take values in suitable co-quantales (in the way as it was proposed in [Fla97]). By assuming suitable conditions (e.g., being co-divisible, co-Girard and a V-domain), we provide, as test questions, a proof of a version of the Tarski-Vaught test (Proposition 4.2) and Łoś Theorem (Theorem 5.27) in our setting. Iovino proved in [Iov01] that there is no any logic extending (equivalent logics to) Continuous Logic satisfying both Countable Tarski-Vaught chain Theorem and Compactness Theorem. Since [0, 1] satisfies all of the assumptions given above, we get new logics by dropping any of those assumptions.

A characterization of Continuous Logic by using quantale-valued logics

Abstract

In this paper, we propose a generalization of Continuous Logic ([BBHU08]) where the distances take values in suitable co-quantales (in the way as it was proposed in [Fla97]). By assuming suitable conditions (e.g., being co-divisible, co-Girard and a V-domain), we provide, as test questions, a proof of a version of the Tarski-Vaught test (Proposition 4.2) and Łoś Theorem (Theorem 5.27) in our setting. Iovino proved in [Iov01] that there is no any logic extending (equivalent logics to) Continuous Logic satisfying both Countable Tarski-Vaught chain Theorem and Compactness Theorem. Since [0, 1] satisfies all of the assumptions given above, we get new logics by dropping any of those assumptions.

Paper Structure

This paper contains 24 sections, 41 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Value co-quantales and Continuity Spaces
  3. Value lattices
  4. Value co-quantales
  5. Co-Girard value co-quantales.
  6. Continuity spaces
  7. The topology of a $\mathbf{V}$-continuity space.
  8. V-domains
  9. Value co-quantale logics
  10. Modulus of uniform continuity
  11. Uniform continuity of $\bigwedge$ and $\bigvee$.
  12. Some basic notions.
  13. Continuous structures.
  14. Languages for continuous structures.
  15. L-structures
  16. ...and 9 more sections

Key Result

Lemma 2.3

(Fla97; Lemma 1.2) Let $L$ be a complete lattice, then for all $x,y,z \in L$ we have that

Theorems & Definitions (137)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • Definition 2.8
  • Definition 2.10
  • Proposition 2.11
  • ...and 127 more