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Bases for Structures and Theories II
| dc.abstract.en | In Part I of this paper (Ketland in Logica Universalis 14:357–381, 2020), I assumed we begin with a (relational) signature P = {Pi} and the corresponding language LP , and introduced the following notions: a definition system dΦ for a set of new predicate symbols Qi, given by a set Φ = {φi} of defining LP -formulas (these definitions have the form: ∀x(Qi(x) ↔ φi)); a corresponding translation function τΦ : LQ → LP ; the corresponding definitional image operator DΦ, applicable to LP - structures and LP -theories; and the notion of definitional equivalence itself: for structures A + dΦ ≡ B + dΘ; for theories, T1 + dΦ ≡ T2 + dΘ. Some results relating these notions were given, ending with two characterizations for definitional equivalence. In this second part, we explain the notion of a representation basis. Suppose a set Φ = {φi} of LP -formulas is given, and Θ = {θi} is a set of LQ-formulas. Then the original set Φ is called a representation basis for an LP -structure A with inverse Θ iff an inverse explicit definition ∀x(Pi(x) ↔ θi) is true in A + dΦ, for each Pi. Similarly, the set Φ is called a representation basis for a LP -theory T with inverse Θ iff each explicit definition ∀x(Pi(x) ↔ θi) is provable in T +dΦ. Some results about representation bases, the mappings they induce and their relationship with the notion of definitional equivalence are given. In particular, we show that T1 (in LP ) is definitionally equivalent to T2 (in LQ), with respect to Φ and Θ, if and only if Φ is a representation basis for T1 with inverse Θ and T2 ≡ DΦT1. |
| dc.affiliation | Uniwersytet Warszawski |
| dc.contributor.author | Ketland, Jeffrey |
| dc.date.accessioned | 2024-01-24T18:33:12Z |
| dc.date.available | 2024-01-24T18:33:12Z |
| dc.date.copyright | 2020-09-21 |
| dc.date.issued | 2020 |
| dc.description.accesstime | BEFORE_PUBLICATION |
| dc.description.finance | Publikacja bezkosztowa |
| dc.description.number | 4 |
| dc.description.version | FINAL_PUBLISHED |
| dc.description.volume | 14 |
| dc.identifier.doi | 10.1007/S11787-020-00261-2 |
| dc.identifier.issn | 1661-8297 |
| dc.identifier.uri | https://repozytorium.uw.edu.pl//handle/item/102036 |
| dc.identifier.weblink | http://link.springer.com/content/pdf/10.1007/s11787-020-00261-2.pdf |
| dc.language | eng |
| dc.pbn.affiliation | philosophy |
| dc.relation.ispartof | Logica Universalis |
| dc.relation.pages | 461-479 |
| dc.rights | Other |
| dc.sciencecloud | nosend |
| dc.subject.en | Definitional equivalence |
| dc.subject.en | Theories |
| dc.subject.en | Definability |
| dc.title | Bases for Structures and Theories II |
| dc.type | JournalArticle |
| dspace.entity.type | Publication |