Lindenbaum and Pair Extension Lemma in Infinitary Logics
Result description
The abstract Lindenbaum lemma is a crucial result in algebraic logic saying that the prime theories form a basis of the closure systems of all theories of an arbitrary given logic. Its usual formulation is however limited to finitary logics, i.e., logics with Hilbert-style axiomatization using finitary rules only. In this contribution, we extend its scope to all logics with a countable axiomatization and a well-behaved disjunction connective. We also relate Lindenbaum lemma to the Pair extension lemma, other well-known result with many applications mainly in the theory of non-classical modal logics. While a restricted form of this lemma (to pairs with finite right-hand side) is, in our context, equivalent to Lindenbaum lemma, we show a perhaps surprising result that in full strength it holds for finitary logics only. Finally we provide examples demonstrating both limitations and applications of our results.
Keywords
Lindenbaum lemmaPair extension lemmaInfinitary logicInfinitary deduction ruleStrong disjunctionPrime theory
The result's identifiers
Result code in IS VaVaI
Alternative codes found
RIV/67985807:_____/18:00491981 RIV/00216208:11210/18:10385612
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
Lindenbaum and Pair Extension Lemma in Infinitary Logics
Original language description
The abstract Lindenbaum lemma is a crucial result in algebraic logic saying that the prime theories form a basis of the closure systems of all theories of an arbitrary given logic. Its usual formulation is however limited to finitary logics, i.e., logics with Hilbert-style axiomatization using finitary rules only. In this contribution, we extend its scope to all logics with a countable axiomatization and a well-behaved disjunction connective. We also relate Lindenbaum lemma to the Pair extension lemma, other well-known result with many applications mainly in the theory of non-classical modal logics. While a restricted form of this lemma (to pairs with finite right-hand side) is, in our context, equivalent to Lindenbaum lemma, we show a perhaps surprising result that in full strength it holds for finitary logics only. Finally we provide examples demonstrating both limitations and applications of our results.
Czech name
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Czech description
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Classification
Type
D - Article in proceedings
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Article name in the collection
Logic, Language, Information and Computation
ISBN
978-3-662-57668-7
ISSN
0302-9743
e-ISSN
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Number of pages
15
Pages from-to
130-144
Publisher name
Springer
Place of publication
Berlin
Event location
Bogotá
Event date
Jul 24, 2018
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
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Basic information
Result type
D - Article in proceedings
OECD FORD
Pure mathematics
Year of implementation
2018