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IKT! and Łukasiewicz-models

Fjellstad, Andreas and Olsen, Jan Fredrik LU (2021) In Notre Dame Journal of Formal Logic 62(2). p.247-256
Abstract

In this note, we show that the first-order logic IK! is sound with regard to the models obtained from continuum-valued Łukasiewicz-models for first-order languages by treating the quantifiers as infinitary strong disjunction/conjunction rather than infinitary weak disjunction/conjunction. Moreover, we show that these models cannot be used to provide a new consistency proof for the theory of truth IKT! obtained by expanding IK! with transparent truth, because the models are inconsistent with transparent truth. Finally, we show that whether or not this inconsistency can be reproduced in the sequent calculus for IKT! depends on how vacuous quantification is treated.

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author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
!-inconsistency, Inconsistency, Infinitary sequents, Multiplicative quantifiers, Non-contractive truth, Soundness, Vacuous quantification
in
Notre Dame Journal of Formal Logic
volume
62
issue
2
pages
10 pages
publisher
Duke University Press
external identifiers
  • scopus:85108725140
ISSN
0029-4527
DOI
10.1215/00294527-2021-0012
language
English
LU publication?
yes
id
11423b61-4a63-45ae-b1e1-08a4f02a60c5
date added to LUP
2021-08-23 09:02:02
date last changed
2022-04-27 03:20:32
@article{11423b61-4a63-45ae-b1e1-08a4f02a60c5,
  abstract     = {{<p>In this note, we show that the first-order logic IK<sup>!</sup> is sound with regard to the models obtained from continuum-valued Łukasiewicz-models for first-order languages by treating the quantifiers as infinitary strong disjunction/conjunction rather than infinitary weak disjunction/conjunction. Moreover, we show that these models cannot be used to provide a new consistency proof for the theory of truth IKT<sup>!</sup> obtained by expanding IK<sup>!</sup> with transparent truth, because the models are inconsistent with transparent truth. Finally, we show that whether or not this inconsistency can be reproduced in the sequent calculus for IKT<sup>!</sup> depends on how vacuous quantification is treated.</p>}},
  author       = {{Fjellstad, Andreas and Olsen, Jan Fredrik}},
  issn         = {{0029-4527}},
  keywords     = {{!-inconsistency; Inconsistency; Infinitary sequents; Multiplicative quantifiers; Non-contractive truth; Soundness; Vacuous quantification}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{247--256}},
  publisher    = {{Duke University Press}},
  series       = {{Notre Dame Journal of Formal Logic}},
  title        = {{IKT<sup>!</sup> and Łukasiewicz-models}},
  url          = {{http://dx.doi.org/10.1215/00294527-2021-0012}},
  doi          = {{10.1215/00294527-2021-0012}},
  volume       = {{62}},
  year         = {{2021}},
}