IKT! and Łukasiewicz-models
(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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https://lup.lub.lu.se/record/11423b61-4a63-45ae-b1e1-08a4f02a60c5
- author
- Fjellstad, Andreas and Olsen, Jan Fredrik LU
- organization
- publishing date
- 2021-05
- 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}}, }