TY - JOUR
T1 - An expansion of first-order Belnap-Dunn logic
AU - Sano, Katsuhiko
AU - Omori, Hitoshi
N1 - Funding Information:
The authors would like to thank the anonymous referees for their valuable and detailed comments. In particular, their comments on the historical background of Belnap–Dunn Logic were quite useful to improve the earlier version of this article. Moreover, the argument of Remark 5.5 is due to one of the referees. The first author would like to thank Shunsuke Yatabe for pointing out the relevance of our △-operator with Baaz’s delta operator. The work of the first author was partially supported by JSPS KAKENHI, Grant-in-Aid for Young Scientists (B) 24700146. The second author is a postdoctoral fellow of Japan Society for the Promotion of Science (JSPS), and the present work was partially supported by a Grant-in-Aid for JSPS Fellows.
PY - 2014/6
Y1 - 2014/6
N2 - This article proposes to expand first-order Belnap-Dunn logic with a new operator, whose intuitive meaning is '- is designated' or '- has designated values'. It amounts to considering all of D'Ottaviano's possibility connective of J3, Baaz's Delta operator of infinite-valued Gödel logic and Kachi's determinate operator, in the different context of Belnap-Dunn logic. As for proof theory, we employ a natural deduction calculus with a slot for weakly definable connectives. Our main theorem is a general completeness result with respect to P({0,1})-valued semantics, which enables us to derive new completeness results for first-order extensions of Kachi's SPL and Omori and Waragai's BS4 and the known completeness results of D'Ottaviano and da Costa's J3 and Carnielli, Marcos and de Amo's LFI1.
AB - This article proposes to expand first-order Belnap-Dunn logic with a new operator, whose intuitive meaning is '- is designated' or '- has designated values'. It amounts to considering all of D'Ottaviano's possibility connective of J3, Baaz's Delta operator of infinite-valued Gödel logic and Kachi's determinate operator, in the different context of Belnap-Dunn logic. As for proof theory, we employ a natural deduction calculus with a slot for weakly definable connectives. Our main theorem is a general completeness result with respect to P({0,1})-valued semantics, which enables us to derive new completeness results for first-order extensions of Kachi's SPL and Omori and Waragai's BS4 and the known completeness results of D'Ottaviano and da Costa's J3 and Carnielli, Marcos and de Amo's LFI1.
KW - Baaz's delta operator
KW - Belnap-Dunn logic
KW - Four-valued logic
KW - Paraconsistent logic
KW - Łukasiewicz's implication
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U2 - 10.1093/jigpal/jzt044
DO - 10.1093/jigpal/jzt044
M3 - Article
AN - SCOPUS:84903126973
SN - 1367-0751
VL - 22
SP - 458
EP - 481
JO - Logic Journal of the IGPL
JF - Logic Journal of the IGPL
IS - 3
ER -