Abstract
In this paper, we investigate an extended first-order Belnap-Dunn logic with classical negation. We introduce a Gentzen-type sequent calculus FBD+ for this logic and prove theorems for syntactically and semantically embedding FBD+ into a Gentzen-type sequent calculus for first-order classical logic. Moreover, we show the cut-elimination theorem for FBD+ and prove the completeness theorems with respect to both valuation and many-valued semantics for FBD+.
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Notes
- 1.
Another system which is equivalent to BD+ is PŁ4 of Méndez and Robles (cf. [11]).
- 2.
Belnap-Dunn logic with \(\triangle \) is equivalent to the expansion of Belnap-Dunn logic by what is sometimes called exclusion negation (cf. [5, p. 829]).
- 3.
By using the strong equivalence substitution property, we can show the Herbrand theorem for FBD+, although we omit the details due to space limitations.
- 4.
Another interesting property of BD+ is the maximality with respect to the set of theorems (but not with respect to the rules of inference) which is proved in [5, Sect. 3.3]. Maximality does not hold for Nelson logics (even for “classical” extensions) since there are extensions, obtained by adding some axioms, that are not classical logic.
- 5.
In FBD+, we can replace the multiplicative (context splitting) type inference rules (cut) and (\(\rightarrow \)left) with their additive (non context splitting) type modifications. But, we adopt the multiplicative type inference rules since these are compatible in the system LK (for the classical logic) presented in [17].
- 6.
Note that \(\varGamma \not \vdash _\mathrm{FBD+} \varPi \) is defined as \(\varGamma \not \vdash _\mathrm{FBD+} \alpha _1 \, \vee \dots \vee \alpha _n\) for some \(\alpha _1, \dots , \alpha _n\in \varPi \).
- 7.
For connexive logic in general, see [21].
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Acknowledgments
We would like to thank the anonymous referees for their valuable comments. Norihiro Kamide was partially supported by JSPS KAKENHI Grant Number JP26330263. Hitoshi Omori is a Postdoctoral Research Fellow of Japan Society for the Promotion of Science (JSPS), and was partially supported by JSPS KAKENHI Grant Number JP16K16684.
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Kamide, N., Omori, H. (2017). An Extended First-Order Belnap-Dunn Logic with Classical Negation. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_6
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