Abstract
A logic \(\mathbf{L}\) is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \(\mathbf{L}\)-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, the famous Dunn–Belnap four-valued logic has (up to the choice of the primitive connectives) exactly one self-extensional four-valued extension which has an implication. We also investigate the main properties of this logic, determine the expressive power of its language (in the four-valued context), and provide a cut-free Gentzen-type proof system for it.
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Notes
However, in [6] it is shown that there is a unique self-extensional three-valued paraconsistent logic in the language of \(\{\lnot ,\wedge ,\vee \}\) for which \(\vee \) is a disjunction, and \(\wedge \) is a conjunction.
Most probably, \(\mathbf{BN4}^\rightarrow \) is the logic that Brady had in mind, since \(\rightarrow \) has in it important properties of a relevant entailment.
Conversely, Brady’s \(\rightarrow \) can be defined in the language of 4CL by
$$\begin{aligned} \varphi \rightarrow \psi =_{Df}(\varphi \supset \psi )\wedge (\lnot \psi \supset \lnot \varphi ) \end{aligned}$$Hence \(\mathbf {L}_{\mathcal {M}_4}\) and 4CL are equivalent. These connections between \(\supset \) and \(\rightarrow \) have first been given in [2]. Note that [2] provided also a Hilbert-type system that is strongly sound and complete for 4CL. (See also [8].) That system has MP for \(\supset \) as its sole rule of inference. BN4, in contrast, has four rules of inference. Still, it is straightforward to show that BN4 and the system for 4CL given in [2, 8] are equivalent.
For \(\tilde{\supset }\) this easily follows from the characterization given in Note 3.9.
The proof is very similar to the proof of Theorem 6.9 in [6].
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This research was supported by the Israel Science Foundation under Grant Agreement No. 817/15.
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Avron, A. The Normal and Self-extensional Extension of Dunn–Belnap Logic. Log. Univers. 14, 281–296 (2020). https://doi.org/10.1007/s11787-020-00254-1
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DOI: https://doi.org/10.1007/s11787-020-00254-1