Abstract
A logic L has the “variable-sharing property” (VSP) if in all L-theorems of conditional form antecedent and consequent share at least a propositional variable. Anderson and Belnap consider the VSP as a necessary property any relevance logic has to fulfil. Now, among relevance logicians, Brady’s logic BN4 is widely viewed as the adequate implicative 4-valued logic. But BN4 does not have the VSP. The aim of this paper is to define a variant of BN4 having, in addition to the VSP, some properties that do not support its consideration as a mere artificial construct.
This work is supported by the Spanish Ministry of Science and Innovation under Grant [PID2020-116502GB-I00]. We thank three referees of CLAR 2021 for their comments and suggestions on a previous draft of this paper.
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Notes
- 1.
Actually, in [21] it is stated that BN4 has the “quasi relevance property” (QRP) The QRP reads: if \(A\,{\rightarrow }\,B\) is a theorem, then either A and B share at least a propositional variable or both \(\lnot A\) and B are theorems. But BN4 lacks the VSP: \(\lnot (A\,{\rightarrow }\,A)\rightarrow (B\,{\rightarrow }{} \mathbf{B})\) is MBN4-valid.
- 2.
Notice that the rule VEQ encloses an infinity of paradoxes of relevance, the simplest of which may be \(q \rightarrow (p\rightarrow p)\). Consequently, it cannot be a rule of a logic with the VSP.
- 3.
A referee of CLAR 2021 worries about many-valued extensions because of the presence of such formulas as A8. This is an interesting question we cannot discuss in detail here. Let us only remark that the type of formulas the referee is concerned about is not the only fault of many-valued extensions; actually, they do not seem to collide with the VSP. For example, it is shown in “A general characterization of the variable-sharing property by means of logical matrices” (G. Robles and J. M. Méndez, Notre Dame Journal of Formal Logic, 53(2), 223–244, 2012) that relatively strong logics with the VSP have Dummett’s axiom for the intermediate logic LC (i.e., \((A\rightarrow B)\vee (B\rightarrow A)\)) as one of their theorems. The conclusion seems inescapable: the VSP and the disjunction property are independent of each other.
- 4.
A referee of CLAR 2021 remarks that this result can be obtained as a special case of the general procedure described in §2 of “Generalizing functional completeness in Belnap-Dunn logic” (H. Omori and K. Sano, Studia Logica, 103(5), 883–917, 2015).
- 5.
A referee of CLAR 2021 remarks that there are \(2^{17}\) different matrices with natural conditionals (in the sense of Definition 5) complying with the VSP. Well then, since this paper was written, we have pursued the topic it introduces and almost all the \(2^{17}\) variants are non-significant in the sense that they lack one or more of the properties MBN4\(^{\text {VSP}}\) exhibits (actually, only 24 of said matrices share the properties MBN4\(^{\text {VSP}}\) has).
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Robles, G. (2021). A Variant with the Variable-Sharing Property of Brady’s 4-Valued Implicative Expansion BN4 of Anderson and Belnap’s Logic FDE. In: Baroni, P., Benzmüller, C., Wáng, Y.N. (eds) Logic and Argumentation. CLAR 2021. Lecture Notes in Computer Science(), vol 13040. Springer, Cham. https://doi.org/10.1007/978-3-030-89391-0_20
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