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An Unexpected Boolean Connective

Abstract

We consider a 2-valued non-deterministic connective \({\wedge \!\!\!\!\!\vee }\) defined by the table resulting from the entry-wise union of the tables of conjunction and disjunction. Being half conjunction and half disjunction we named it platypus. The value of \({\wedge \!\!\!\!\!\vee }\) is not completely determined by the input, contrasting with usual notion of Boolean connective. We call non-deterministic Boolean connective any connective based on multi-functions over the Boolean set. In this way, non-determinism allows for an extended notion of truth-functional connective. Unexpectedly, this very simple connective and the logic it defines, illustrate various key advantages in working with generalized notions of semantics (by incorporating non-determinism), calculi (by allowing multiple-conclusion rules) and even of logic (moving from Tarskian to Scottian consequence relations). We show that the associated logic cannot be characterized by any finite set of finite matrices, whereas with non-determinism two values suffice. Furthermore, this logic is not finitely axiomatizable using single-conclusion rules, however we provide a very simple analytic multiple-conclusion axiomatization using only two rules. Finally, deciding the associated multiple-conclusion logic is \(\mathbf {coNP}\)-complete, but deciding its single-conclusion fragment is in \({\mathbf {P}}\).

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Notes

  1. In this paper exclude the possibility of a multi-function outputting the empty set however there are situations where this option is desirable as we mention in the end of Sect. 3.

  2. I thank Carlos Caleiro for suggesting platypus and thus steering me away from using such boring alternatives.

  3. This notion is usually presented over Tarskian consequence relations. In such setting, due to the asymmetry in the very notion of logic, the two notions are not symmetric. We will not enter in details here but just state that for positive logics \(\vartriangleright \) where we have that \(\vartriangleright \) is left-(right-)inclusion logic iff \(\vdash _\vartriangleright \) is. We point to [9, 12] for more details.

  4. Rules with empty set of conclusion discontinue the branch of the node where it is applied. We have that \(\Gamma \vartriangleright _R \Delta \) whenever there is a R-derivation departing from \(\Gamma \) where the leaf of each non discontinued branch must be a formula in \(\Delta \).

  5. Using \(\mathbf{2}^\omega \approx \wp ({{\mathbb {N}}})\approx \{({{\mathbb {N}}},1)\}\cup \{(0,X):X\subsetneq {{\mathbb {N}}}\}\).

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Correspondence to Sérgio Marcelino.

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So they cut him to pieces, wrote a thesis

A cranium of deceit, he’s prone to lie and cheat

It’s no wonder – a blunder from down under

Duckbill, watermole, duckmole!

Mr. Bungle, Platypus.

Research funded by FCT/MCTES through national funds and when applicable co-funded by EU under the project UIDB/50008/2020.

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Marcelino, S. An Unexpected Boolean Connective. Log. Univers. 16, 85–103 (2022). https://doi.org/10.1007/s11787-021-00280-7

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Keywords

  • Boolean-connectives
  • Two-valued logics
  • Generalized truth-functionality
  • Non-deterministic semantics
  • Multiple-conclusion rules
  • Axiomatizability
  • Analytic calculi

Mathematics Subject Classification

  • Primary 03B50
  • Secondary 03B35