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Quasi-Nelson; Or, Non-involutive Nelson Algebras

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Algebraic Perspectives on Substructural Logics

Part of the book series: Trends in Logic ((TREN,volume 55))

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Abstract

We introduce a generalisation of Nelson algebras having a not necessarily involutive negation. We suggest dubbing this class quasi-Nelson algebras, in analogy with quasi-De Morgan lattices, which are a non-involutive generalisation of De Morgan lattices due to Sankappanavar. We show that, similarly to the involutive case, our new class of algebras can be equivalently presented as (1) quasi-Nelson residuated lattices, i.e. models of the well-known Full Lambek calculus with exchange and weakening, extended with the Nelson axiom; (2) non-involutive twist-structures, i.e. special binary products of Heyting algebras, which generalise a well-known construction for representing algebraic models of Nelson’s constructive logic with strong negation; (3) quasi-Nelson algebras, i.e. models of non-involutive Nelson logic, viewed as an axiomatic expansion of the negation-free fragment of intuitionistic logic; (4) the class of bounded commutative integral residuated lattices which satisfy a universal algebraic property introduced in a previous paper, and called \((\mathbf {0}, \mathbf {1})\)-congruence orderability therein. The equivalence of the four presentations, and in particular the extension of the twist-structure representation to the non-involutive case, is the main technical result of the paper. We hope, however, that the main impact of the paper will lie in the possibility of opening new ways to: (1) obtain deeper insights into the distinguishing feature of Nelson’s logic (i.e. the Nelson axiom) and its algebraic counterpart (the Nelson identity); and (2) be able to investigate certain purely algebraic properties (such as 3-potency and \((\mathbf {0}, \mathbf {1})\)-congruence orderability) in a more general setting.

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Notes

  1. 1.

    By an extension (of a given logic) we mean a stronger logic over the same language; by an expansion we mean a logic obtained by adding new connectives.

  2. 2.

    In the sense of this paper, the algebraic counterpart of a finitary and finitely (Blok-Pigozzi) algebraisable deductive system \(\mathbf {S}\) is the equivalent quasivariety semantics of \(\mathbf {S}\). All deductive systems considered in this paper, including \(\mathbf {N3}\), are Blok-Pigozzi algebraisable.

  3. 3.

    Formally, by ‘quasi-Nelson logic’ we mean the (necessarily unique) deductive system whose algebraic counterpart is the class of quasi-Nelson algebras. For a finite (Hilbert-style) presentation of this deductive system, see [24]. See also Remark 3.

  4. 4.

    Loosely speaking, by this we mean that the three definitions can be seen as alternative presentations of the “same” class of algebras in a different algebraic language, analogous to the presentation of Boolean algebras as Boolean rings or that of MV-algebras as certain lattice-ordered groups.

  5. 5.

    It is also useful to keep in mind that, since we are dealing with algebraisable logics, the most relevant features of logical systems are reflected by (and can therefore be derived from) their algebraic models.

  6. 6.

    Throughout the paper it will be useful for the reader to bear in mind that Nelson residuated lattices are term equivalent to the above-mentioned classes of Nelson algebras and to \(\mathbf {N3}\)-lattices.

  7. 7.

    Bounded implicative meet-semilattices are the \(\langle \wedge , \rightarrow , 0 \rangle \)-subreducts of Heyting algebras, corresponding to the conjunction-implication-negation fragment of intuitionistic logic. Thus, in particular, the \(\langle \wedge , \rightarrow , 0 \rangle \)-reduct of every Heyting algebra is a bounded implicative meet-semilattice.

  8. 8.

    This and the other unexplained terms are from Sankappanavar’s terminology: see  [37, Definition 2.1] and  [37, Definition 2.6].

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Authors and Affiliations

  1. Departamento de Informática e Matemática Aplicada, Universidade Federal do Rio Grande do Norte, Natal, Brazil

    Umberto Rivieccio

  2. Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, Cagliari, Italy

    Matthew Spinks

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  1. Umberto Rivieccio
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Correspondence to Umberto Rivieccio .

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Editors and Affiliations

  1. Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, Via Is Mirrionis, Cagliari, Italy

    Davide Fazio

  2. Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, Via Is Mirrionis, Cagliari, Italy

    Antonio Ledda

  3. Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, Via Is Mirrionis, Cagliari, Italy

    Francesco Paoli

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Rivieccio, U., Spinks, M. (2021). Quasi-Nelson; Or, Non-involutive Nelson Algebras. In: Fazio, D., Ledda, A., Paoli, F. (eds) Algebraic Perspectives on Substructural Logics. Trends in Logic, vol 55. Springer, Cham. https://doi.org/10.1007/978-3-030-52163-9_8

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