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Unorthodox Algebras and their associated Unorthodox Logics

Hanamantagouda P. Sankappanavar

TL;DR

This work introduces five unorthodox De Morgan semi-Heyting algebras and the generated variety $\mathbb{RUNO}1$, addressing whether nonclassical implication can differ from orthodox intuition. It establishes that $\mathbb{RUNO}1 = \mathbb{V}(\mathbf{A1},\dots,\mathbf{A5})$ and that the variety is a discriminator, with each algebra primal and every subvariety satisfying AP and ES, yielding a Boolean lattice of 32 subvarieties. The authors develop a corresponding algebraizable logic, $\mathcal{RUNO}1$, built as an extension of $\mathcal{DMSH}$ with unorthodox axioms, and derive 32 axiomatic extensions (logics) all decidable; they also provide bases for these extensions and prove that they are all algebraizable. The results give a detailed structural and logical landscape for unorthodox algebras, including bases for subvarieties, identities, and a clear correspondence between subvarieties and axiomatic extensions, while highlighting open problems about dualities and simple-algebra characterizations.

Abstract

This paper grew out of our investigation into a simple, but natural, question: Can 'F implies T' be distinct from F and T? To this end, we introduce five 'unorthodox' algebras that will play a major role, not only in providing a positive answer to the question, but also in their similarity to the 2-element Boolean algebra 2. Yet, they are remarkably dissimilar from 2 in many respects. In this paper, we will examine these five algebras both algebraically and logically. We define, and initiate an investigation into, a subvariety, called RUNO1, of the variety of De Morgan semi-Heyting algebras and show that RUNO1 is, in fact, the variety generated by the five algebras. Then several applications of this theorem are given. It is shown that RUNO1 is a discriminator variety and that all five algebras are primal. It is also shown that every subvariety of RUNO1 satisfies the Strong Amalgamation Property and the property that epimorphisms are surjective (ES). It is shown that the lattice of subvarieties of RUNO1 is a Boolean lattice of 32 elements. The bases for all the subvarieties of RUNO1 are also given. We introduce a new logic called mathcal{RUNO1} and show that it is algebraizable with the variety RUNO1 as its equivalent algebraic semantics. We then present axiomatizations for all 32 axiomatic extensions of mathcal{RUNO1} and deduce that all the axiomatic extensions are decidable. The paper ends with some open problems.

Unorthodox Algebras and their associated Unorthodox Logics

TL;DR

This work introduces five unorthodox De Morgan semi-Heyting algebras and the generated variety , addressing whether nonclassical implication can differ from orthodox intuition. It establishes that and that the variety is a discriminator, with each algebra primal and every subvariety satisfying AP and ES, yielding a Boolean lattice of 32 subvarieties. The authors develop a corresponding algebraizable logic, , built as an extension of with unorthodox axioms, and derive 32 axiomatic extensions (logics) all decidable; they also provide bases for these extensions and prove that they are all algebraizable. The results give a detailed structural and logical landscape for unorthodox algebras, including bases for subvarieties, identities, and a clear correspondence between subvarieties and axiomatic extensions, while highlighting open problems about dualities and simple-algebra characterizations.

Abstract

This paper grew out of our investigation into a simple, but natural, question: Can 'F implies T' be distinct from F and T? To this end, we introduce five 'unorthodox' algebras that will play a major role, not only in providing a positive answer to the question, but also in their similarity to the 2-element Boolean algebra 2. Yet, they are remarkably dissimilar from 2 in many respects. In this paper, we will examine these five algebras both algebraically and logically. We define, and initiate an investigation into, a subvariety, called RUNO1, of the variety of De Morgan semi-Heyting algebras and show that RUNO1 is, in fact, the variety generated by the five algebras. Then several applications of this theorem are given. It is shown that RUNO1 is a discriminator variety and that all five algebras are primal. It is also shown that every subvariety of RUNO1 satisfies the Strong Amalgamation Property and the property that epimorphisms are surjective (ES). It is shown that the lattice of subvarieties of RUNO1 is a Boolean lattice of 32 elements. The bases for all the subvarieties of RUNO1 are also given. We introduce a new logic called mathcal{RUNO1} and show that it is algebraizable with the variety RUNO1 as its equivalent algebraic semantics. We then present axiomatizations for all 32 axiomatic extensions of mathcal{RUNO1} and deduce that all the axiomatic extensions are decidable. The paper ends with some open problems.

Paper Structure

This paper contains 20 sections, 47 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. "Unorthodox Algebras"
  3. Preliminaries
  4. The variety $\mathbb{RUNO}1$
  5. On the height of algebras of $\mathbb{RUNO}1$ satisfying (SC)
  6. A Characterization of Subdirectly irreducible Algebras in $\mathbb{RUNO}1$
  7. Discriminator
  8. Properties AP, SAP, ES
  9. The lattice of subvarieties of $\mathbb{RUNO}1$
  10. Bases for subvarieties of $\mathbb{RUNO}1$
  11. More Identities
  12. Toward a logic for the variety $\mathbb{RUNO}1$
  13. Abstract Algebraic Logic: Basic definitions and results
  14. Algebraizability of a logic
  15. Algebraizabilty of Implicative Logics
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let $\mathbf{L} = \langle L, \land, \lor, \to, 0, 1\rangle$ be a semi-Heyting algebra. For $a,b,c \in L, a \land b \leq c$ if and only if $a \leq b \to (b \land c)$.

Theorems & Definitions (61)

  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Theorem 4.1
  • ...and 51 more