Algebraic semantics for Nelson’s logic S

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(1)Thiago Nascimento da Silva. Algebraic semantics for Nelson’s logic 𝒮. Natal, RN, Brasil 2017.

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(3) Thiago Nascimento da Silva. Algebraic semantics for Nelson’s logic 𝒮. Dissertação de Mestrado apresentada ao Programa de Pós-Graduação em Sistemas e Computação como requisito parcial para obtenção do título de Mestre em Sistemas e Computação.. Universidade Federal do Rio Grande do Norte – UFRN Departamento de Informática e Matemática Aplicada – DIMAp Programa de Pós-Graduação em Sistemas e Computação – PPgSC. Supervisor: Umberto Rivieccio Co-supervisor: João Marcos de Almeida. Natal, RN, Brasil 2017.

(4) Catalogação da Publicação na Fonte. UFRN / SISBI / Biblioteca Setorial Especializada do Centro de Ciências Exatas e da Terra – CCET. Silva, Thiago Nascimento da. Algebraic semantics for Nelson’s logic S / Thiago Nascimento da Silva. – 2017. 64 f. Dissertação (mestrado) – Universidade Federal do Rio Grande do Norte, Centro de Ciências Exatas e da Terra, Departamento de Informática e Matemática Aplicada, Programa de Pós-Graduação em Sistemas e Computação. Natal, RN, 2017. Orientador: Umberto Rivieccio. Coorientador: João Marcos de Almeida. 1. Lógica algébrica – Dissertação. 2. Semântica – Dissertação. 3. Lógicas de Nelson – Dissertação. I. Rivieccio, Umberto. II. Almeida, João Marcos de. III. Título. RN/UF/BSE-CCET. CDU 510.6. Elaborado por Joseneide Ferreira Dantas - CRB-15/324.

(5) Thiago Nascimento da Silva. Algebraic semantics for Nelson’s logic 𝒮. Dissertação de Mestrado apresentada ao Programa de Pós-Graduação em Sistemas e Computação como requisito parcial para obtenção do título de Mestre em Sistemas e Computação.. Trabalho aprovado. Natal, RN, Brasil, 25 de janeiro de 2018:. Prof. Dr. Umberto Rivieccio Orientador - Presidente. Prof. Dr. João Marcos de Almeida Co-orientador. Prof. Dr. Hugo Luiz Mariano Membro Externo. Natal, RN, Brasil 2017.

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(7) To my family, for their endless support..

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(9) Acknowledgements In conclusion of these two years of studying I am grateful for everything I experienced and I owe gratitude to some special people who contributed to my journey. To my advisers João Marcos (In the first year) and Umberto Rivieccio (In the second year) - thanks for the guidance and patience. To my friends from the Logic group (Carol (in memoriam), Evelyn, Hudson, João Daniel, Patrick, Raquel and Sanderson) for their continuous contributions both logic and english, as well as philosophy. To my friends from UFES (Aaron, Bruno, Eneas and José) for the years that we spent studying together. For all the effort to guarantee my admission in the master’s degree I would like to thanks Professor João Marcos, Helida and Professor Uirá. I am also grateful to the professors Elaine Pimentel and Matthew Spinks for their valued contribution to my work. I would like to express gratitude to my family for all the indispensable support. Napoleão and Rony, without their support I couldn´t continue my studies. Nilson and Luciene, and Fran, thanks for helping with my moving to Natal. Finally, to Ana Paula, thanks for your daily help, support and comprehension..

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(11) “And you shall know the truth, and the truth shall set you free.” — John 8:32, Bible.

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(13) Resumo Além da mais conhecida lógica de Nelson (𝒩 3) e da lógica paraconsistente de Nelson (𝒩 4), David Nelson introduziu no artigo de 1959 "Negation and separation of concepts in constructive systems", com motivações de aritmética e construtividade, a lógica que ele chamou de "𝒮". Naquele trabalho, a lógica é definida por meio de um cálculo (que carece crucialmente da regra de contração) tendo infinitos esquemas de regras, e nenhuma semântica é fornecida. Neste trabalho nós tomamos o fragmento proposicional de 𝒮, mostrando que ele é algebrizável (de fato, implicativo) no sentido de Blok & Pigozzi com respeito a uma classe de reticulados residuados involutivos. Assim, fornecemos a primeira semântica para 𝒮 (que chamamos de 𝒮-álgebras), bem como um cálculo estilo Hilbert finito equivalente à apresentação de Nelson. Fornecemos um algoritmo para construir 𝒮-álgebras a partir de 𝒮-álgebras ou reticulados implicativos e demonstramos alguns resultados sobre a classe de álgebras que introduzimos. Nós também comparamos 𝒮 com outras lógicas da família de Nelson, a saber, 𝒩 3 e 𝒩 4.. Palavras-chaves: Lógica, Lógicas de Nelson, Lógicas construtivistas, Negação forte, Lógica de Nelson paraconsistente, Lógicas subestruturais, Reticulados residuados trêspotente, Lógica algébrica..

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(15) Abstract Besides the better-known Nelson logic (𝒩 3) and paraconsistent Nelson logic (𝒩 4), in Negation and separation of concepts in constructive systems (1959) David Nelson introduced a logic that he called 𝒮, with motivations of arithmetic and constructibility. The logic was defined by means of a calculus (crucially lacking the contraction rule) having infinitely many rule schemata, and no semantics was provided for it. We look in the present dissertation at the propositional fragment of 𝒮, showing that it is algebraizable (in fact, implicative) in the sense of Blok and Pigozzi with respect to a class of involutive residuated lattices. We thus provide the first known algebraic semantics for 𝒮(we call them of 𝒮-algebras) as well as a finite Hilbert-style calculus equivalent to Nelson’s presentation. We provide an algorithm to make 𝒮-algebras from 𝒮-algebras or implicative lattices and we prove some results about the class of algebras which we have introduced. We also compare 𝒮 with other logics of the Nelson family, that is, 𝒩 3 and 𝒩 4.. Key-words: Logic, Nelson’s logics, Constructive logics, Strong negation, Paraconsistent Nelson logic, Substructural logics, Three-potent residuated lattices, Algebraic logic..

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(17) Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1. NELSON’S LOGIC 𝒮 . . . . . . . . . . . . . . . . . . . . . . . . . . 19. 1.1. A brief introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. 1.2. Nelson’s logic 𝒮 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. 2. ALGEBRAIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. 2.1. 𝒮 is algebraizable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. 2.2. 𝒮-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28. 2.3. 𝒮-algebras as residuated lattices . . . . . . . . . . . . . . . . . . . . . 31. 3. CALCULUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35. 3.1. A finite Hilbert-style calculus for 𝒮 ′ . . . . . . . . . . . . . . . . . . . 35. 3.2. Deduction theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37. 4. MORE ON 𝒮-ALGEBRAS. 4.1. ℒ3 algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. 4.2. Making 𝒮-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. 5. RELATION WITH 𝒩 3 AND 𝒩 4 . . . . . . . . . . . . . . . . . . . . 47. 5.1. 𝒩 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47. 5.2. 𝒩 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. 6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57. 7. FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59. BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . 43. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61. Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.

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(19) 17. Preface In order to study the notion of constructible falsity, David Nelson introduced a few systems of non-classical logic. Nelson’s systems combine an intuitionistic approach to truth and a dual-intuitionistic treatment of falsity. The logics introduced by Nelson (𝒮, 𝒩 3, and 𝒩 4) accept some important theorems of classical logic, like ¬¬𝜑 ⇔ 𝜑, while rejecting others, like (𝜑 ⇒ (𝜑 ⇒ 𝜓)) ⇒ (𝜑 ⇒ 𝜓) and (𝜑 ∧ ¬𝜑) ⇒ 𝜓. Nelson introduced these logics with the aim of studying constructive proofs in Number Theory. To such an end he gave a definition of truth [Nelson, 1949, Definition 1] (analogous to Kleene’s [Kleene, 1945, p. 112]) according to which a formula could be either affirmed or denied. The main logic which we are going to study in this work does not respect the formula (𝑝 ∧ ¬𝑝) ⇒ 𝑞. In logics respecting this formula, every theory containing a formula and its negation is trivial, that is, all formulas are in the theory. Paraconsistents logics are a weakening of classical logic lacking this formula, we can have a theory that has a conjunction of a formula and its negation, but is not trivial. Besides the philosophical and mathematical interest about these, there is aplicational interest for computer science. As an example we can mention a database having a considerable amount of information. It is plausible to consider that the database has information that contradicts each other, either in storing incorrect information or in having more than one source. One can fix the problem by eliminating the contradictory information, but in doing that we may lose importation information. Another way is to use some paraconsistent logic to infer results. Results about the application of paraconsistent logics in computer science can be seen in [Abe et al., 2015, Chapter 6]. The first system for studying constructible falsity is introduced in [Nelson, 1949] and is nowadays known as Nelson’s logic, called 𝒩 3 following [Odintsov, 2008]. A paraconsistent version of 𝒩 3, called nowadays 𝒩 4, was introduced in [Almukdad and Nelson, 1984]. In fact 𝒩 3 is an axiomatic extension of 𝒩 4 by the axiom ¬𝜑 → (𝜑 → 𝜓). The logic 𝒩 3 is well studied, both in a proof-theoretic approach and through algebraic methods [Odintsov, 2008]. For 𝒩 4 the results are more recent and due to the work of Odintsov [Odintsov, 2008] we know that 𝒩 4 (and also 𝒩 3) is algebraizable in the sense of Blok and Pigozzi [Blok and Pigozzi, 1989]. Besides 𝒩 3 and 𝒩 4, Nelson introduced a logic which he called 𝒮 [Nelson, 1959],.

(20) 18. CONTENTS. also aimed at the study of the concept of realizability. The original presentation of 𝒮 by Nelson has infinite rule schemata, no algebraic semantics is offered, and it is not clear whether 𝒩 3 is weaker or stronger than 𝒮. Unlike 𝒩 3 and 𝒩 4, the logic 𝒮 received little attention and some basic questions about 𝒮 were still open. Can 𝒮 be finitely axiomatized? What is the exact relation between 𝒮, 𝒩 3 and 𝒩 4? Are the formulas that Nelson claims not to be derivable in 𝒮 [Nelson, 1959, p. 213] indeed not derivable? We will here use the modern techniques of algebraic logic to answer these questions. In addition, employing an algebraic semantics based on distributive lattices we will show that the (corrected version of) Nelson’s presentation of 𝒮 has redundant rules. The dissertation is organized as follows. In Chapter 1 we give a short introduction remembering the basic concepts about algebraic logic, we present the logic 𝒮 and derive some formulas which will later be used to establish its algebraizability. In Chapter 2 we prove that 𝒮 is implicative and as consequence of this it is also algebraizable and we present its equivalent algebraic semantics. We give for 𝒮 two presentations for its semantics. The first one is given directly from its algebraizability thanks to the algorithm of [Blok and Pigozzi, 1989, Theorem 2.17], the second one is given thanks to the result that the class of algebras which are a semantics for 𝒮 is a variety. In Chapter 3 we provide another calculus for 𝒮 (which is an axiomatic extension of 𝐹 𝐿𝑒𝑤 [Spinks and Veroff, 2008b]), one that has finite axiom schemata and only one rule schema, modus ponens. Having only one rule it is much easier to prove the Deduction Metatheorem using induction over derivations. In Chapter 4 we provide an algorithm to producing 𝒮-algebras from 𝒮-algebras or implicative lattices and give concrete 𝒮-algebras which provide counter-examples for the formulas that Nelson claimed to be invalid in 𝒮. In Chapter 5 we prove that 𝒩 3 is a proper axiomatic extension of 𝒮 and that 𝒮 and 𝒩 4 are not comparable, we show that we can obtain 𝒩 3 from 𝒮 by adding one axiom. Finishing the dissertation, we show that the variety of 𝒮-algebras is not finitely generated..

(21) 19. 1 Nelson’s logic 𝒮 Studying algebraic logic, we establish using algebraic methods a bridge between proof theoretic view and algebraic semantics view. In this chapter we define the basics concepts that we use to make this bridge, recall Nelson’s original presentation of the propositional fragment of 𝒮 [Nelson, 1959] and prove some theorems in 𝒮 that will be used later to establish its algebraizability. The logic which we are going to study in this dissertation it is a fixed version of 𝒮. Nelson’s original presentation has many typos, we have fixed the typos and told why we thought that our version is correct.. 1.1 A brief introduction Definition 1. A language is a pair L = ⟨𝐿, ar⟩ where 𝐿 ̸= ∅ is a set and ar : 𝐿 −→ 𝜔 is a function. Take 𝑠 ∈ 𝐿, we say that ar(𝑠) is the arity of 𝑠.. ∙ Take 𝑠 ∈ 𝐿, if ar(𝑠) = 0, then 𝑠 is said to be a constant. ∙ If ar(𝑠) = 𝑛, where 𝑛 > 0, then 𝑛 is said to be a connective. If ar(𝑠) = 1, then 𝑠 is a unary connective. If ar(𝑠) = 2, then 𝑠 is a binary connective.. In our case, two constants will have constant use: ⊥ (bottom) and ⊤ (top), which represent falsity and truth, respectively. Their interpretation in the algebra will be minimum (0) and maximum (1) of the algebra, respectively. Definition 2. Let Var be an infinite set and L = ⟨𝐿, ar⟩ be a language with Var ∩ 𝐿 = ∅. The L-algebra of formulas Fm above Var is given by the following clausules: 1. Var ⊆ Fm. 2. If 𝑠 ∈ 𝐿 is such that ar(𝑠) = 0, then 𝑠 ∈ Fm. 3. If 𝑠 ∈ 𝐿 is such that ar(𝑠) = 𝑛 > 0, then 𝑠(𝜙1 , · · · , 𝜙𝑛 ) ∈ Fm, where 𝜙𝑖 ∈ Fm. Definition 3. An evaluation is any homomorphism 𝑣 : Fm −→ A, where A is an algebra with same similarity of Fm, that is, of type 𝐿..

(22) Chapter 1. Nelson’s logic 𝒮. 20. Definition 4. Let Fm be a set of formulas, henceforth the set of equations of the language L is denoted by E𝑞 and is defined as E𝑞 := Fm × Fm. We write 𝜑 ≈ 𝜓 rather than (𝜑, 𝜓). We say that an evaluation 𝜐 satisfies 𝜑 ≈ 𝜓 in A when 𝜐(𝜑) = 𝜐(𝜓). We say that an L-algebra A satisfies 𝜑 ≈ 𝜓 when all evaluations in A satisfies it, we use the notation A |= 𝜑 ≈ 𝜓. Definition 5. A substitution is any endomorphism 𝜌 : Fm −→ Fm. Definition 6. A logic is a pair ℒ ⊢ℒ ⊆. =. ⟨L, ⊢ℒ ⟩ where L is a language and. P(Fm) × Fm is a relation satisfying the following properties, for all. Γ ∪ Δ ∪ {𝜑} ⊆ Fm: (I). Identity:. If 𝜑 ∈ Γ, then (Γ, 𝜑) ∈ ⊢ℒ .. (M) Monotonicity: (Γ, 𝜑) ∈ ⊢ℒ and Γ ⊆ Δ, then (Δ, 𝜑) ∈ ⊢ℒ . (T). Transitivity:. (Γ, 𝜑) ∈ ⊢ℒ and (Δ, 𝜓) ∈⊢ℒ for every 𝜓 ∈ Γ, then (Δ, 𝜑) ∈⊢ℒ .. (S). Structurality:. If (Γ, 𝜑) ∈ ⊢ℒ , then (𝜌Γ, 𝜌𝜑) ∈ ⊢ℒ for every substituition 𝜌.. We are going to write Γ ⊢ℒ 𝜑 to designate (Γ, 𝜑) ∈ ⊢ℒ and a formula 𝜑 such that ∅ ⊢ℒ 𝜑 is said to be a theorem.. 1.2 Nelson’s logic 𝒮 Definition 7. Nelson’s logic 𝒮 = ⟨Fm, ⊢𝒮 ⟩ is the sentential logic in the language ⟨∧, ∨, ⇒, ¬, ⊥⟩ of type ⟨2, 2, 2, 1, 0⟩ defined by the Hilbert-style calculus with the rule schemata in Table 1 and the following axiom schemata. Hereupon we shall use 𝜑 ⇔ 𝜓 as an abbreviation for (𝜑 ⇒ 𝜓) ∧ (𝜓 ⇒ 𝜑). Axioms (A1) 𝜑 ⇒ 𝜑 (A2) ⊥ ⇒ 𝜑 (A3) ¬𝜑 ⇒ (𝜑 ⇒ ⊥) (A4) ¬⊥ (A5) (𝜑 ⇒ 𝜓) ⇔ (¬𝜓 ⇒ ¬𝜑)..

(23) 1.2. Nelson’s logic 𝒮. 21. Rules Γ ⇒ (𝜑 ⇒ (𝜓 ⇒ 𝛾)) (P) Γ ⇒ (𝜓 ⇒ (𝜑 ⇒ 𝛾)). 𝜑 ⇒ (𝜑 ⇒ (𝜑 ⇒ 𝛾)) (C) 𝜑 ⇒ (𝜑 ⇒ 𝛾). Γ⇒𝜑 𝜑⇒𝛾 (E) Γ⇒𝛾. Γ⇒𝜑 𝜓⇒𝛾 (⇒ l) Γ ⇒ ((𝜑 ⇒ 𝜓) ⇒ 𝛾). 𝛾 (⇒ r) 𝜑⇒𝛾. 𝜑⇒𝛾 (∧l1) (𝜑 ∧ 𝜓) ⇒ 𝛾. 𝜓⇒𝛾 (∧l2) (𝜑 ∧ 𝜓) ⇒ 𝛾. Γ⇒𝜑 Γ⇒𝜓 (∧r) Γ ⇒ (𝜑 ∧ 𝜓). 𝜑⇒𝛾 𝜓⇒𝛾 (∨l1) (𝜑 ∨ 𝜓) ⇒ 𝛾. (𝜑 ⇒2 𝛾) (𝜓 ⇒2 𝛾) (∨l2) ((𝜑 ∨ 𝜓) ⇒2 𝛾). Γ⇒𝜑 (∨r1) Γ ⇒ (𝜑 ∨ 𝜓). Γ⇒𝜓 (∨r2) Γ ⇒ (𝜑 ∨ 𝜓). (𝜑 ∧ ¬𝜓) ⇒ 𝛾 (¬ ⇒ l) ¬(𝜑 ⇒ 𝜓) ⇒ 𝛾. Γ ⇒2 (𝜑 ∧ ¬𝜓) (¬ ⇒ r) Γ ⇒2 ¬(𝜑 ⇒ 𝜓). (¬𝜑 ∨ ¬𝜓) ⇒ 𝛾 (¬ ∧ l) ¬(𝜑 ∧ 𝜓) ⇒ 𝛾. Γ ⇒ (¬𝜑 ∨ ¬𝜓) (¬ ∧ r) Γ ⇒ ¬(𝜑 ∧ 𝜓). (¬𝜑 ∧ ¬𝜓) ⇒ 𝛾 (¬ ∨ l) ¬(𝜑 ∨ 𝜓) ⇒ 𝛾. Γ ⇒ (¬𝜑 ∧ ¬𝜓) (¬ ∨ r) Γ ⇒ ¬(𝜑 ∨ 𝜓). 𝜑⇒𝛾 (¬¬l) ¬¬𝜑 ⇒ 𝛾. Γ⇒𝜑 (¬¬r) Γ ⇒ ¬¬𝜑 Table 1 – Rules of 𝒮. In Table 1, following Nelson’s notation, Γ = (𝜑1 , . . . , 𝜑𝑛 ) is an arbitrary finite list of formulas, and the following abbreviations are used: Γ ⇒ 𝜑 := 𝜑1 ⇒ (𝜑2 ⇒ (. . . ⇒ (𝜑𝑛 ⇒ 𝜑) . . .)). In case Γ is empty, then Γ ⇒ 𝜑 is just 𝜑. Moreover, we let 𝜑 ⇒2 𝜓 := 𝜑 ⇒ (𝜑 ⇒ 𝜓) and Γ ⇒2 𝜑 := 𝜑1 ⇒2 (𝜑2 ⇒2 (. . . ⇒2 (𝜑𝑛 ⇒2 𝜑) . . .)). We have fixed obvious typos in the rules (∧l2), (∧r) and (¬ ⇒ r) as they appear in [Nelson, 1959, p. 214-5]. The original rule (∧l2) in Nelson’s paper was: (𝜑 ∧ 𝜓) ⇒ 𝛾 (∧l2) 𝜓⇒𝛾 This clearly makes the logic inconsistent. Indeed, taking 𝜑 = 𝛾, we have: (𝜑 ∧ 𝜓) ⇒ 𝜑 (∧l2) 𝜓⇒𝜑.

(24) Chapter 1. Nelson’s logic 𝒮. 22. As (𝜑 ∧ 𝜓) ⇒ 𝜑 is a theorem (Proposition 1.3), 𝜓 ⇒ 𝜑 is a theorem too. Choosing 𝜓 as an axiom, we have that 𝜑 is a theorem for all formulas 𝜑. The original rule (∧r) in Nelson’s paper was: Γ ⇒ (𝜑 ∧ 𝜓) (∧r) Γ⇒𝜑 Γ⇒𝜓 This rule does not seem consistent with the notation used by Nelson for the others (Nelson was presenting rules from introduction, this rule is eliming the conjunction). Moreover, it can be proved from other rules of 𝒮. Γ ⇒ (𝜑 ∧ 𝜓) (𝜑 ∧ 𝜓) ⇒ 𝜑 (E) Γ⇒𝜑 The other case is analogous. The rule (¬ ⇒ r) appeared in Nelson’s paper as follows: Γ ⇒2 ¬(𝜑 ⇒ 𝜓) (¬ ⇒ r) Γ ⇒2 (𝜑 ∧ 𝜓) This rule is not even classically valid. Following Nelson’s notation, the correct rule would have been: Γ ⇒2 ¬(𝜑 ⇒ 𝜓) (¬ ⇒ r) Γ ⇒2 (𝜑 ∧ ¬𝜓) This rule does not seem consistent with the notation used by Nelson for the others (the ⇒ connective should appear on the right-hand side at the bottom, and the ∧ connective at the top); moreover, it is redundant. To see this notice that, using (¬ ⇒ l), we can show that the formula ¬(𝜑 ⇒ 𝜓) ⇒ (𝜑 ∧ ¬𝜓) is a theorem of 𝒮 (that is, it is derivable from an empty set of premises): (A1) (𝜑 ∧ ¬𝜓) ⇒ (𝜑 ∧ ¬𝜓) (¬ ⇒ l) ¬(𝜑 ⇒ 𝜓) ⇒ (𝜑 ∧ ¬𝜓) Now using (E) we have: Γ ⇒2 ¬(𝜑 ⇒ 𝜓) ¬(𝜑 ⇒ 𝜓) ⇒ (𝜑 ∧ ¬𝜓) Γ ⇒2 (𝜑 ∧ ¬𝜓).

(25) 1.2. Nelson’s logic 𝒮. 23. Rule (C), called weak condensation by Nelson, replaces (and is indeed a weaker form of) the standard contraction rule:. 𝜑 ⇒ (𝜑 ⇒ 𝜓) 𝜑⇒𝜓. (C) is also known in the literature as 3-2 contraction (see, e.g., [Restall, 1993, p. 389]) and corresponds to the algebraic property of 3-potency (see Section 2.3). Notice also that the usual rule of modus ponens is an instance of (E) for Γ = ∅: 𝜑 𝜑⇒𝜓 𝜓. (MP). Lastly, we observe that every rule schema involving Γ is actually a shorthand for a denumerably infinite set of rule schemata. For instance, the schema Γ⇒𝜑 𝜓⇒𝛾 Γ ⇒ ((𝜑 ⇒ 𝜓) ⇒ 𝛾). (⇒ l). stands for the following collection of rules: 𝜑 𝜓⇒𝛾 (𝜑 ⇒ 𝜓) ⇒ 𝛾 𝛾1 ⇒ 𝜑 𝜓 ⇒ 𝛾 𝛾1 ⇒ ((𝜑 ⇒ 𝜓) ⇒ 𝛾) ··· 𝛾1 ⇒ (𝛾2 ⇒ (𝛾3 ⇒ 𝜑)) 𝜓 ⇒ 𝛾 𝛾1 ⇒ (𝛾2 ⇒ (𝛾3 ⇒ ((𝜑 ⇒ 𝜓) ⇒ 𝛾))) ··· One of the crucial steps to prove that a logic is algebraizable (in the sense of Blok and Pigozzi [Blok and Pigozzi, 1989, Definition 2.2]) is to prove that it satisfies certain congruence properties. In our context, this entails checking that 𝜑 ⇔ 𝜓 ⊢𝒮 ¬𝜑 ⇔ ¬𝜓 and {𝜑1 ⇔ 𝜓1 , 𝜑2 ⇔ 𝜓2 } ⊢𝒮 (𝜑1 ∙ 𝜑2 ) ⇔ (𝜓1 ∙ 𝜓2 ) for each connective ∙ ∈ {∧, ∨, ⇒}. The following result will be used to prove it in the next chapter. Example 1. The following formulas are theorems of 𝒮: 1. 𝜑 ⇒ (𝜑 ∨ 𝜓)..

(26) Chapter 1. Nelson’s logic 𝒮. 24. 2. 𝜓 ⇒ (𝜑 ∨ 𝜓). 3. (𝜑 ∧ 𝜓) ⇒ 𝜑. 4. (𝜑 ∧ 𝜓) ⇒ 𝜓. 5. (𝜑 ⇒ (𝜓 ⇒ 𝛾)) ⇔ (𝜓 ⇒ (𝜑 ⇒ 𝛾)). Proof. All derivations are straightforward. We illustrate the first item as an example. Taking Γ = 𝜑 in rule (∨r1), we have 𝜑⇒𝜑 𝜑 ⇒ (𝜑 ∨ 𝜓) Since 𝜑 ⇒ 𝜑 is an axiom, it follows that 𝜑 ⇒ (𝜑 ∨ 𝜓) is a theorem of 𝒮..

(27) 25. 2 Algebraization In this chapter we prove that 𝒮 is algebraizable in the sense of Blok and Pigozzi (and, in fact, is implicative [Font, 2016, Definition 2.3]), and we give two equivalent presentations for its equivalent algebraic semantics (to be called 𝒮-algebras). The first one is obtained via the algorithm of [Blok and Pigozzi, 1989, Theorem 2.17], while the second one is closer to the usual axiomatizations of classes of residuated lattices, which are the algebraic counterpart of many logics in the substructural family. The second presentation of 𝒮-algebras will allow us to see at a glance that they form an equational class, and also makes it easier to compare them with other known classes of algebras of substructural logics.. 2.1 𝒮 is algebraizable Definition 8. An implicative logic is a logic ℒ in a language Σ having a term 𝛼(𝑝, 𝑞) in two variables that satisfies the following: IL1. ⊢ℒ 𝛼(𝑝, 𝑝). IL2. 𝛼(𝑝, 𝑞), 𝛼(𝑞, 𝑟) ⊢ℒ 𝛼(𝑝, 𝑟). IL3. ⋃︀𝑛. IL4. 𝑝, 𝛼(𝑝, 𝑞) ⊢ℒ 𝑞. IL5. 𝑞 ⊢ℒ 𝛼(𝑝, 𝑞). 𝑖=1 {𝛼(𝑝𝑖 , 𝑞𝑖 ), 𝛼(𝑞𝑖 , 𝑝𝑖 )}. ⊢ℒ 𝛼(𝜆𝑝1 . . . 𝑝𝑛 , 𝜆𝑞1 . . . 𝑞𝑛 ) for each 𝑛-ary 𝜆 in Σ. Clarifying the notation in IL3, if ar(𝜆) = 1, then the notation 𝛼(𝜆𝜑𝑖 · · · 𝜑𝑛 , 𝜆𝜓𝑖 · · · 𝜓𝑛 ) means that 𝛼(𝜆𝜑, 𝜆𝜓). If ar(𝜆) = 2, then 𝛼(𝜑1 𝜆𝜑2 , 𝜓1 𝜆𝜓2 ). Defining the relations: 𝑝≡𝑞. if and only if ∅ ⊢ℒ 𝛼(𝑝, 𝑞) 𝑎𝑛𝑑 ∅ ⊢ℒ 𝛼(𝑞, 𝑝) 𝑝 ≤ℒ 𝑞. if and only if ∅ ⊢ℒ 𝛼(𝑝, 𝑞). ∙ According items IL1 and IL2, ≤ℒ is a quasiorder in ℒ and that the relation ≤ℒ defines an order ≤ in the algebra quotient Fm/≡..

(28) 26. Chapter 2. Algebraization. ∙ Item IL3 tell us that the relation ≤ℒ is a congruence of the formula algebra Fm and the quotient Fm/≡ is an algebra of type Σ. ∙ Item IL4 tell us that the set of theorems is an upset, that is, if ∅ ⊢ℒ 𝑝 and 𝑝 ≤ℒ 𝑞, then ∅ ⊢ℒ 𝑞 and the canonical projection of Fm into Fm/≡ has the following property: For all 𝑝 ∈ Fm, ∅ ⊢ℒ 𝑝 if and only if 𝑝 ∈ Thm/≡, where Thm is the set of theorems of ℒ. Moreover, Thm/≡ is an upset of order ≤ in Fm/≡. ∙ Finalizing, item IL5 tell us that all theorems are in the same class in the algebra Fm/≡. Definition 9. Let ℒ be an implicative logic in the language Σ. An ℒ-algebra is an algebra A of type Σ whose carrier has an element 1 ∈ 𝐴 with the following features:. (LALG1) For all Γ ∪ {𝜑} ⊆ Fm and all valuation 𝜐 , if Γ ⊢ℒ 𝜑 and 𝜐(Γ) ⊆ {1}, then 𝜐(𝜑) = 1. (LALG2) For all 𝑎, 𝑏 ∈ 𝐴, if 𝛼(𝑝, 𝑞) = 1 and 𝛼(𝑞, 𝑝) = 1, then 𝑎 = 𝑏.. The class of ℒ-algebras is denoted by Alg*ℒ. It will be convenient for us to work with the following definition of algebraizable logic, which is not the original one [Blok and Pigozzi, 1989, Definition 2.1] but an equivalent so-called intrinsic characterization [Blok and Pigozzi, 1989, Theorem 3.21]. Definition 10. A logic ℒ is algebraizable if and only if there are finite and non-empty set of equations E(𝑥) ⊆ E𝑞 and formulas Δ(𝑥, 𝑦) ⊆ 𝐹 𝑚 such that ℒ satisfies the following five conditions:. (R). ⊢ℒ Δ(𝜑, 𝜑). (Sym). Δ(𝜑, 𝜓) ⊢ℒ Δ(𝜓, 𝜑). (Trans). Δ(𝜑, 𝜓) ∪ Δ(𝜓, 𝛾) ⊢ℒ Δ(𝜑, 𝛾). (Re). ⋃︀𝑛. (Alg3). 𝜑 ⊣⊢ℒ Δ(E(𝜑)). 𝑖=1. Δ(𝜑𝑖 , 𝜓𝑖 ) ⊢ℒ Δ(𝜆𝜑1 · · · 𝜑𝑛 , 𝜆𝜓1 · · · 𝜓𝑛 ) for each 𝑛-ary 𝜆 ∈ Σ.

(29) 2.1. 𝒮 is algebraizable. 27. Then E(𝑥) is said to be the set of defining equations and Δ(𝑥, 𝑦) is the set of equivalence formulas. Every implicative logic ℒ is algebraizable with respect to the class Alg*ℒ [Font, 2016, Proposition 3.15], and the algebraizability is witnessed by the defining equation 𝜏 (𝜑) := {𝜑 ≈ 𝛼(𝜑, 𝜑)} and the equivalence formulas 𝜌(𝜑 ≈ 𝜓) := {𝛼(𝜑, 𝜓), 𝛼(𝜓, 𝜑)}. Since the term 𝛼(𝜑, 𝜑) denotes an algebraic constant in all ℒ-algebras, it is often denoted simply by ⊤ or 1 (as we shall also do below). In the case of 𝒮, the term 𝛼(𝑝, 𝑞) can be chosen to be 𝑝 ⇒ 𝑞. Theorem 1. The calculus ⊢𝒮 is implicative , and thus algebraizable. Proof. IL1 follows immediately from axiom (A1), while IL2 follows from rule (E). We have to prove that ⇒ respects IL3 for each connective ∙ ∈ {∧, ∨, ⇒, ¬}. (¬) {𝜑 ⇔ 𝜓} ⊢𝒮 ¬𝜑 ⇔ ¬𝜓 holds by axiom (A5) and (MP). (∧) We must prove that {(𝜑1 ⇔ 𝜓1 ) ∧ (𝜑2 ⇔ 𝜓2 )} ⊢𝒮 (𝜑1 ∧ 𝜑2 ) ⇔ (𝜓1 ∧ 𝜓2 ). From Example 1.3 and 1.4 we have ⊢𝒮 (𝜑1 ∧ 𝜑2 ) ⇒ 𝜑1 and ⊢𝒮 (𝜑1 ∧ 𝜑2 ) ⇒ 𝜑2 . From rule (E) we have: (𝜑1 ∧ 𝜑2 ) ⇒ 𝜑1 𝜑1 ⇒ 𝜓1 (E) (𝜑1 ∧ 𝜑2 ) ⇒ 𝜓1 and again by (E): (𝜑1 ∧ 𝜑2 ) ⇒ 𝜑2 𝜑2 ⇒ 𝜓2 (E) (𝜑1 ∧ 𝜑2 ) ⇒ 𝜓2 Now, by rule (∧r), we have: (𝜑1 ∧ 𝜑2 ) ⇒ 𝜓1 (𝜑1 ∧ 𝜑2 ) ⇒ 𝜓2 (∧r) (𝜑1 ∧ 𝜑2 ) ⇒ (𝜓1 ∧ 𝜓2 ) The remainder of the proof is similar. (∨) We must prove that {(𝜑1 ⇔ 𝜓1 ) ∧ (𝜑2 ⇔ 𝜓2 )} ⊢𝒮 (𝜑1 ∨ 𝜑2 ) ⇔ (𝜓1 ∨ 𝜓2 ). From Example 1.1 and 1.2, 𝜓1 ⇒ (𝜓1 ∨ 𝜓2 ) and 𝜓2 ⇒ (𝜓1 ∨ 𝜓2 ) are derivable. From the (E) rule we have: 𝜑1 ⇒ 𝜓1 𝜓1 ⇒ (𝜓1 ∨ 𝜓2 ) (E) 𝜑1 ⇒ (𝜓1 ∨ 𝜓2 ).

(30) 28. Chapter 2. Algebraization. and also: 𝜑2 ⇒ 𝜓2 𝜓2 ⇒ (𝜓1 ∨ 𝜓2 ) (E) 𝜑2 ⇒ (𝜓1 ∨ 𝜓2 ) Now from rule (∨l1) we have: 𝜑1 ⇒ (𝜓1 ∨ 𝜓2 ) 𝜑2 ⇒ (𝜓1 ∨ 𝜓2 ) (∨l1) (𝜑1 ∨ 𝜑2 ) ⇒ (𝜓1 ∨ 𝜓2 ) The remainder of the proof is similar. (⇒) We must prove that {(𝜃 ⇔ 𝜑) ∧ (𝜓 ⇔ 𝛾)} ⊢𝒮 (𝜃 ⇒ 𝜓) ⇔ (𝜑 ⇒ 𝛾). This time, using rule (⇒ l) we have: 𝜑⇒𝜃 𝜓⇒𝛾 (⇒ l) 𝜑 ⇒ ((𝜃 ⇒ 𝜓) ⇒ 𝛾) Now, from Example 1.5 and (MP), we have: 𝜑 ⇒ ((𝜃 ⇒ 𝜓) ⇒ 𝛾) (𝜑 ⇒ ((𝜃 ⇒ 𝜓) ⇒ 𝛾)) ⇒ ((𝜃 ⇒ 𝜓) ⇒ (𝜑 ⇒ 𝛾)) (MP) (𝜃 ⇒ 𝜓) ⇒ (𝜑 ⇒ 𝛾) The remainder of the proof is analogous. Finally, IL4 follows from (MP) and IL5 follows from (⇒ r).. 2.2 𝒮-algebras By Blok-Pigozzi’s algorithm ([Blok and Pigozzi, 1989, Theorem 2.17], we have that the equivalent algebraic semantics of 𝒮 is the class of algebras given in Definition 11 below. Let Ax the set of axioms of 𝒮 and Inf R the set the inference rules of 𝒮, we define the algebraic semantics of 𝒮 as follows: Definition 11. An 𝒮-algebra is a structure A = ⟨𝐴, ∧, ∨, ⇒, ¬, 0, 1⟩ of type ⟨2, 2, 2, 1, 0, 0⟩ that satisfies the following equations and quasiequations: 1. E(𝜙), for each 𝜙 ∈ Ax 2. E(Δ(𝜙, 𝜙)) 3.. 𝑛 ⋃︀. E(𝛾𝑖 ) implies E(𝜙) for each 𝛾1 , · · · , 𝛾𝑛 ⊢𝒮 𝜙 ∈ Inf R. 𝑖=1. 4. E(Δ(𝜙, 𝜓)) implies 𝜙 ≈ 𝜓.

(31) 2.2. 𝒮-algebras. 29. We shall henceforth denote by E(An) the equation given from Definition 11.1 for the axiom An (for 1 ≤ n ≤ 5) of 𝒮, and by Q(R) the quasiequation given from Definition 11.3 for the rule R of 𝒮. We will also use the following abbreviations: 𝑎 * 𝑏 := ¬(𝑎 ⇒ ¬𝑏), 𝑎2 := 𝑎*𝑎 and 𝑎𝑛 := 𝑎*(𝑎𝑛−1 ) for 𝑛 > 2. As the notation suggests, the defined connective * is intended as a “strong conjunction” in the sense of substructural logics (alternative to the “weak conjunction” ∧) that will be interpreted as a monoid operation on 𝒮-algebras, and having the implication ⇒ as residuum. We shall now prove a few properties of 𝒮algebras that will indeed allow us to view them as a class of residuated structures. Proposition 1. Let 𝐴 be an 𝒮-algebra and 𝑎, 𝑏, 𝑐 ∈ 𝐴. Then, 1. 𝑎 ⇒ 𝑎 = 𝑏 ⇒ 𝑏 = 1. 2. The relation ≤ defined by 𝑎 ≤ 𝑏 iff 𝑎 ⇒ 𝑏 = 1, is a partial order with maximum 1 and minimum 0. 3. 𝑎 ⇒ 𝑏 = ¬𝑏 ⇒ ¬𝑎. 4. 𝑎 ⇒ (𝑏 ⇒ 𝑐) = 𝑏 ⇒ (𝑎 ⇒ 𝑐). 5. ¬¬𝑎 = 𝑎 and 𝑎 ⇒ 0 = ¬𝑎. 6. ⟨𝐴, *, 1⟩ is a commutative monoid. 7. (𝑎 * 𝑏) ⇒ 𝑐 = 𝑎 ⇒ (𝑏 ⇒ 𝑐). 8. The pair ⟨*, ⇒⟩ is residuated with respect to ≤, i.e., 𝑎 * 𝑏 ≤ 𝑐 iff 𝑏 ≤ 𝑎 ⇒ 𝑐. 9. 𝑎2 ≤ 𝑎3 . 10. ⟨𝐴, ∧, ∨⟩ is a lattice with order ≤. 11. (𝑎 ∨ 𝑏)2 ≤ 𝑎2 ∨ 𝑏2 . Proof. 1. Follows from the fact that 𝒮 is an implicative logic, see [Font, 2016, Lemma 2.6]. In particular, ¬0 = 0 ⇒ 0 = 1.. 2. By E(A2) we have that 0 is the minimum element with respect to the order ≤. The rest easily follows from the fact that 𝒮 is implicative..

(32) 30. Chapter 2. Algebraization. 3. Follows from E(A5) and item 2 above.. 4. By Q(P) and 2 above, we have that 𝑑 ≤ 𝑎 ⇒ (𝑏 ⇒ 𝑐) implies 𝑑 ≤ 𝑏 ⇒ (𝑎 ⇒ 𝑐) for all 𝑑 ∈ 𝐴. Then, taking 𝑑 = 𝑎 ⇒ (𝑏 ⇒ 𝑐), we have 𝑎 ⇒ (𝑏 ⇒ 𝑐) ≤ 𝑏 ⇒ (𝑎 ⇒ 𝑐) which easily implies the desired result.. 5. ¬¬𝑎 = 𝑎 follows from item 2 above together with Q(¬¬l) and Q(¬¬r). By item 3 above, 𝑎 ⇒ 0 = ¬0 ⇒ ¬𝑎 = 1 ⇒ ¬𝑎 = ¬𝑎. The last equality holds because, on the one hand, by Q(⇒ l) we have that 1 ≤ 1 and ¬𝑎 ≤ ¬𝑎 implies 1 ⇒ ¬𝑎 ≤ ¬𝑎. On the other, by item 1 we have ¬𝑎 ⇒ ¬𝑎 ≤ 1 and so we can apply Q(⇒ r) to obtain 1 ⇒ (¬𝑎 ⇒ ¬𝑎) = 1. By item 4, we have 1 ⇒ (¬𝑎 ⇒ ¬𝑎) = ¬𝑎 ⇒ (1 ⇒ ¬𝑎), hence we conclude that ¬𝑎 ⇒ (1 ⇒ ¬𝑎) = 1 and so, by item 2, ¬𝑎 ≤ 1 ⇒ ¬𝑎.. 6. As to commutativity, using items 3 and 5 above, we have 𝑎 * 𝑏 = ¬(𝑎 ⇒ ¬𝑏) = ¬(¬¬𝑏 ⇒ ¬𝑎) = ¬(𝑏 ⇒ ¬𝑎) = 𝑏 * 𝑎. As to associativity, using 3, 5, Q(¬¬r) and Q(¬¬l), we have (𝑎 * 𝑏) * 𝑐 = ¬(¬(𝑎 ⇒ ¬𝑏) ⇒ ¬𝑐)) = ¬(¬¬𝑐 ⇒ ¬¬(𝑎 ⇒ ¬𝑏)) = ¬(𝑐 ⇒ (𝑎 ⇒ ¬𝑏)) = ¬(𝑎 ⇒ (𝑐 ⇒ ¬𝑏)) = ¬(𝑎 ⇒ (𝑏 ⇒ ¬𝑐)) = ¬(𝑎 ⇒ ¬¬(𝑏 ⇒ ¬𝑐)) = 𝑎 * (𝑏 * 𝑐). As to 1 being the neutral element, using item 5 above, we have 𝑎 * 1 = 𝑎 * ¬0 = ¬(𝑎 ⇒ ¬¬0) = ¬(𝑎 ⇒ 0) = ¬¬𝑎 = 𝑎.. 7. Using items 2, 3, 5 and 6 above, we have (𝑎 * 𝑏) ⇒ 𝑐 = ¬(𝑎 ⇒ ¬𝑏) ⇒ 𝑐 = ¬𝑐 ⇒ ¬¬(𝑎 ⇒ ¬𝑏) = ¬𝑐 ⇒ (𝑎 ⇒ ¬𝑏) = 𝑎 ⇒ (¬𝑐 ⇒ ¬𝑏) = 𝑎 ⇒ (¬¬𝑏 ⇒ ¬¬𝑐) = 𝑎 ⇒ (𝑏 ⇒ 𝑐).. 8. By item 2 above, we have 𝑎 * 𝑏 ≤ 𝑐 iff (𝑎 * 𝑏) ⇒ 𝑐 = 1 iff, by item 7, 𝑎 ⇒ (𝑏 ⇒ 𝑐) = 1 iff, by 6, 𝑏 ⇒ (𝑎 ⇒ 𝑐) = 1 iff, by 2 again, 𝑏 ≤ 𝑎 ⇒ 𝑐.. 9. By Q(C) we have that 𝑎3 ≤ 𝑐 implies 𝑎2 ≤ 𝑐 for all 𝑐 ∈ 𝐴. Then, taking 𝑐 = 𝑎3 , we have 𝑎2 ≤ 𝑎3 .. 10. We check that 𝑎 ∧ 𝑏 is the infimum of the set {𝑎, 𝑏} with respect to ≤. First of all, we have 𝑎 ∧ 𝑏 ≤ 𝑎 and 𝑎 ∧ 𝑏 ≤ 𝑏 by Q(∧l1), Q(∧l2) and item 2 above. Then, assuming 𝑐 ≤ 𝑎 and 𝑐 ≤ 𝑏, we have 𝑐 ≤ 𝑎 ∧ 𝑏 by Q(∧r). A similar reasoning, using Q(∨r1), Q(∨r2) and Q(∨l1) shows that 𝑎 ∨ 𝑏 is the supremum of {𝑎, 𝑏}..

(33) 2.3. 𝒮-algebras as residuated lattices. 31. 11 By 10 we have that 𝑎2 ≤ 𝑎2 ∨ 𝑏2 and 𝑏2 ≤ 𝑎2 ∨ 𝑏2 . Hence, by item 8, we have 𝑎 ≤ 𝑎 ⇒ (𝑎2 ∨ 𝑏2 ) and 𝑏 ≤ 𝑏 ⇒ (𝑎2 ∨ 𝑏2 ). By item 2 we have then 𝑎 ⇒ (𝑎 ⇒ (𝑎2 ∨ 𝑏2 ) = 𝑏 ⇒ (𝑏 ⇒ (𝑎2 ∨𝑏2 ) = 1, hence we can use Q(∨l2) to obtain (𝑎∨𝑏) ⇒ ((𝑎∨𝑏) ⇒ (𝑎2 ∨𝑏2 ) = 1. Then items 2 and 8 give us (𝑎 ∨ 𝑏)2 ≤ 𝑎2 ∨ 𝑏2 as required.. 2.3 𝒮-algebras as residuated lattices In this section we introduce an equivalent presentation of 𝒮-algebras which takes precisely the properties in Proposition 1 as postulates. We begin with the following wellknown definitions (see e.g. [Galatos et al., 2007, p. 185]). Definition 12. A commutative integral bounded residuated lattice (CIBRL) is an algebra A = ⟨𝐴, ∧, ∨, *, ⇒, 0, 1⟩ of type ⟨2, 2, 2, 2, 0, 0⟩ such that:. 1. ⟨𝐴, ∧, ∨, 0, 1⟩ is a bounded lattice with ordering ≤, minimum element 0 and maximum 1. 2. ⟨𝐴, *, 1⟩ is a commutative monoid. 3. The pair ⟨*, ⇒⟩ is residuated with respect to ≤, i.e., 𝑎 * 𝑏 ≤ 𝑐 iff 𝑏 ≤ 𝑎 ⇒ 𝑐.. Letting ¬𝑥 := 𝑥 ⇒ 0, we say that a residuated lattice is involutive [Galatos and Raftery, 2004, p. 186] when ¬¬𝑎 = 𝑎 (in such a case, it follows that 𝑎 ⇒ 𝑏 = ¬𝑏 ⇒ ¬𝑎)[Ono, 2010, Lemma 3.1]. We say that a residuated lattice is 3-potent when it satisfies the equation 𝑥2 ≤ 𝑥3 . We have defined earlier * from ⇒, and now * is a primitive operation, but we can show that every CIBRL satisfies 𝑥 * 𝑦 := ¬(𝑥 ⇒ ¬𝑦), see [Galatos and Raftery, 2004, Lemma 5.1]. Definition 13. An 𝒮 ′ -algebra is a three-potent involutive CIBRL. Since CIBRLs form an equational class [Galatos et al., 2007, Theorem 2.7], it is clear that 𝒮 ′ -algebras are also an equational class; by contrast, it is far from obvious from Definition 11 whether 𝒮-algebras are equationally axiomatizable or not. By Proposition 1, we immediately have the following result..

(34) 32. Chapter 2. Algebraization. Proposition 2. Let A = ⟨𝐴, ∧, ∨, ⇒, ¬, 0, 1⟩ be an 𝒮-algebra. Letting 𝑥*𝑦 := ¬(𝑥 ⇒ ¬𝑦), we have that A = ⟨𝐴, ∧, ∨, * ⇒, 0, 1⟩ is an 𝒮 ′ -algebra. Lemma 1.. 1. Any CIBRL satisfies the equation (𝑥 ∨ 𝑦) * 𝑧 ≈ (𝑥 * 𝑧) ∨ (𝑦 * 𝑧).. 2. Any CIBRL satisfies 𝑥2 ∨ 𝑦 2 ≈ (𝑥2 ∨ 𝑦 2 )2 . 3. Any 3-potent CIBRL satisfies (𝑥 ∨ 𝑦 2 )2 ≈ (𝑥 ∨ 𝑦)2 . 4. Any 3-potent CIBRL satisfies (𝑥 ∨ 𝑦)2 ≈ 𝑥2 ∨ 𝑦 2 . Proof. 1. See [Galatos et al., 2007, Lemma 2.6].. 2. Let 𝐿 be a CIBRL and 𝑎, 𝑏 arbitrary elements of this lattice. From 𝑎2 ≤ 𝑎2 ∨ 𝑏2 and 𝑏2 ≤ 𝑎2 ∨ 𝑏2 , using monotonicity of *, we have 𝑎4 ≤ (𝑎2 ∨ 𝑏2 )2 and 𝑏4 ≤ (𝑎2 ∨ 𝑏2 )2 . Using 3-potency, the latter inequalities simplify to 𝑎2 ≤ (𝑎2 ∨ 𝑏2 )2 and 𝑏2 ≤ (𝑎2 ∨ 𝑏2 )2 . Thus, 𝑎2 ∨ 𝑏2 ≤ (𝑎2 ∨ 𝑏2 )2 .. 3. We have 𝑎 ∨ 𝑏2 ≤ 𝑎 ∨ 𝑏 from monotonicity of * and supremum of ∨, therefore (𝑎 ∨ 𝑏2 )2 ≤ (𝑎 ∨ 𝑏)2 . For the converse, we have that 𝑎 * 𝑏 ≤ 𝑎, whence 𝑎 * 𝑏 ≤ 𝑎 ∨ 𝑏2 . Also, as 𝑎2 ≤ 𝑎 and 𝑏2 ≤ 𝑏 from monotonicity of *, we have that 𝑎2 ≤ 𝑎 ∨ 𝑏2 and 𝑏2 ≤ 𝑎 ∨ 𝑏2 . By supremum of ∨, 𝑎2 ∨ 𝑎 * 𝑏 ∨ 𝑏2 ≤ 𝑎 ∨ 𝑏2 . But 𝑎2 ∨ 𝑎 * 𝑏 ∨ 𝑏2 = (𝑎 ∨ 𝑏)2 by Lemma 1, so (𝑎 ∨ 𝑏)2 ≤ 𝑎 ∨ 𝑏2 . Using monotonicity of *, (𝑎 ∨ 𝑏)4 ≤ (𝑎 ∨ 𝑏2 )2 and from 3-potency (𝑎 ∨ 𝑏)2 ≤ (𝑎 ∨ 𝑏2 )2 .. 4. From Lemma 1.2 we have (𝑎2 ∨ 𝑏2 ) = (𝑎2 ∨ 𝑏2 )2 , and from Lemma 1.3 we have (𝑎2 ∨ 𝑏2 )2 = (𝑎2 ∨ 𝑏)2 = (𝑏 ∨ 𝑎2 )2 = (𝑏 ∨ 𝑎)2 . At this point we are in a position to show that, as anticipated, 𝒮 ′ -algebras and 𝒮-algebras can be viewed as two presentations (in slightly different languages) of the same class of structures. For this, we need to check that any 𝒮 ′ -algebra satisfies all (quasi)equations introduced in Definition 11. Proposition 3. Let A = ⟨𝐴, ∧, ∨, *, ⇒, 0, 1⟩ be an 𝒮 ′ -algebra. Letting ¬𝑥 := 𝑥 ⇒ 0, we have that A = ⟨𝐴, ∧, ∨, ⇒, ¬, 0, 1⟩ is an 𝒮-algebra. Proof. Let A be an 𝒮 ′ -algebra. We first consider the equations corresponding to the axioms of 𝒮. As 𝑎 ≤ 𝑏 iff 𝑎 ⇒ 𝑏 ≈ 1, we usually write 𝑎 ≤ 𝑏 rather than 𝑎 ⇒ 𝑏 ≈ 1..

(35) 2.3. 𝒮-algebras as residuated lattices. 33. Equations The equation E(A1) easily follows from integrality. We have E(A2) from the fact that 0 is the minimum element of A. From the definition of ¬ in 𝒮 ′ and from E(A1) follows E(A3). We know that 1 := ¬0 therefore we have E(A4). As A is involutive, follows E(A5). Remains to prove the equation E(Δ(𝜙, 𝜙)), see that we need to prove the following identity (𝜑 ⇒ 𝜑) ∧ (𝜑 ⇒ 𝜑) ≈ 1, we already know that 𝜑 ⇒ 𝜑 ≈ 1, therefore also (𝜑 ⇒ 𝜑) ∧ (𝜑 ⇒ 𝜑) ≈ 1.. In the next items, for each rule schema involving Γ the proof must be by induction over the lenght of the finite list (𝜑1 , . . . , 𝜑𝑛 ). From the monotonicty of * in the rules of type Γ ⇒ 𝜑 we can take Γ = (𝛾) and from 3-potency in the rules of type Γ ⇒2 𝛾 we can take Γ = (𝛾, 𝛾). Quasiequations. Q(P) follows from the commutativity of * and from the identity (𝑎*𝑏) ⇒ 𝑐 ≈ 𝑎 ⇒ (𝑏 ⇒ 𝑐).. Q(C) follows from 3-potency: since 𝑎2 ≤ 𝑎3 , we have that 𝑎3 ⇒ 𝑏 ≈ 1 implies 𝑎2 ⇒ 𝑏 ≈ 1.. Q(E) follows from the fact that A comprises a partial order ≤ that is determined by the implication ⇒.. To prove Q(⇒ l), suppose 𝑎 ≤ 𝑏 and 𝑐 ≤ 𝑑. From 𝑐 ≤ 𝑑, as 𝑏 ⇒ 𝑐 ≤ 𝑏 ⇒ 𝑐, using residuation we have that 𝑏 * (𝑏 ⇒ 𝑐) ≤ 𝑐 ≤ 𝑑, therefore 𝑏 * (𝑏 ⇒ 𝑐) ≤ 𝑑 and therefore 𝑏 ⇒ 𝑐 ≤ 𝑏 ⇒ 𝑑. Note that as 𝑎 ≤ 𝑏, using residuation we have that 𝑎 * (𝑏 ⇒ 𝑑) ≤ 𝑏 * (𝑏 ⇒ 𝑑) ≤ 𝑑, therefore 𝑏 ⇒ 𝑑 ≤ 𝑎 ⇒ 𝑑 and thus 𝑏 ⇒ 𝑐 ≤ 𝑎 ⇒ 𝑑. Now, since 𝑏 ⇒ 𝑐 ≤ 𝑎 ⇒ 𝑑 iff 𝑎 * (𝑏 ⇒ 𝑐) ≤ 𝑑 iff 𝑎 ≤ (𝑏 ⇒ 𝑐) ⇒ 𝑑, we obtain the desired result.. For Q(⇒ r) we need to prove that if 𝑑 ≈ 1, then 𝑏 ⇒ 𝑑 ≈ 1. This follows immediately from integrality.. Quasiequations Q(∧l1), Q(∧l2), Q(∧r), Q(∨l1), Q(∨r1) and Q(∨r2) follow straight-.

(36) 34. Chapter 2. Algebraization. forwardly from the fact that A is partially ordered and the order is determined by the implication.. To prove Q(∨l2), notice that (𝑏 ∨ 𝑐)2 ≤ 𝑏2 ∨ 𝑐2 by Lemma 1.4. Suppose 𝑏2 ≤ 𝑑 and 𝑐2 ≤ 𝑑, then since A is a lattice, we have 𝑏2 ∨ 𝑐2 ≤ 𝑑 and as (𝑏 ∨ 𝑐)2 ≤ 𝑏2 ∨ 𝑐2 we conclude that (𝑏 ∨ 𝑐)2 ≤ 𝑑 and thus (𝑏 ∨ 𝑐)2 ⇒ 𝑑 ≈ 1.. As to Q(¬ ⇒ l), by integrality we have 𝑏 * 𝑐 ≤ 𝑏 and 𝑏 * 𝑐 ≤ 𝑐. Thus 𝑏 * 𝑐 ≤ 𝑏 ∧ 𝑐. Now, if 𝑏 ∧ 𝑐 ≤ 𝑑, then 𝑏 * 𝑐 ≤ 𝑑.. To prove Q(¬ ⇒ r), suppose 𝑑2 ≤ 𝑏 ∧ 𝑐. Using monotonicity of *, we have 𝑑2 * 𝑑2 ≤ (𝑏 ∧ 𝑐) * (𝑏 ∧ 𝑐), i.e., 𝑑4 ≤ (𝑏 ∧ 𝑐)2 . Using 3-potency, we have 𝑑4 ≈ 𝑑2 , therefore 𝑑2 ≤ (𝑏 ∧ 𝑐)2 . Now, see that 𝑏∧𝑐 ≤ 𝑏 and 𝑏∧𝑐 ≤ 𝑐, using monotonicity of * we have (𝑏∧𝑐)*(𝑏∧𝑐) ≤ 𝑏*𝑐. Since (𝑏 ∧ 𝑐)2 ≤ 𝑏 * 𝑐, we have 𝑑2 ≤ (𝑏 ∧ 𝑐)2 ≤ (𝑏 * 𝑐), i.e., 𝑑2 ≤ (𝑏 * 𝑐).. Q(¬ ∧ l), Q(¬ ∧ r), Q(¬ ∨ l) and Q(¬ ∨ l) follow from the De Morgan’s Laws [Galatos et al., 2007, Lemma 3.17].. Finally, we have Q(¬¬l) and Q(¬¬r) from A being involutive.. Remain to prove the quasiequation E(Δ(𝜙, 𝜓)) implies 𝜙 ≈ 𝜓, that is, if ((𝜙 ⇒ 𝜓) ∧ (𝜓 ⇒ 𝜙)) ≈ 1, then 𝜙 ≈ 𝜓. As 1 is the maximum of the algebra, we have that (𝜙 ⇒ 𝜓) ≈ 1 and (𝜓 ⇒ 𝜙) ≈ 1, therefore 𝜙 ≤ 𝜓 and 𝜓 ≤ 𝜙, as ≤ is an order relation, follows 𝜙 ≈ 𝜓. From Propositions 2 and 3 above we obtain the desired result: Theorem 2. The classes of 𝒮-algebras and of 𝒮 ′ -algebras are term equivalent.1 An advantage of the presentation given in Definition 13 is that it makes it straightforward to check that, for instance, the three-element MV-algebra [Cignoli et al., 2000] is a model of Nelson’s logic 𝒮. This in turn allows one to prove that the formulas which Nelson claims not to be derivable in 𝒮 [Nelson, 1959, p. 213] are indeed not valid.. 1. See [Spinks and Veroff, 2008a, p. 329] for a formal definition of term equivalence..

(37) 35. 3 Calculus We are now going to introduce a finite Hilbert-style calculus and prove that it is algebraizable with respect to the class of 𝒮 ′ -algebras (hence, w.r.t 𝒮-algebras). This will give us a finite presentation of 𝒮 that is equivalent to Nelson’s calculus of Section 1.2, but has the advantage of involving only a finite number of axiom schemata. Our calculus is an axiomatic extension of the full Lambek calculus with exchange and weakening (𝐹 𝐿𝑒𝑤 ) of [Spinks and Veroff, 2008b], which will allow us to obtain algebraizability of 𝒮 as an easy extension of the corresponding result about 𝐹 𝐿𝑒𝑤 .. 3.1 A finite Hilbert-style calculus for 𝒮 ′ Definition 14. The logic 𝒮 ′ = ⟨Fm, ⊢𝒮 ′ ⟩ is the sentential logic in the language ⟨∧, ∨, ⇒, *, ⊥, ⊤⟩ of type ⟨2, 2, 2, 2, 0, 0⟩ defined by the Hilbert-style calculus with the following axiom schemata and modus ponens (from 𝜑 and 𝜑 ⇒ 𝜓, infer 𝜓) as the only rule:. (S1). (𝜙 ⇒ 𝜓) ⇒ ((𝜓 ⇒ 𝛾) ⇒ (𝜙 ⇒ 𝛾)). (S2). (𝜙 ⇒ (𝜓 ⇒ 𝛾)) ⇒ (𝜓 ⇒ (𝜙 ⇒ 𝛾)). (S3). 𝜙 ⇒ (𝜓 ⇒ 𝜙). (S4). 𝜙 ⇒ (𝜓 ⇒ (𝜙 * 𝜓)). (S5). (𝜙 ⇒ (𝜓 ⇒ 𝛾)) ⇒ ((𝜙 * 𝜓) ⇒ 𝛾). (S6). (𝜙 ∧ 𝜓) ⇒ 𝜙. (S7). (𝜙 ∧ 𝜓) ⇒ 𝜓. (S8). (𝜙 ⇒ 𝜓) ⇒ ((𝜙 ⇒ 𝛾) ⇒ (𝜙 ⇒ (𝜓 ∧ 𝛾))). (S9). 𝜙 ⇒ (𝜙 ∨ 𝜓). (S10). 𝜓 ⇒ (𝜙 ∨ 𝜓). (S11). (𝜙 ⇒ 𝛾) ⇒ ((𝜓 ⇒ 𝛾) ⇒ ((𝜙 ∨ 𝜓) ⇒ 𝛾)).

(38) 36. Chapter 3. Calculus. (S12). ⊤. (S13). ⊥⇒𝜙. (S14). ((𝜙 ⇒ ⊥) ⇒ ⊥) ⇒ 𝜙. (S15). (𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓))) ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓)). Example 2. We can deduce from ⊢𝒮 ′ the following formula (S1’). (𝜓 ⇒ 𝛾) ⇒ ((𝜙 ⇒ 𝜓) ⇒ (𝜙 ⇒ 𝛾)). 1. (𝜙 ⇒ 𝜓) ⇒ ((𝜓 ⇒ 𝛾) ⇒ (𝜙 ⇒ 𝛾)). 2. (𝜙 ⇒ 𝜓) ⇒ ((𝜓 ⇒ 𝛾) ⇒ (𝜙 ⇒ 𝛾)) ⇒ (𝜓 ⇒ 𝛾) ⇒ ((𝜙 ⇒ 𝜓) ⇒ (𝜙 ⇒ 𝛾)) (S2). 3. (𝜓 ⇒ 𝛾) ⇒ ((𝜙 ⇒ 𝜓) ⇒ (𝜙 ⇒ 𝛾)). (S1). (MP)(1, 2). Example 3. If ∅ ⊢𝒮 ′ 𝜙 then ∅ ⊢𝒮 ′ 𝜓 ⇒ 𝜙. 1. 𝜙. hypothesis. 2. 𝜙 ⇒ (𝜓 ⇒ 𝜙). (S3). 3. (𝜓 ⇒ 𝜙). (MP)(1, 2). Axioms from (S1) to (S13) are those that axiomatize 𝐹 𝐿𝑒𝑤 as presented in [Spinks and Veroff, 2008b, Section 5], where it is proven that 𝐹 𝐿𝑒𝑤 is algebraizable. This allows us to immediately obtain the following. Theorem 3. The calculus 𝒮 ′ is algebraizable (with the same defining equation and equivalence formulas as 𝒮) with respect to the class of 𝒮 ′ -algebras.. Proof. We know from [Spinks and Veroff, 2008b, Lemma 5.2] that 𝐹 𝐿𝑒𝑤 is algebraizable with respect to the class of commutative integral bounded residuated lattices, see [Spinks and Veroff, 2008b, Lemma 5.3]. 𝒮 ′ is an axiomatic extension of 𝐹 𝐿𝑒𝑤 , therefore, by [Font, 2016, Proposition 3.31], it is also algebraizable with the same defining equation and equivalence formulas. The corresponding class of algebras is a subvariety of 𝐶𝐼𝐵𝑅𝐿 that can be axiomatized by adding the equations corresponding to the new axioms. It is easy to check that these equations imply precisely that the equivalent semantics of 𝒮 ′ is the class of all involutive (S14) and three-potent (S15) 𝐶𝐼𝐵𝑅𝐿, i.e., the class of 𝒮 ′ -algebras..

(39) 3.2. Deduction theorem. 37. Although the logics 𝒮 and 𝒮 ′ have been initially defined over different propositional languages (namely ⟨∧, ∨, ⇒, ¬, ⊥⟩ for 𝒮 and ⟨∧, ∨, ⇒, *, ⊥, ⊤⟩ for 𝒮 ′ ), we can obviously expand the language of 𝒮 to include the connectives ⊤ and * defined by ⊤ := ¬⊥ and 𝜑 * 𝜓 := ¬(𝜑 ⇒ ¬𝜓). This allows us to state the following. Corollary 1. The calculi 𝒮 (in the above-defined extended language) and 𝒮 ′ define the same logic. Proof. The result follows straightforwardly from the fact that 𝒮 and 𝒮 ′ are algebraizable ( with the same defining equation and equivalence formulas) with respect to the same class of algebras. To be more formal one can invoke the algorithm of [Font, 2016, Proposition 3.47] which allows one to obtain an axiomatization of an algebraizable logic from a presentation of the corresponding class 𝐾 that is the equivalent algebraic semantics; notice that the algorithm uses only the (quasi)equations that axiomatize 𝐾 and the defining equations and equivalence formulas witnessing algebraizability.. 3.2 Deduction theorem Working with Nelson’s original presentation of 𝒮, it can be hard to prove some version of Deduction Metatheorem. Indeed, if we prove it using induction over the structure of the derivation, we need to apply induction over each rule of system. The advantage of 𝒮 ′ , then, is that it has only one rule. Theorem 4. If Γ ∪ {𝜙} ⊢ 𝜓, then Γ ⊢ 𝜙 ⇒ (𝜙 ⇒ 𝜓). Proof. We are going to prove it applying the principle of induction on the structure of the proof of Γ ∪ {𝜙} ⊢ 𝜓. Base case In this case, our derivation of 𝜓 from Γ ∪ {𝜙} is a sequence with a single formula, therefore, this formula is 𝜓, so we have 3 cases: 𝜓 is an axiom: In this case, using the axiom (S3) from 𝒮 ′ , we have 𝜓 ⇒ (𝜙 ⇒ 𝜓) and as 𝜓 is axiom, by modus ponens we have 𝜙 ⇒ 𝜓. Now, using axiom (S3) again, we have (𝜙 ⇒ 𝜓) ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓)) and as we have 𝜙 ⇒ 𝜓 it follows that (𝜙 ⇒ (𝜙 ⇒ 𝜓)). Notice that we do not use any formula of Γ, therefore Γ ⊢ 𝜙 ⇒ (𝜙 ⇒ 𝜓)..

(40) 38. Chapter 3. Calculus. 𝜓 ∈ Γ: This case is like the case above. 𝜓 ∈ {𝜙} : In this case, 𝜓 = 𝜙 and we need to prove 𝜓 ⇒ (𝜓 ⇒ 𝜓) and this is the axiom (S3). Inductive step Supposing the result is valid for any derivation sequence of 𝑘 ≤ 𝑛 steps with 𝑛 > 1, we will prove that the result is valid for any derivation sequence with 𝑛 + 1 formulas. If 𝜓 ∈ Γ ∪ {𝜙} or 𝜓 is an axiom, the proof is as above in the base case. Suppose then that 𝜓 can be proved from previous formulas in the sequence through an aplication of modus ponens, and so this previous formulas are of type 𝜃 and 𝜃 ⇒ 𝜓, from the inductive hypothesis we have Γ ⊢ 𝜙 ⇒ (𝜙 ⇒ 𝜃) and Γ ⊢ 𝜙 ⇒ (𝜙 ⇒ (𝜃 ⇒ 𝜓)), thus we have the following proof of Γ ⊢ 𝜙 ⇒ (𝜙 ⇒ 𝜓): 𝑖 𝑖𝑖 .. . 𝑘 .. .. 𝜙 ⇒ (𝜙 ⇒ 𝜃). hypothesis. 𝑙. 𝜙 ⇒ (𝜙 ⇒ (𝜃 ⇒ 𝜓)). hypothesis. 1. (𝜙 ⇒ (𝜃 ⇒ 𝜓)) ⇒ (𝜃 ⇒ (𝜙 ⇒ 𝜓)). (S2). 2. ((𝜙 ⇒ (𝜃 ⇒ 𝜓)) ⇒ (𝜃 ⇒ (𝜙 ⇒ 𝜓))) ⇒ (𝜙 ⇒ (𝜙 ⇒ (𝜃 ⇒ 𝜓))) ⇒ (𝜙 ⇒ (𝜃 ⇒ (𝜙 ⇒ 𝜓))). (S1’). 3. (𝜙 ⇒ (𝜙 ⇒ (𝜃 ⇒ 𝜓))) ⇒ (𝜙 ⇒ (𝜃 ⇒ (𝜙 ⇒ 𝜓))). (MP)(1, 2). 4. 𝜙 ⇒ (𝜃 ⇒ (𝜙 ⇒ 𝜓)). (MP)(1, 3). 5. (𝜙 ⇒ (𝜃 ⇒ (𝜙 ⇒ 𝜓))) ⇒ (𝜃 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓))). (S2). 6. 𝜃 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓)). (MP)(4, 5). 7. ((𝜃 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓)))) ⇒ ((𝜙 ⇒ 𝜃) ⇒ (𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓))). (S1’). 8. ((𝜙 ⇒ 𝜃) ⇒ (𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓))). (MP)(6, 7). 9. ((𝜙 ⇒ 𝜃) ⇒ (𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓))) ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜃)) ⇒ (𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓)))). (S1’). 10. (𝜙 ⇒ (𝜙 ⇒ 𝜃)) ⇒ (𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓)))). (MP)(8, 9). 11. 𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓))). (MP)(𝑘, 10). 12. 𝜙 ⇒ (𝜙 ⇒ (𝜙 ⇒ 𝜓)). (S15). 13. 𝜙 ⇒ (𝜙 ⇒ 𝜓). (S15). Theorem 4 suggests that, defining 𝜙 → 𝜓 := 𝜙2 ⇒ 𝜓, one may obtain in 𝒮 a new implication-type connective → that enjoys the classical Deduction-Detachment Theorem. This is precisely what happens in Nelson’s logic 𝒩 3, where in fact → is usually taken as the primitive implication and ⇒ as a defined one (see Subsection 5.2). Whether a similar inter-definability may hold for 𝒮 as well is actually an interesting open question, which we shall discuss in chapter 7. For now what we can say is that the above-defined term → does indeed behave as an implication operator on 𝒮-algebras, at least in the sense that.

(41) 3.2. Deduction theorem. 39. it is an example of a weak relative pseudo complementation according to the terminology introduced by Blok, Köhler and Pigozzi [Blok et al., 1984]. This last paper is the second of a series devoted to varieties of algebras of nonclassical logics, focusing in particular those varieties that enjoy a property called equationally definable principal congruences or EDPC for short [Blok et al., 1984, p. 338]. It is well known that a logic that is algebraizable with respect to some variety of algebras enjoys a (generalized) Deduction-Detachment Theorem1 if and only its associated variety has EDPC [Font, 2016, Corollary 3.86]. This applies, in particular, to our logic 𝒮 and to 𝒮-algebras. In the context of varieties of non-classical logic having EDPC, the authors of [Blok et al., 1984] single out those that possess term-definable operations which can be viewed as generalizations of intuitionistic conjunction, implication and bi-implication. These operations are called, respectively, weak meet, weak relative pseudo complementation and Gödel equivalence. Algebras having them are called weak Brouwerian semilattices with filter preserving operations or WBSO for short [Blok et al., 1984, Definition 2.1]. Definition 15. Let A be an algebra of type F and Θ an equivalence relation. Then Θ is congruence on A if it satisfies the following property: For each 𝜆 ∈ F and 𝑎𝑖 , 𝑏𝑖 ∈ A. If 𝑎𝑖 Θ𝑏𝑖 then 𝜆(𝑎1 , · · · , 𝑎𝑛 )Θ𝜆(𝑏1 , · · · , 𝑏𝑛 ) In the next definiton 𝐶𝑝 A denote the lattice of congruences on A which are finitely generated. Definition 16. A binary operation ⊕ is called dual relative pseudo complementation if for every Θ ∈ 𝐶𝑝 A, it holds the condition: Φ ⊕ Ψ ⊆ Θ iff Ψ ⊆ Φ ∨ Θ. In the next definition, Θ(𝑎, 𝑏) denotes the principal congruence generated by the pair 𝑎, 𝑏. Definition 17. Let A an arbitrary algebra and 1 ∈ 𝐴. Each following term is defined with respect to 1 and it is called: 1. See e.g. [Font, 2016, Definition 3.76] for a precise definition of generalized Deduction-Detachment Theorem..

(42) 40. Chapter 3. Calculus. · Weak meet if for all 𝑎, 𝑏 ∈ 𝐴 Θ(𝑎 · 𝑏, 1) = Θ(𝑎, 1) ∨ Θ(𝑏, 1) → Weak relative pseudo complementation if ⟨𝐶𝑝 A, ∨, 𝐼⟩ is dually relatively pseudo complemented and Θ(𝑎 → 𝑏, 1) = Θ(𝑎, 1) ⊕ Θ(𝑏, 1) ∇ Gödel equivalence if for all 𝑎, 𝑏 ∈ 𝐴 Θ(𝑎, 𝑏) = Θ(𝑎∇𝑏, 1) In the next definition we shall adopt → instead 𝛼(𝑥, 𝑦) to not overload notation, but keeping in mind that → is just a notation, → can be any binary term having the properties in definition below. The same case applies to the other binary terms. Definition 18. V is a variety of weak Brouwerian semilattices with filter preserving operations (WBSO) if there exist binary terms →, · and ∇ and a constant 1 such that the followings identities and quasi-identities hold in V.. 1. 𝑥 → 𝑥 = 1 2. 𝑥 → 1 = 1 3. (𝑥 → 𝑦) → ((𝑦 → 𝑧) → (𝑥 → 𝑧)) = 1 4. 1 · 1 = 1 5. (𝑧 → 𝑥) → ((𝑧 → 𝑦) → (𝑧 → (𝑥 · 𝑦))) = 1 6. (𝑥 · 𝑦) → 𝑥 = 1,. (𝑥 · 𝑦) → 𝑦 = 1. 7. 𝑥∇𝑥 = 1 8. (𝑥∇1) → 𝑥 = 1,. 𝑥 → (𝑥∇1) = 1. 9. (𝑥∇𝑦) → (𝑦∇𝑥) = 1 10. ((𝑥∇𝑦) · (𝑦∇𝑧)) → (𝑥∇𝑧) = 1.

(43) 3.2. Deduction theorem. 41. 11. For every fundamental operation 𝜆 of V, (· · · ((𝑥0 ∇𝑦0 ) · (𝑥1 ∇𝑦1 )) · · · ) · (𝑥𝑚−1 ∇𝑦𝑚−1 ) → 𝜆(𝑥0 , · · · , 𝑥𝑚−1 )∇𝜆(𝑦0 , · · · , 𝑦𝑚−1 ) 12. 1 → 𝑥 = 1 implies 𝑥 = 1 13. 𝑥∇𝑦 = 1 implies 𝑥 = 𝑦. In order to relate the latest two definition above, see that [Blok et al., 1984, Lemma 2.7] shows the following lemma: Lemma 2. V is a WBSO variety iff it has terms →, ·, ∇ such that → is a weak relative pseudo complement, · is a weak meet, and ∇ is a Gödel equivalence term, all with respect to 1. As observed in [Blok et al., 1984, p. 358], the algebraic counterpart of Nelson’s logic 𝒩 3 is a WBSO variety. The same is true for 𝒮-algebras. Theorem 5. 𝒮-algebras are a WBSO variety in which a weak meet is given by ∧ (or, equivalently, by *), weak relative pseudo complementation is given by the term 𝑥2 ⇒ 𝑦 and Gödel equivalence is 𝑥 ⇔ 𝑦. Proof. Items 1., 2. and 8. follow from Example 3 for being 𝛼2 ⇒ 𝛼 and 𝛼2 ⇒ 1 theorems. Items 4. and 6. follow from lattice properties. 3. We must prove that (𝑥2 ⇒ 𝑦)2 ⇒ ((𝑦 2 ⇒ 𝑧)2 ⇒ (𝑥2 ⇒ 𝑧) = 1, from deduction theorem, it equations holds iff 𝑥2 ⇒ 𝑦 ⊢𝒮 ((𝑦 2 ⇒ 𝑧)2 ⇒ (𝑥2 ⇒ 𝑧) iff {𝑥2 ⇒ 𝑦, 𝑦 2 ⇒ 𝑧} ⊢𝒮 𝑥2 ⇒ 𝑧 iff {𝑥2 ⇒ 𝑦, 𝑦 2 ⇒ 𝑧, 𝑥} ⊢𝒮 𝑧 and the last deduction follows from modus ponens. 5. In order to prove (𝑧 2 ⇒ 𝑥)2 ⇒ ((𝑧 2 ⇒ 𝑦)2 ⇒ (𝑧 2 ⇒ 𝑥 ∧ 𝑦)) = 1, we use deduction theorem, 𝑧 2 ⇒ 𝑥 ⊢𝒮 ((𝑧 2 ⇒ 𝑦)2 ⇒ (𝑧 2 ⇒ 𝑥 ∧ 𝑦)) iff {𝑧 2 ⇒ 𝑥, 𝑧 2 ⇒ 𝑦} ⊢𝒮 𝑧 2 ⇒ 𝑥 ∧ 𝑦 iff {𝑧 2 ⇒ 𝑥, 𝑧 2 ⇒ 𝑦, 𝑧} ⊢𝒮 𝑥 ∧ 𝑦. The last deduction follows from modus ponens. 7. We know that the partial order is determined by ⇒ and as 𝑥 ≤ 𝑥, follows the equation. 9. It follows from commutativity of ∧ and ⇔. 10. See that ((𝑥 ⇔ 𝑦) ∧ (𝑦 ⇔ 𝑧))2 ⇒ (𝑥 ⇔ 𝑧) = 1 iff (𝑥 ⇔ 𝑦) ∧ (𝑦 ⇔ 𝑧) ⊢𝒮 ′ (𝑥 ⇔ 𝑧). The last deduction follows from (S3). 11. See Theorem 1, this item is just the case IL3. 12. It follows from the fact that 1 is the maximum of the lattice. 13. Finishing, it follows from Proposition 1.2..

(44) 42. Chapter 3. Calculus. We have proved it for ∧, the proof for * is in a similar way, observe that 𝑥 * 𝑦 ⇒ 𝑥 ∧ 𝑦 = 1, therefore we can use it with the rule {𝑥 ⇒ 𝑦, 𝑦 ⇒ 𝑧} ⊢𝒮 𝑥 ⇒ 𝑧 for proving the theorem to *. Finishing the chapter, we remind that as happens with weak meet, weak relative pseudo complementationd and Gödel equivalence need not be unique. For example, one can prove that choosing 𝑥2 * 𝑦 as weak meet, (𝑥2 ⇒ (𝑥2 * 𝑦))2 as weak relative pseudo complementation and (𝑥 ⇔ 𝑦)2 as Gödel equivalence, they satisfy the Definition 18..

(45) 43. 4 More on 𝒮-algebras In this chapter we are going to show some examples of 𝒮-algebras that will allow us to prove that the formulas which Nelson claims without proof not to be derivable in 𝒮 [Nelson, 1959, p. 213] are actually not valid. To this end, an advantage of the presentation of Definition 13 is that it makes it easier to check that, for instance, the three-element MV-chain presented below (and indeed every algebra in the variety it generates), is a model of 𝒮. 4.1 ℒ3 algebra The three-element linearly ordered MV-algebra [Cignoli et al., 2000, Definition 1.1.1], that we shall call Ł3 (for Łukasiewicz three-valued logic), is defined as follows. The universe is {0, 21 , 1} in the language {∧, ∨, *, ⇒, ¬, 0, 1}, the operations are given by: ⇒. 0. 1 2. 1. *. 0. 1 2. 1. ¬. 0. 1. 1. 1. 0. 0. 0. 0. 0. 1. 1 2. 1 2. 1. 1. 1 2. 0. 0. 1 2. 1 2. 1 2. 1. 0. 1 2. 1. 1. 0. 1 2. 1. 1. 0. It is well-known that Ł3 is an involutive CIBRL (see, e.g., [Cignoli et al., 2000, Lemma 1.1.4 and Proposition 1.1.5]). It is also easy to check that Ł3 is three-potent, therefore it is an 𝒮-algebra. We can thus use it as a counter-model to show that the formulas below are not 𝒮-theorems. Proposition 4. The following formulas are not true in ℒ3 with designated set 𝒟 = {1}, consequently not true in 𝒮-algebras and thus can not be proved in the logic 𝒮. 1. 𝑝 ∨ ¬𝑝 2. ¬(𝑝 ∧ ¬𝑝) 3. (𝑝 ∧ ¬𝑝) ⇒ 𝑞 4. (𝑝 ⇒ (𝑝 ⇒ 𝑞)) ⇒ (𝑝 ⇒ 𝑞) 5. (𝑝 ⇒ (𝑞 ⇒ 𝑟)) ⇒ ((𝑝 ∧ 𝑞) ⇒ 𝑟).

(46) Chapter 4. More on 𝒮-algebras. 44. 6. (𝑝 ∧ ¬𝑞) ⇒ ¬(𝑝 ⇒ 𝑞) Proof. It is sufficient to find, for each formula 𝜑 above, a valuation 𝜐 : Fm −→ Ł3 such that 𝜐(𝜑) ̸= 1. The result will then follow by our completeness result (Theorem 3).. 1. Letting 𝜐(𝑝) = 12 , we have 𝜐(𝑝 ∨ ¬𝑝) = 𝜐(𝑝) ∨ ¬𝜐(𝑝) =. 1 2. ∨ ¬ 12 =. 1 2. ∨. 1 2. = 12 .. 2. Letting 𝜐(𝑝) = 12 , we have 𝜐(¬(𝑝 ∧ ¬𝑝)) = ¬(𝜐(𝑝) ∧ ¬𝜐(𝑝)) = ¬( 21 ∧ ¬ 12 ) = ¬( 12 ∧ 12 ) = ¬ 12 = 12 . 3. Let 𝜐(𝑝) =. 1 2. 4. Let 𝜐(𝑝) =. 1 2. and 𝜐(𝑞) = 0. Then 𝜐(𝑝 ⇒ 𝑞) = 𝜐(𝑝) ⇒ 𝜐(𝑞) = 12 ⇒ 0 = 21 . and 𝜐(𝑞) = 0. Then 𝜐((𝑝 ⇒ (𝑝 ⇒ 𝑞)) ⇒ (𝑝 ⇒ 𝑞)) = (𝜐(𝑝) ⇒ (𝜐(𝑝) ⇒. 𝜐(𝑞)) ⇒ (𝜐(𝑝) ⇒ 𝜐(𝑞)) = ( 21 ⇒ ( 12 ⇒ 0)) ⇒ ( 21 ⇒ 0) = ( 12 ⇒ 12 ) ⇒ 5. Let 𝜐(𝑝) = 𝜐(𝑞) =. 1 2. 1 2. = (1 ⇒ 12 ) = 12 .. and 𝜐(𝑟) = 0, then 𝜐((𝑝 ⇒ (𝑞 ⇒ 𝑟)) ⇒ ((𝑝 ∧ 𝑞) ⇒ 𝑟)) = ( 12 ⇒. ( 12 ⇒ 0)) ⇒ (( 12 ∧ 21 ) ⇒ 0) = ( 12 ⇒ ( 12 ⇒ 0)) ⇒ ( 12 ⇒ 0) = ( 12 ⇒ 12 ) ⇒. 1 2. =1⇒. 1 2. = 12 .. 6. Let 𝜐(𝑝) = 𝜐(𝑞) = 12 , then 𝜐((𝑝 ∧ ¬𝑞) ⇒ ¬(𝑝 ⇒ 𝑞)) = (𝜐(𝑝) ∧ ¬𝜐(𝑞)) ⇒ ¬(𝜐(𝑝) ⇒ 𝜐(𝑞)) = ( 21 ∧ ¬ 12 ) ⇒ ¬( 12 ⇒ 12 ) =. 1 2. ⇒ ¬1 =. 1 2. ⇒ 0 = 21 .. 4.2 Making 𝒮-algebras We. now. present. an. adaptation. of. the. construction. introduced. in. [Galatos and Raftery, 2004, Section 6] to embed a CIRL into one having an involutive negation. This will give us a simple way to construct 𝒮-algebras. Definition 19. Given a CIRL A = ⟨𝐴, ∧, ∨, *, ⇒, 1⟩, let 𝐴′ = {𝑎′ : 𝑎 ∈ 𝐴} be a disjoint copy of 𝐴, and let 𝐴* = 𝐴 ∪ 𝐴′ . We extend the lattice order of A to 𝐴* as follows. For all 𝑎, 𝑏 ∈ 𝐴: 1. 𝑎 ≤𝐴* 𝑏 iff 𝑎 ≤𝐴 𝑏. 2. 𝑎′ ≤𝐴* 𝑏..

(47) 4.2. Making 𝒮-algebras. 45. 3. 𝑎′ ≤𝐴* 𝑏′ iff 𝑏 ≤𝐴 𝑎. For each 𝑥 ∈ 𝐴 ∪ 𝐴′ , we define (𝑎′ )′ = 𝑎. The behavior of the lattice operations is fixed according to De Morgan’s laws: 𝑎′ ∧ 𝑏′ := (𝑎 ∨ 𝑏)′ and 𝑎′ ∨ 𝑏′ := (𝑎 ∧ 𝑏)′ . The operations * and ⇒ are extended to 𝐴* as follows para todo a b in A: 𝑎 * 𝑏′ := (𝑎 ⇒ 𝑏)′ 𝑎 ⇒ 𝑏′ := (𝑎 * 𝑏)′. 𝑎′ * 𝑏′ := 1′ .. 𝑎′ ⇒ 𝑏′ := 𝑏 ⇒ 𝑎 𝑎′ ⇒ 𝑏 := 1.. It is shown in [Galatos and Raftery, 2004, Section 6] that, if A is a CIRL, then A* is an involutive CIBRL into which A is embedded in the obvious way. Moreover, we have the following. Proposition 5. A* is an 𝒮-algebra if and only if A is a three-potent CIRL. Proof. One direction is immediate: if A* is an 𝒮-algebra, then it is three-potent, hence so is A as a {∧, ∨, *, ⇒, 1}-subalgebra of A* . Conversely, if A is a three-potent CIRL, since we already know that A* is CIBRL, it remains to show that 𝑎2 ≤ 𝑎3 for all 𝑎 ∈ A* . For 𝑎 ∈ 𝐴 the result follows from 3-potency of A. If 𝑎 ∈ 𝐴′ , then by Definition 19 we have 𝑎2 = 1′ = 𝑎3 . Corollary 2. If A is an implicative lattice or an 𝒮-algebra, then A* is an 𝒮-algebra. In fact, it is not difficult to check that if A is an implicative lattice, then A* is a special 𝒮-algebra known as an 𝒩 3-lattice (we shall deal with these structures in Section 5.2). Example 4. The above construction is useful to produce examples of 𝒮-algebras that witness the failure of further formulas that are not valid in 𝒮. For example, the algebra (Ł3 )* , where Ł3 is the above-introduced MV-algebra, witnesses the failure of ((𝑝 ⇒ 𝑞) ⇒ 𝑞) ⇒ ((𝑞 ⇒ 𝑝) ⇒ 𝑝). (4.1). which is however valid in three-valued Łukasiewicz logic. The behaviour of implication in (Ł3 )* is shown in the table below. To show that the above-mentioned formula is not valid, let 𝜐 : Fm −→ (Ł3 )* be a valuation such that 𝜐(𝑝) = 0 and 𝜐(𝑞) = 0′ . We have 𝜐(((𝑝 ⇒ 𝑞) ⇒ 𝑞) ⇒ ((𝑞 ⇒ 𝑝) ⇒.

(48) Chapter 4. More on 𝒮-algebras. 46. ⇒ 1′. 1′ 1. 1′ 2 ′. 1 2. 0 0. 0 0′. 1 2. 1′ 2 ′. 1. 1. 1′ 2. 0′ 1 1 1 1 1 1 2 0′ 0′ 0′ 0′ 1′ 0′ 2. 0 1 1 1 1 1 2. 0. 1 2. 1 1 1 1 1 1 2. 1 1 1 1 1 1 1. 𝑝))) = (𝜐(𝑝) ⇒ 𝜐(𝑞)) ⇒ 𝜐(𝑞)) ⇒ ((𝜐(𝑞) ⇒ 𝜐(𝑝)) ⇒ 𝜐(𝑝)) = ((0 ⇒ 0′ ) ⇒ 0′ ) ⇒ ((0′ ⇒ 0) ⇒ 0) = (0′ ⇒ 0′ ) ⇒ (1 ⇒ 0) = 1 ⇒ 0 = 0. Together with our previous considerations about Ł3 , the preceding example entails that Łukasiewicz three-valued logic is a proper extension of 𝒮: in fact it is easy to check that it is precisely the axiomatic extension of 𝒮 obtained by adding the axiom schema corresponding to (4.1) above. On the other hand, no other logic in the Łukasiewicz family is comparable with 𝒮, because they all lack 3-potency, whereas 𝒮 does not satisfy (4.1) which is valid in all of them..

(49) 47. 5 Relation with 𝒩 3 and 𝒩 4 As mentioned earlier, David Nelson is remembered for having introduced, besides 𝒮, two better-known logics: 𝒩 3, which is usually called just Nelson logic [Nelson, 1949], and 𝒩 4 which is known as paraconsistent Nelson logic [Almukdad and Nelson, 1984]. Both logics are algebraizable with respect to classes of residuated structures (called, respectively, 𝒩 3-lattices, or Nelson algebras, and 𝒩 4-lattices). The question then arises of what is precisely the relation between 𝒮 and these other logics, or equivalently between 𝒮-algebras and 𝒩 3 and 𝒩 4-lattices). Can we meaningfully say that one is stronger than the other? By looking at their algebraic models, it will not be difficult to show that 𝒩 3 (which is known to be an axiomatic extension of 𝒩 4) can also be viewed as an axiomatic extension of 𝒮, while 𝒩 4 and 𝒮 do not seem to be comparable in any meaningful way.. 5.1 𝒩 4 Definition 20. 𝒩 4 = ⟨Fm, ⊢𝒩 4 ⟩ is the sentential logic in the language ⟨∧, ∨, →, ¬⟩ of type ⟨2, 2, 2, 1⟩ defined by the Hilbert-style calculus with the following axiom schemata and modus ponens (from 𝜑 and 𝜑 → 𝜓, infer 𝜓) as the only rule schema. (N1). 𝜑 → (𝜓 → 𝜑). (N2). (𝜑 → (𝜓 → 𝛾)) → ((𝜑 → 𝜓) → (𝜑 → 𝛾)). (N3). (𝜑 ∧ 𝜓) → 𝜑. (N4). (𝜑 ∧ 𝜓) → 𝜓. (N5). (𝜑 → 𝜓) → ((𝜑 → 𝛾) → (𝜑 → (𝜓 ∧ 𝛾))). (N6). 𝜑 → (𝜑 ∨ 𝜓). (N7). 𝜓 → (𝜑 ∨ 𝜓). (N8). (𝜑 → 𝛾) → ((𝜓 → 𝛾) → ((𝜑 ∨ 𝜓) → 𝛾)). (N9). ¬¬𝜑 ↔ 𝜑.

(50) Chapter 5. Relation with 𝒩 3 and 𝒩 4. 48. (N10). ¬(𝜑 ∨ 𝜓) ↔ (¬𝜑 ∧ ¬𝜓). (N11). ¬(𝜑 ∧ 𝜓) ↔ (¬𝜑 ∨ ¬𝜓). (N12). ¬(𝜑 → 𝜓) ↔ (𝜑 ∧ ¬𝜓). Here 𝜑 ↔ 𝜓 abbreviates (𝜑 → 𝜓) ∧ (𝜓 → 𝜑). The implication → in 𝒩 4 is usually called weak implication, in contrast to the strong implication ⇒ that is defined by the following term: 𝜑 ⇒ 𝜓 := (𝜑 → 𝜓) ∧ (¬𝜓 → ¬𝜑). Example 5. We can deduce from ⊢𝒩 4 the following formula 𝜑 → 𝜑. Example 6. We can deduce from ⊢𝒩 4 the following formula (𝜓 → 𝜓) → (𝜑 → 𝜑). As the notation suggests, it is the strong implication and not the weak that we shall compare with the implication of 𝒮. This appears indeed to be the more meaningful choice, as we shall explain below. A prominent feature of the weak implication of 𝒩 4 is that on the one hand (unlike the implication of 𝒮) it enjoys the Deduction-Detachment Theorem in its classical form: Γ ∪ {𝜙} ⊢𝒩 4 𝜓 if and only if Γ ⊢𝒩 4 𝜙 → 𝜓. On the other hand contraposition fails (𝜙 → 𝜓 ̸⊢𝒩 4 ¬𝜓 → ¬𝜙) which entails that the above-defined bi-implication does not satisfy the following congruence property (again unlike 𝒮, as mentioned just before our Example 1): ⊢𝒩 4 𝜙 ↔ 𝜓 does not imply ⊢𝒩 4 ¬𝜙 ↔ ¬𝜓. By contrast, the strong implication of 𝒩 4 does not have the Deduction-Detachment Theorem but satisfies contraposition (like the implication of 𝒮), and the associated bi-implication (𝜑 ⇒ 𝜓) ∧ (𝜓 ⇒ 𝜑) does enjoy the congruence property. These considerations apply to the logic 𝒩 3 considered in the next subsection. It is well known [Rivieccio, 2011, Theorem 2.6] that 𝒩 4 is algebraizable (though not implicative) with defining equation E(𝜑) := {𝜑 ≈ 𝜑 → 𝜑} and equivalence formulas Δ(𝜑, 𝜓) := {𝜑 ⇒ 𝜓, 𝜓 ⇒ 𝜑}. The implication in E(𝜑) could as well be taken to be the strong one, so E(𝜑) := {𝜑 ≈ 𝜑 ⇒ 𝜑} would work too. By contrast, letting Δ(𝜑, 𝜓) := {𝜑 → 𝜓, 𝜓 → 𝜑} or the equivalent Δ(𝜑, 𝜓) := {𝜑 ↔ 𝜓} would not work precisely because of the failure of the above-mentioned congruence property..

(51) 5.1. 𝒩 4. 49. The equivalent algebraic semantics of 𝒩 4 is the class of 𝒩 4-lattices defined below [Odintsov, 2008, Definition 8.4.1]. Definition 21. An algebra A = ⟨𝐴, ∨, ∧, →, ¬⟩ is an 𝒩 4-lattice if it satisfies the following properties: 1. ⟨𝐴, ∨, ∧, ¬⟩ is a De Morgan lattice. 2. The relation ⪯ defined, for all 𝑎, 𝑏 ∈ 𝐴, by 𝑎 ⪯ 𝑏 iff (𝑎 → 𝑏) → (𝑎 → 𝑏) = (𝑎 → 𝑏) is a pre-order on A. 3. The relation ≡ defined, for all 𝑎, 𝑏 ∈ 𝐴 as 𝑎 ≡ 𝑏 iff 𝑎 ⪯ 𝑏 and 𝑏 ⪯ 𝑎 is a congruence relation with respect to ∧, ∨, → and the quotient-algebra A◁▷ := ⟨𝐴, ∨, ∧, →⟩/≡ is an implicative lattice. 4. For any 𝑎, 𝑏 ∈ 𝐴, ¬(𝑎 → 𝑏) ≡ 𝑎 ∧ ¬𝑏. 5. For any 𝑎, 𝑏 ∈ 𝐴, 𝑎 ≤ 𝑏 iff 𝑎 ⪯ 𝑏 and ¬𝑏 ⪯ ¬𝑎, where ≤ is a lattice order of A. Example 7. A most simple example of an 𝒩 4-lattice is the four-element one A4 whose lattice reduct is the four-element diamond De Morgan algebra.. 1 𝑛. 𝑏 0. The table of A4 is given by:. →. 0. n. b. 1. ¬. 0. 1. 1. 1. 1. 0. 1. n. 1. 1. 1. 1. n. n. b. 0. n. b. 1. b. b. 1. 0. n. b. 1. 1. 1. One can check that A4 satisfies all properties of Definition 21 (the quotient A4 /≡ mentioned in Definition 21.3 being the two-element Boolean algebra)..