We present two discrete dualities for double Stone algebras. Each of these dualities involves a different class of frames and a different definition of a complex algebra. We discuss relationships between these classes of frames and show that one of them is a weakening of the other. We propose a logic based on double Stone algebras.

Brain damage may doubly dissociate cognitive modules, but the practice of revealing dissociations is predicated on modularity being true (T. Shallice, 1988). This article questions the utility of assuming modularity, as it examines a paradigmatic double dissociation of reading modules. Reading modules illustrate two general problems. First, modularity fails to converge on a fixed set of exclusionary criteria that define pure cases. As a consequence, competing modular theories force perennial quests for purer cases, which simply perpetuates growth in the (...) list of exclusionary criteria. The first problem leads, in part, to the second problem. Modularity fails to converge on a fixed set of pure cases. The second failure perpetuates unending fractionation into more modules. (shrink)

In the Laws of Thought, Boole establishes a theory of secondary propositions based upon the notion of time. This temporal interpretation of secondary propositions has historically been met with wide disapproval and is usually dismissed in the modern literature as a philosophical non-starter. What was Boole thinking? This paper attempts to give an answer to this question. Specifically, it provides an account according to which Boole’s temporal interpretation follows from his psychologistic conception of logic, in addition to certain background assumptions (...) regarding the psychological necessity of the concept of time. Once Boole’s psychological premises are laid bare, it becomes clearer how he might have viewed the temporal interpretation to be an essential feature of his theory of secondary propositions. (shrink)

Kai Draper’s War and Individual Rights: The Foundations of Just War Theory seeks to “give birth to an alternative approach” to traditional just war theory. This review seeks to analyse and evaluate this alternative approach. Draper’s approach to just war theory differs from other approaches in three ways. First, it is “highly individualistic.” Second, Draper’s approach avoids reliance upon the principle of double effect. Third, this approach is “largely rights-based”—it seeks “to understand the ethics of war mostly by way (...) of understanding certain fundamental moral rights”. This review will argue that Draper succeeds in offering an alternative to traditional just war theory that is both rights-based and capable of dispensing with the needless complexities of the principle of double effect. However this review will argue that Draper falls short of his ultimate goal of refuting Rousseau’s claim that war is “a relationship between State and State, in which individuals are enemies only by accident”. (shrink)

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [ 48 ] and [ 50 ] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for (...) a unary expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH . We also use it to prove that the variety DQSSH of dually quasi-Stone semi- Heyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH . Finally, we apply these results to give bases for “small” subvarieties of BDQDSH , DQSSH , and DblSH. (shrink)

The purpose of this paper is to define and investigate the new class of quasi-Stone algebras . Among other things we characterize the class of simple QSA's and the class of subdirectly irreducible QSA's. It follows from this characterization that the subdirectly irreducible QSA's form an elementary class and that the variety of QSA's is locally finite. Furthermore we prove that the lattice of subvarieties of QSA's is an -chain. MSC: 03G25, 06D16, 06E15.

The variety O2 of double Ockham algebras consists of the algebras (A ∨, ∧, f,g 0,1) of type (2,2,1,1,0,0) where (A; ∨, ∧,f, 0,1) and (A; ∨, ∧,g 0,1) are Ockham algebras. In [16], M. Sequeira introduced several subvarieties of O2. In this paper we give a construction of free double Ockham algebras on a partially ordered set. We also describe free objects for the subvarieties of O2 considered in [16].

In this paper we shall introduce the variety WQS of weak-quasi-Stone algebras as a generalization of the variety QS of quasi-Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬-lattices to give a duality for WQS. We prove that a weak-quasi-Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also (...) determine the simple and subdirectly irreducible algebras. (shrink)

In conventional generalization of the main results of classical measure theory to Stone algebra valued measures, the values that measures and functions can take are Booleanized, while the classical notion of a σ-field is retained. The main purpose of this paper is to show by abundace of illustrations that if we agree to Booleanize the notion of a σ-field as well, then all the glorious legacy of classical measure theory is preserved completely.

In this paper we describe the Priestley space of a quasi-Stone algebra and use it to show that the class of finite quasi-Stone algebras has the amalgamation property. We also describe the Priestley space of the free quasi-Stone algebra over a finite set.

In this paper we shall introduce the variety WQS of weak-quasi-Stone algebras as a generalization of the variety QS of quasi-Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬-lattices to give a duality for WQS. We prove that a weak-quasi-Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also (...) determine the simple and subdirectly irreducible algebras. (shrink)

In this note we give equational bases for varieties of quasi-Stone algebras.

The notion of a context in formal concept analysis and that of an approximation space in rough set theory are unified in this study to define a Kripke context. For any context (G,M,I), a relation on the set G of objects and a relation on the set M of properties are included, giving a structure of the form ((G,R), (M,S), I). A Kripke context gives rise to complex algebras based on the collections of protoconcepts and semiconcepts of the underlying (...) context. On abstraction, double Boolean algebras (dBas) with operators and topological dBas are defined. Representation results for these algebras are established in terms of the complex algebras of an appropriate Kripke context. As a natural next step, logics corresponding to classes of these algebras are formulated. A sequent calculus is proposed for contextual dBas, modal extensions of which give logics for contextual dBas with operators and topological contextual dBas. The representation theorems for the algebras result in a protoconcept-based semantics for these logics. (shrink)

This paper is a study of duality in the absence of canonicity. Specifically it concerns double quasioperator algebras, a class of distributive lattice expansions in which, coordinatewise, each operation either preserves both join and meet or reverses them. A variety of DQAs need not be canonical, but as has been shown in a companion paper, it is canonical in a generalized sense and an algebraic correspondence theorem is available. For very many varieties, canonicity (as traditionally defined) and correspondence (...) lead on to topological dualities in which the topological and correspondence components are quite separate. It is shown that, for DQAs, generalized canonicity is sufficient to yield, in a uniform way, topological dualities in the same style as those for canonical varieties. However topology and correspondence are no longer separable in the same way. (shrink)

Involutive Stone algebras were introduced by R. Cignoli and M. Sagastume in connection to the theory of $n$-valued Łukasiewicz–Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras named Six. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds (...) of truth values, and hence not only preserving the value $1$. Among other things, we prove that Six is a many-valued logic that can be determined by a finite number of matrices. Besides, we show that Six is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of Six providing a Gentzen calculus for it, which is sound and complete with respect to the logic. (shrink)

An endomorphism on an algebra \ is said to be strong if it is compatible with every congruence on \; and \ is said to have the strong endomorphism kernel property if every congruence on \, other than the universal congruence, is the kernel of a strong endomorphism on \. Here we characterise the structure of those double MS-algebras that have this property by the way of Priestley duality.

We show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, A is a countable structure with finite signature, and d is a degree, we say that A has αth-jump degree d if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of A with universe ω (...) in which the functions and relations have degree at most c. We show that every degree d ≥ 0 (ω) is the ωth jump degree of a Boolean algebra, but that for $n no Boolean algebra has nth-jump degree $\mathbf{d} > 0^{(n)}$ . The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties. (shrink)

We investigate the class of those algebras in which is a de Morgan algebra, is a quasi-Stone algebra, and the operations \ and \ are linked by the identity x**º = x*º*. We show that such an algebra is subdirectly irreducible if and only if its congruence lattice is either a 2-element chain or a 3-element chain. In particular, there are precisely eight non-isomorphic subdirectly irreducible Stone de Morgan algebras.

Review of the paper mentioned in the title.

In this paper, we introduce a variety bdO of Ockham algebras with balanced double pseudocomplementation, consisting of those algebras of type where is an Ockham algebra, is a double p -algebra, and the operations and are linked by the identities [ f ( x )]* = [ f ( x )] + = f 2 ( x ), f ( x *) = x ** and f ( x + ) = x ++ . We give (...) a description of the congruences on the algebras, and show that there are precisely nine non-isomorphic subdirectly irreducible members in the class of the algebras via the Priestley duality. We also describe all axioms in the variety bdO , and provide a characterization of all subvarieties of bdO determined by 12 none-equivalent axioms, identifying therein the biggest subvariety in which every principal congruence is complemented. (shrink)

This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these asvariant semanticsand present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we (...) demonstrate that failure via specific finite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstra’s criteria in the corresponding substructural logical setting. (shrink)

In A. Kumar, & M. Banerjee [(2012). Definable and rough sets in covering-based approximation spaces. In T. Li. (eds.), Rough sets and knowledge technology (pp. 488–495). Springer-Verlag], A. Kumar, & M. Banerjee [(2015). Algebras of definable and rough sets in quasi order-based approximation spaces. Fundamenta Informaticae, 141(1), 37–55], authors proposed a pair of lower and upper approximation operators based on granules generated by quasiorders. This work is an extension of algebraic results presented therein. A characterisation has been presented for (...) those quasiorder-generated covering-based approximation spaces whose corresponding collections of definable and rough sets form Stone algebras. The notion of rough lattice was proposed in A. Kumar, & M. Banerjee [(2015). Algebras of definable and rough sets in quasi order-based approximation spaces. Fundamenta Informaticae, 141(1), 37–55], A. Kumar [(2020). A Study of Algebras and Logics of Rough Sets Based on Classical and Generalized Approximation Spaces. In Transactions on Rough Sets XXII, LNCS (Vol. 12485, pp. 123–251). Springer]. Some special rough lattices are introduced in this work, viz. rough Stone algebra, ∼1-complemented and ∼2-complemented rough lattices. Representations of these algebras in terms of rough sets are obtained. Moreover, logics for these algebras are shown to be sound and complete with respect to rough set semantics. (shrink)

Let A be an algebra. We say that the functions f 1 , . . . , f m : A n → A are algebraic on A provided there is a finite system of term-equalities $${{\bigwedge t_{k}(\overline{x}, \overline{z}) = s_{k}(\overline{x}, \overline{z})}}$$ satisfying that for each $${{\overline{a} \in A^{n}}}$$, the m -tuple $${{(f_{1}(\overline{a}), \ldots , f_{m}(\overline{a}))}}$$ is the unique solution in A m to the system $${{\bigwedge t_{k}(\overline{a}, \overline{z}) = s_{k}(\overline{a}, \overline{z})}}$$. In this work we present a collection of general (...) tools for the study of algebraic functions, and apply them to obtain characterizations for algebraic functions on distributive lattices, Stone algebras, finite abelian groups and vector spaces, among other well known algebraic structures. (shrink)

Algebraic proofs of the cut-elimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the double-negation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ)nf is provable in minimal logic, where θnf denotes the negation-normal form of θ. The translation is used to show that cut-elimination theorems for classical logic can be viewed as (...) special cases of the cut-elimination theorems for intuitionistic logic. (shrink)

Let ${\mathbb{BRL}}$ denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety ${\mathbb{V}}$ of ${\mathbb{BRL}}$ is a unary term t in the language of bounded residuated lattices such that for every ${{\bf A} \in \mathbb{V}, t^{A}}$ , the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense that there is (...) a variety ${\mathbb{V}_{t} \subsetneq \mathbb{BRL}}$ such that a variety ${\mathbb{V} \subsetneq \mathbb{BRL}}$ admits the unary term t as a Boolean retraction term if and only if ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ . Moreover, the equation s(x) = t(x) holds in ${\mathbb{V}_{s} \cap \mathbb{V}_{t}}$ . The radical of ${{\bf A} \in \mathbb{BRL}}$ , with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ is associated a variety ${\mathbb{V}^{r}}$ formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in ${\mathbb{V}}$ . Each free algebra in such ${\mathbb{V}}$ is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety ${\mathbb{V}^{r}}$ , also with the same number of free generators.A hierarchy of subvarieties of ${\mathbb{BRL}}$ admitting Boolean retraction terms is exhibited. (shrink)

We generalize the double negation construction of Boolean algebras in Heyting algebras to a double negation construction of the same in Visser algebras. This result allows us to generalize Glivenko’s theorem from intuitionistic propositional logic and Heyting algebras to Visser’s basic propositional logic and Visser algebras.

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in (...) all Heyting algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be "completed" conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives. (shrink)

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space (...) form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras. (shrink)