In this paper, the Boolean representation of R0-algebras are investigated. In particular, we show that directly indecomposable R0-algebras are equivalent to local R0-algebras and any nontrivial R0-algebra is representable as a weak Boolean product of local R0-algebras.

A framework for the mind-matter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive non-Boolean description of a world without an a priori mind-matter distinction. Such a description in terms of a locally Boolean but globally non-Boolean structure makes allowance for the fact that Boolean descriptions play a privileged role in science. If we accept (...) the insight that there are no ultimate building blocks, the existence of holistic correlations between contextually chosen parts is a natural consequence. The main problem of a genuinely non-Boolean description is to find an appropriate partition of the universe of discourse. If we adopt the idea that all fundamental laws of physics are invariant under time translations, then we can consider a partition of the world into a tenseless and a tensed domain.In the sense of a regulative principle, the material domain is defined as the tenseless domain with its homogeneous time. The tensed domain contains the mental domain with a tensed time characterized by a privileged position, the Now. Since this partition refers to two complementary descriptions which are not given a priori we have to expect correlations between these two domains. In physics it corresponds to Newton 's separation of universal laws of nature and contingent initial conditions Both descriptions have a non-Boolean structure and can be encompassed into a single non-Boolean description. Tensed and tenseless time can be synchronized by holistic correlations. (shrink)

We establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, complete f-rings are "boundedly algebraically compact" in the language $(+,-,\cdot,\wedge,\vee,\leq)$ , and the positive cone of a complete l-group with infinity adjoined is algebraically compact in the language (+, ∨, ≤). We also give an example with (...) any first-order language. The proofs can be translated into "naive set theory" in a uniform way. (shrink)

A regularity theory of causation analyses type-level causation in terms of Boolean difference-making. The essential ingredient that helps this theoretical framework overcome the problems of Hume’s and Mill’s classical accounts is a principle of non-redundancy: only Boolean dependency structures from which no elements can be eliminated track causation. The first part of this article argues that the recent regularity-theoretic literature has not consistently implemented this principle, for it disregarded an important type of redundancies: structural redundancies. Moreover, it is (...) shown that a regularity theory needs to be underwritten by a hitherto neglected metaphysical background assumption stipulating that the world’s causal makeup is not ambiguous. Against that background, the second part then develops a new regularity theory that does justice to all types of redundancies and, thereby, provides the first all-inclusive notion of Boolean difference-making. (shrink)

A regularity theory of causation analyses type-level causation in terms of Boolean difference-making. The essential ingredient that helps this theoretical framework overcome the problems of Hume’s and Mill’s classical accounts is a principle of non-redundancy: only Boolean dependency structures from which no elements can be eliminated track causation. The first part of this paper argues that the recent regularity theoretic literature has not consistently implemented this principle, for it disregarded an important type of redundancies: structural redundancies. Moreover, it (...) is shown that a regularity theory needs to be underwritten by a hitherto neglected metaphysical background assumption stipulating that the world's causal makeup is not ambiguous. Against that background, the second part then develops a new regularity theory that does justice to all types of redundancies and, thereby, provides the first all-inclusive notion of Boolean difference-making. (shrink)

We consider the theory Thprin of Boolean algebras with a principal ideal, the theory Thmax of Boolean algebras with a maximal ideal, the theory Thac of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Thsa of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Thprin and Thsa. If T is a theory in a first order language and (...) α is a linear order with least element, then we let Sentalg be the Lindenbaum-Tarski algebra with respect to T, and we let intalg be the interval algebra of α. Using rank diagrams, we show that Sentalg ⋍ intalg, Sentalg ⋍ intalg ⋍ Sentalg, and Sentalg ⋍ intalg. For Thmax and Thac we use Ershov's elementary invariants of these theories. We also show that the algebra of formulas of the theory Tx of Boolean algebras with finitely many ideals is atomic. (shrink)

We continue the investigation, initiated in Salibra et al. (Found Sci, 2020), of Boolean-like algebras of dimension n (nBA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\textrm{BA}$$\end{document}s), algebras having n constants e1,⋯,en\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf e_1,\dots,\mathsf e_n$$\end{document}, and an (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document}-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of nBA\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\textrm{BA}$$\end{document}s share many remarkable properties with the variety of Boolean algebras and with primal varieties. Putting to good use the concept of a central element, we extend the Boolean power construction to that of a semiring power and we prove two representation theorems: (i) Any pure nBA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\textrm{BA}$$\end{document} is isomorphic to the algebra of n-central elements of a Boolean vector space; (ii) Any member of a variety of nBAs with one generator is isomorphic to a Boolean power of this generator. This yields a new proof of Foster’s theorem on primal varieties. (shrink)

In this article, I consider the positive logic of Boolean groups (i.e. Abelian groups where every non-identity element has order 2), where these are taken as frames for an operational semantics à la Urquhart. I call this logic BG. It is shown that the logic over the smallest nontrivial Boolean group, taken as a frame, is identical to the positive fragment of a quasi-relevance logic that was developed by Robles and Méndez (an extension of this result where (...) negation is included is also discussed). It is proved that BG satisfies the variable sharing property and is Halldén complete, while failing to satisfy the disjunction property. Sound and complete subscripted tableaux are presented for BG. An axiomatization (in the positive language extended with fusion) is presented, which is sound with respect to BG and, with respect to a related ternary relational semantics, complete; the problem of identifying a complete axiomatization for BG itself is left open. Connections between BG and the quasi-relevance logic KR are also discussed. (shrink)

Given an infinite Boolean algebra B, we find a natural class of equation image-definable equivalence relations equation image such that every imaginary element from Beq is interdefinable with an element from a sort determined by some equivalence relation from equation image. It follows that B together with the family of sorts determined by equation image admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author's earlier results concerning definable equivalence relations and weak elimination of (...) imaginaries for Boolean algebras, obtained in 10. (shrink)

For a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B1 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then $\langle M_T, \leq\rangle$ is well-quasi-ordered. ■ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s (...) such that $B\upharpoonright a$ is an atomic Boolean algebra and $B\upharpoonright s$ is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that $\langle A, \leq\rangle$ is a partial well-quasi-ordering, if it is a partial quasi-ordering and for every $\{a_i\mid i \in \omega\} \subseteq A$ , there are $i < j < \omega$ such that ai ≤ aj. Theorem 2. $\langle M_{T_\omega}, \leq\rangle$ contains a subset M such that the partial orderings $\langle M, \leq \upharpoonright M \rangle$ and $\langle\mathscr{P}(\omega), \subseteq\rangle$ are isomorphic. ■ Let M'0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M'0, let B1 ≤' B2 mean that B1 is embeddable in B2. Remark. $\langle M'_0, \leq'\rangle$ is well-quasi-ordered. ■ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering. (shrink)

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FLw-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras (...) are represented as Esakia products of simple n-potent MV-algebras. (shrink)

The Boolean many-valued approach to vagueness is similar to the infinite-valued approach embraced by fuzzy logic in the respect in which both approaches seek to solve the problems of vagueness by assigning to the relevant sentences many values between falsity and truth, but while the fuzzy-logic approach postulates linearly-ordered values between 0 and 1, the Boolean approach assigns to sentences values in a many-element complete Boolean algebra. On the modal-precisificational approach represented by Kit Fine, if a (...) sentence is indeterminate in truth value in some world, it is taken to be true in one precisified world accessible from that world and false in another. This paper points to a way to unify these two approaches to vagueness by showing that Fine’s version of the modal-precisificational approach can be combined with the Boolean many-valued approach instead of supervaluationism, one of the most popular approaches to vagueness. (shrink)

We prove weak elimination of imaginary elements for Boolean orderings with finitely many atoms. As a consequence we obtain equivalence of the two notions of o-minimality for Boolean ordered structures, introduced by C. Toffalori. We investigate atoms in Boolean algebras induced by algebraically closed subsets of Boolean ordered structures. We prove uniqueness of prime models in strongly o-minimal theories of Boolean ordered structures.

If κω is a cardinal, a complete Boolean algebra is called κ-dependent if for each sequence bβ: β<κ of elements of there exists a partition of the unity, P, such that each pP extends bβ or bβ′, for κ-many βκ. The connection of this property with cardinal functions, distributivity laws, forcing and collapsing of cardinals is considered.

The game is played on a complete Boolean algebra in ω-many moves. At the beginning White chooses a non-zero element p of and, in the nth move, White chooses a positive pn

Boolean algebra (...) carries a strictly positive Maharam submeasure or contains a countable dense subset, then Black has a winning strategy in the game played on . A Suslin algebra on which the game is undetermined is constructed and the game is compared with the well-known cut-and-choose games , and introduced by Jech. (shrink)

Heidorf, L., Moderate families in Boolean algebras, Annals of Pure and Applied Logic 57 217–250. A subset F of a Boolean algebra B will be called moderate if no element of B splits infinitely many elements of F . Disjoint moderate sets occur in connection with a product construction that is systematically studied in this paper. In contrast to the usual full direct product, these so-called moderate products preserve many properties of their factors. This can be used, (...) for example, to construct big retractive Boolean algebras that are not embeddable into interval algebras. The paper is also concerned with those Boolean algebras whose ideals are generated by moderate sets. The results may be summarized by saying that in many respects these algebras behave like countable ones. (shrink)

A structural theorem for Kleene algebras is proved, showing that an element of a Kleene algebra can be looked upon as an ordered pair of sets, and that negation with the Kleene property is describable by the set-theoretic complement. The propositional logic \ of Kleene algebras is shown to be sound and complete with respect to a 3-valued and a rough set semantics. It is also established that Kleene negation can be considered as a modal operator, due to a (...) perp semantics of \. Moreover, another representation of Kleene algebras is obtained in the class of complex algebras of compatibility frames. One concludes with the observation that \ can be imparted semantics from different perspectives. (shrink)

The main notion dealt with in this article is where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion.Some of the main results of the paper are:• (1) If there is a maximal (...) family of size λ≥κ of pairwise almost disjoint subsets of κ each of size κ, then there is a maximal family of size λ of pairwise almost disjoint subsets of κ+ each of size κ.• (2) A characterization of the class of all cardinalities of partitions of 1 in a product in terms of such classes for the factors; and a similar characterization for weak products.• (3) A cardinal number characterization of sets of cardinals with a largest element which are for some BA the set of all cardinalities of partitions of 1 of that BA.• (4) A computation of the set of cardinalities of partitions of 1 in a free product of finite-cofinite algebras. (shrink)

Logics designed to deal with vague statements typically allow algebraic semantics such that propositions are interpreted by elements of residuated lattices. The structure of these algebras is in general still unknown, and in the cases that a detailed description is available, to understand its significance for logics can be difficult. So the question seems interesting under which circumstances residuated lattices arise from simpler algebras in some natural way. A possible construction is described in this paper.Namely, we consider pairs consisting of (...) a Brouwerian algebra and an equivalence relation. The latter is assumed to be in a certain sense compatible with the partial order, with the formation of differences, and with the formation of suprema of pseudoorthogonal elements; we then call it an s-equivalence relation. We consider operations which, under a suitable additional assumption, naturally arise on the quotient set. The result is that the quotient set bears the structure of a residuated lattice. Further postulates lead to dual BL-algebras. In the case that we begin with Boolean algebras instead, we arrive at dual MV-algebras. (shrink)

We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom implies finiteness of Boolean algebras with compact top, whereas the converse fails (...) in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices. (shrink)

The account of causal regularities in the influential INUS theory of causation has been refined in the recent developments of the regularity approach to causation and of the Boolean methods for inference of deterministic causal structures. A key element in the refinement is to strengthen the minimality or non-redundancy condition in the original INUS account. In this paper, we argue that the Boolean framework warrants a further strengthening of the minimality condition. We motivate our stronger condition by (...) showing, first, that a rationale for strengthening the original minimality condition in the INUS theory is also applicable to our proposal to go further, and second, that the new element of the stronger condition is a counterpart to a well-established minimality condition for probabilistic causal models. We also compare the various minimality conditions in terms of the difference-making criteria they imply and argue for the criterion implied by our condition. Finally, we show that putative counterexamples to our proposal can be addressed in the same way that the Boolean theorists defend the current minimality conditions in their framework. (shrink)

In Plato's Parmenides we find on the one hand that the One is denied every property , and on the other hand that the One is attributed every property . In the course of the history of Platonism , these assertions - probably meant by Plato as ontological statements of an entirely formal nature - were repeatedly made the starting points of metaphysical speculations. In the Mystical Theology of the Pseudo-Dionysius they became principles of Christian mysticism and negative theology. I (...) shall show that the two assertions can each be interpreted within the ontological framework of ancient and medieval logic in such a manner that it becomes true, and I shall make plausible that they were understood, with regard to their logical core, by pagan and Christian Platonic metaphysicians just as is indicated by that interpretation. The mentioned ontological framework is basically the Boolean algebra of first-order properties. The main points of the interpretation are on the one hand the identification of the One with the maximal element of the algebra of properties, and on the other hand two alternative intuitively prominent mereological definitions of ontic predication. (shrink)

Belnap and Dunn’s well-known 4-valued logic FDE is an interesting and useful non-classical logic. FDE is defined by using conjunction, disjunction and negation as the sole propositional connectives. Then the question of expanding FDE with an implication connective is of course of great interest. In this sense, some implicative expansions of FDE have been proposed in the literature, among which Brady’s logic BN4 seems to be the preferred option of relevant logicians. The aim of this paper is to define a (...) class of implicative expansions of FDE in whose elements Boolean negation is definable, whence strong logics such as the paraconsistent and paracomplete logic PŁ4 and BN4 itself are definable, in addition to classical propositional logic. (shrink)

A sequence x=xn:nω of elements of a complete Boolean algebra converges to a priori if lim infx=lim supx=b. The sequential topology τs on is the maximal topology on such that x→b implies x→τsb, where →τs denotes the convergence in the space — the a posteriori convergence. These two forms of convergence, as well as the properties of the sequential topology related to forcing, are investigated. So, the a posteriori convergence is described in terms of killing of tall ideals on (...) ω, and it is shown that the a posteriori convergence is equivalent to the a priori convergence iff forcing by does not produce new reals. A property of c.B.a.’s, satisfying -cc -cc and providing an explicit definition of the a posteriori convergence, is isolated. Finally, it is shown that, for an arbitrary c.B.a. , the space is sequentially compact iff the algebra has the property and does not produce independent reals by forcing, and that implies P is the unique sequentially compact c.B.a. in the class of Suslin forcing notions. (shrink)

Let B be a superatomic Boolean algebra (BA). The rank of B (rk(B)), is defined to be the Cantor Bendixon rank of the Stone space of B. If a ∈ B - {0}, then the rank of a in B (rk(a)), is defined to be the rank of the Boolean algebra $B b \upharpoonright a \overset{\mathrm{def}}{=} \{b \in B: b \leq a\}$ . The rank of 0 B is defined to be -1. An element a ∈ B (...) - {0} is a generalized atom $(a \in \widehat{At}(B))$ , if the last nonzero cardinal in the cardinal sequence of B $\upharpoonright$ a is 1. Let a,b $\in\widehat{At}$ (B). We denote a ∼ b, if rk(a) = rk(b) = rk(a · b). A subset H $\subseteq \widehat{At}$ (B) is a complete set of representatives (CSR) for B, if for every a $\in \widehat{At}$ (B) there is a unique h ∈ H such that h ∼ a. Any CSR for B generates B. We say that B is canonically well-generated (CWG), if it has a CSR H such that the sublattice of B generated by H is well-founded. We say that B is well-generated, if it has a well-founded sublattice L such that L generates B. THEOREM 1. Let B be a Boolean algebra with cardinal sequence $\langle\aleph_0: i . If B is CWG, then every subalgebra of B is CWG. A superatomic Boolean algebra B is essentially low (ESL), if it has a countable ideal I such that rk(B/I) ≤ 1. Theorem 1 follows from Theorem 2.9, which is the main result of this work. For an ESL BA B we define a set F B of partial functions from a certain countably infinite set to ω (Definition 2.8). Theorem 2.9 says that if B is an ESL Boolean algebra, then the following are equivalent. (1) Every subalgebra of B is CWG; and (2) F B is bounded. THEOREM 2. If an ESL Boolean algebra is not CWG, then it has a subalgebra which is not well-generated. (shrink)

The general fact of the impossibility of a bivalent, truth-functional semantics for the propositional structures determined by quantum mechanics should be more subtly demarcated according to whether the structures are taken to be orthomodular latticesP L or partial-Boolean algebrasP A; according to whether the semantic mappings are required to be truth-functional or truth-functional ; and according to whether two-or-higher dimensional Hilbert spaceP structures or three-or-higher dimensional Hilbert spaceP structures are being considered. If the quantumP structures are taken to be (...) orthomodular latticesP L, then bivalent mappings which preserve the operations and relations of aP L must be truth-functional . Then as suggested by von Neumann and Jauch-Piron and as proven in this paper, the mere presence of incompatible elements in aP L is sufficient to rule out any semantical or hidden-variable proposal which imposes this strong condition, for anytwo-or-higher dimensional Hilbert spaceP L structure. Thus from the orthomodular lattice perspective, the peculiarly non-classical feature of quantum mechanics and the peculiarly non-Boolean feature of the quantum propositional structures is the existence of incompatible magnitudes and propositions. However, the weaker truth-functionality condition can instead be imposed upon the semantic or hidden-variable mappings on theP L structures, although such mappings ignore the lattice meets and joins of incompatibles and preserve only the partial-Boolean algebra structural features of theP L structures. Or alternatively, the quantum propositional structures can be taken to be partial-Boolean algebrasP A, where bivalent mappings which preserve the operations and relations of aP A need only be truth-functional (c). In either case, the Gleason, Kochen-Specker proofs show that any semantical or hidden variable proposal which imposes this truth-functionality (c) condition is impossible for anythree-or-higher dimensional Hilbert spaceP A orP L structures. But such semantical or hidden-variable proposals are possible for any two dimensional Hilbert spaceP A orP L structures, in spite of the presence of incompatibles in these structures, in spite of the fact that Heisenberg's Uncertainty Principle applies to the incompatible elements in these structures, and in spite of the fact that these structures are non-Boolean in the Piron sense. (shrink)

This recent addition to the Studies in Logic series is a systematic treatise on the set-theoretic, or semantic, approach to mathematical logic and axiomatic method. The basic notions for the discussion are those of different kinds of languages, their realizations, and the models of a formula. The book begins with a preliminary "chapter 0," giving some general theorems about classes of functions defined by finite schemas. These results are directly applicable to the language of truth-functional propositional logic, and such application (...) is accomplished in detail in chapter 1, in which interpolation, finiteness and compactness theorems are proved for propositional calculus. Chapters 2 and 3 deal with the predicate calculus without and with identity, respectively. More advanced topics dealt with in subsequent chapters include elimination of quantifiers. Certain algebraic structures such as real closed fields and some Boolean rings are given in the form of axiomatic systems in which every formula is equivalent to a quantifier-free formula. This leads to completeness. Also covered are type theory and alternate methods for developing predicate logic, the theory of definability, and the theory of principal models and infinite formulas. Appendices deal with some additional matters of philosophic interest, including a detailed discussion of set-theoretic vs. combinatorial foundations of mathematics. The book lacks an index and a bibliography, but these are not serious defects. This is a highly rewarding work.—H. P. K. (shrink)

Abstract. Let REL(O*E) be the relation algebra of binary relations defined on the Boolean algebra O*E of regular open regions of the Euclidean plane E. The aim of this paper is to prove that the canonical contact relation C of O*E generates a subalgebra REL(O*E, C) of REL(O*E) that has infinitely many elements. More precisely, REL(O*,C) contains an infinite family {SPPn, n ≥ 1} of relations generated by the relation SPP (Separable Proper Part). This relation can be used to (...) define point-free concept of connectedness that for the regular open regions of E coincides with the standard topological notion of connectedness, i.e., a region of the plane E is connected in the sense of topology if and only if it has no separable proper part. Moreover, it is shown that the contact relation algebra REL(O*E, C) and the relation algebra REL(O*E, NTPP) generated by the non-tangential proper parthood relation NTPP, coincide. This entails that the allegedly purely topological notion of connectedness can be defined in mereological terms. (shrink)

A set of propositional connectives is said to be functionally complete if all propositional formulae can be expressed using only connectives from that set. In this paper we give sufficient and necessary conditions for a one-element set of propositional connectives to be functionally complete. These conditions provide a simple and elegant characterization of functionally complete one-element sets of propositional connectives.

Disjunctivism, as a theory of visual experience, claims that the mental states involved in a “good case” experience of veridical perception and a “bad case” experience of hallucination differ, even in those cases in which the two experiences are indistinguishable for their subject. Consider the veridical perception of a bar stool and an indistinguishable hallucination; both of these experiences might be classed together as experiences (as) of a bar stool or experiences of seeming to see a bar stool. This might (...) lead us to think that the experiences we undergo in the two cases must be of the same kind, the difference being that the former, but not the latter, is connected to the world in the right kind of way. Such a conjecture has been called a “highest common factor” or “common kind” assumption. At heart, disjunctivism consists in the rejection of this assumption. According the disjunctivist, veridical experiences and hallucinations do not share a common component. There are a host of interesting questions surrounding disjunctivism including: What is involved in the claims that good case and bad case experiences differ? Why might one might want to be a disjunctivist? What kinds of claims can the disjunctivist make about hallucination and illusion? These questions, and problems for the thesis, will be discussed as we proceed. (shrink)

Generalizations of Boolean elements of a BL-algebra L are studied. By utilizing the MV-center MV(L) of L, it is reproved that an element x L is Boolean iff x x * = 1. L is called semi-Boolean if for all x L, x * is Boolean. An MV-algebra L is semi-Boolean iff L is a Boolean algebra. A BL-algebra L is semi-Boolean iff L is an SBL-algebra. A BL-algebra L is called hyper-Archimedean (...) if for all x L, xn is Boolean for some finite n 1. It is proved that hyper-Archimedean BL-algebras are MV-algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV-algebras or BL-algebras. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

In this paper we prove that the free pseudocomplemented residuated lattices are decomposable if and only if they are Stone, i.e., if and only if they satisfy the identity ¬ x ∨ ¬¬ x = 1. Some applications are given.

This paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures with operators—akin to the theory of modal Algebras—RBAs serve as models of modal logics in which truth (...) is relative. In particular, modal RBAs provide semantics for various well-known awareness logics and an alternative view of possibility semantics. (shrink)