This work contributes to the domains of Boolean algebra and of Bayesian probability, by proposing an algebraic extension of Boolean algebras, which implements an operator for the Bayesian conditional inference and is closed under this operator. It is known since the work of Lewis (Lewis' triviality) that it is not possible to construct such conditional operator within the space of events. Nevertheless, this work proposes an answer which complements Lewis' triviality, by the construction of a conditional operator outside the space of events, thus resulting in an algebraic extension. In particular, it is proved that any probability defined on a Boolean algebra may be extended to its algebraic extension in compliance with the multiplicative definition of the conditional probability. In the last part of this paper, a new bivalent logic is introduced on the basis of this algebraic extension, and basic properties are derived.
We introduce a new variant of dependence logic called Boolean dependence logic. In Boolean dependence logic dependence atoms are of the type =(x_1,...,x_n,α), where αis a Boolean variable. Intuitively, with Boolean dependence atoms one can express quantification of relations, while standard dependence atoms express quantification over functions. We compare the expressive power of Boolean dependence logic to dependence logic and first-order logic enriched by partially-ordered connectives. We show that the expressive power of Boolean dependence logic and dependence logic coincide. We define natu…
In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone's representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof of completeness theorem in propositional logic will be given using Stone's theorem from Boolean algebra.
We propose a graph-based extension of Boolean logic called Boolean Graph Logic (BGL). Construing formula trees as the cotrees of cographs, we may state semantic notions such as evaluation and entailment in purely graph-theoretic terms, whence we recover the definition of BGL. Naturally, it is conservative over usual Boolean logic. Our contributions are the following: (1) We give a natural semantics of BGL based on Boolean relations, i.e. it is a multivalued semantics, and show adequacy of this semantics for the corresponding notions of entailment. (2) We show that the complexity of evaluation …
This paper presents an investigation on the structure of conditional events and on the probability measures which arise naturally in this context. In particular we introduce a construction which defines a (finite) {\em Boolean algebra of conditionals} from any (finite) Boolean algebra of events. By doing so we distinguish the properties of conditional events which depend on probability and those which are intrinsic to the logico-algebraic structure of conditionals. Our main result provides a way to regard standard two-place conditional probabilities as one-place probability functions on condit…
We consider a 2-valued non-deterministic connective $\wedge \hskip-5.5pt \vee$ defined by the table resulting from the entry-wise union of the tables of conjunction and disjunction. Being half conjunction and half disjunction we named it platypus. The value of $\wedge \hskip-5.5pt \vee$ is not completely determined by the input, contrasting with usual notion of Boolean connective. We call non-deterministic Boolean connective any connective based on multi-functions over the Boolean set. In this way, non-determinism allows for an extended notion of truth-functional connective. Unexpectedly, this…
Moisil in 1941, while constructing the algebraic models of n-valued Łukasiewicz logic defined the set $B^{[n]}$,where $B$ is a Boolean algebra and `n' being a natural number. Further it was proved by Moisil himself the representations of n-valued Łukasiewicz Moisil algebra in terms of $B^{[n]}$. In this article, structural representation results for Stone, dual Stone and double Stone algebras are proved similar to Moisil's work by showing that elements of these algebras can be looked upon as monotone ordered tuple of sets. 3-valued semantics of logic for Stone algebra, dual Stone algebras and …
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. Further, we show that negation with the Kleene property (called the `Kleene negation') always arises from the set theoretic complement. The corresponding propositional logic is then studied through a 3-valued and rough set semantics. It is also established that Kleene negation can be considered as a modal operator, and enables giving a perp semantics to the logic. One concludes with the observation that all the semantics for this logic are equivalent.
We investigate the possibility of extending the non-functionally complete logic of a collection of Boolean connectives by the addition of further Boolean connectives that make the resulting set of connectives functionally complete. More precisely, we will be interested in checking whether an axiomatization for Classical Propositional Logic may be produced by merging Hilbert-style calculi for two disjoint incomplete fragments of it. We will prove that the answer to that problem is a negative one, unless one of the components includes only top-like connectives.
Christine Tasson introduced an algebraic notion of totality for a denotational model of linear logic in the category of vector spaces. The notion of total boolean function is, in a way, quite intuitive. This note provides a positive answer to the question of completeness of the "boolean centroidal calculus" w.r.t. total boolean functions.
We present a substructural epistemic logic, based on Boolean BI, in which the epistemic modalities are parametrized on agents' local resources. The new modalities can be seen as generalizations of the usual epistemic modalities. The logic combines Boolean BI's resource semantics --- we introduce BI and its resource semantics at some length --- with epistemic agency. We illustrate the use of the logic in systems modelling by discussing some examples about access control, including semaphores, using resource tokens. We also give a labelled tableaux calculus and establish soundness and completene…
Constructing a fuzzy subset logic L with Boolean properties is notoriously difficult because under a handful of "reasonable" conditions, we have the following three debilitating constraints: (1) Bellman and Giertz in 1973 showed that if L is distributive, then it must be idempotent. (2) Dubois and Padre in 1980 showed that if L has the excluded middle or the non-contradiction property or both, then it must be non-idempotent. (3) Bellman and Giertz also demonstrated in 1973 that even if L is idempotent, then the only choice available for the (meet,join) logic operator pair is the (min,max) oper…
The article is a study of two algebraic structures, the `contrapositionally complemented pseudo-Boolean algebra' (ccpBa) and `contrapositionally $\vee$ complemented pseudo-Boolean algebra' (c$\vee$cpBa). The algebras have recently been obtained from a topos-theoretic study of categories of rough sets. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. We study properties of these algebras, give examples, and compare them with relevant existing algebras. `Intuitionistic Log…
The Boolean logic of subsets, usually presented as `propositional logic,' is considered as being "classical" while intuitionistic logic and the many sublogics and off-shoots are "non-classical." But there is another mathematical logic, the logic of partitions, that is at the same mathmatical level as Boolean subset logic since subsets and quotient sets (partitions or equivalence relations) are dual to one another in the category-theoretic sense. Our purpose here is to explore the notions of negation and implication in that other mathematical logic of partitions.
Moisil logic, having as algebraic counterpart Łukasiewicz-Moisil algebras, provide an alternative way to reason about vague information based on the following principle: a many-valued event is characterized by a family of Boolean events. However, using the original definition of Łukasiewicz-Moisil algebra, the principle does not apply for subalgebras. In this paper we identify an alternative and equivalent definition for the $n$-valued Łukasiewicz-Moisil algebras, in which the determination principle is also saved for arbitrary subalgebras, which are characterized by a Boolean algebra and a fa…
Fuzzy logic is a way to argue with boolean predicates for which we only have a confidence value between 0 and 1 rather than a well defined truth value. It is tempting to interpret such a confidence as a probability. We use Markov kernels, parametrised probability distributions, to do just that. As a consequence we get general fuzzy logic connectives from probabilistic computations on products of the booleans, stressing the importance of joint confidence functions. We discuss binary logic connectives in detail and recover the "classic" fuzzy connectives as bounds for the confidence for general …
We introduce quantum Boolean algebras which are the analogue of the Weyl algebras for Boolean affine spaces. We study quantum Boolean algebras from the logical and set theoretical viewpoints.
It is well-known that a Hilbert-style deduction system for first-order classical logic is sound and complete for a model theory built using all Boolean algebras as truth-value algebras if and only if it is sound and complete for a model theory utilizing only irreducible Boolean algebras (which are all isomorphic to the two-element Boolean algebra). In this paper, we prove an analogous result for any first-order logic with an algebraic semantics satisfying certain minimal assumptions, and we then apply our result to first-order quantum logic.
For any first order theory T we construct a Boolean valued model M, in which precisely the T--provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a first order formula. Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory.
We present a correlated \textit{and} gate which may be used to propagate uncertainty and dependence through Boolean functions, since any Boolean function may be expressed as a combination of \textit{and} and \textit{not} operations. We argue that the \textit{and} gate is a bivariate copula family, which has the interpretation of constructing bivariate Bernoulli random variables following a given Pearson correlation coefficient and marginal probabilities. We show how this copula family may be used to propagate uncertainty in the form of probabilities of events, probability intervals, and probab…
Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not…
In this paper a conditional logic is defined and studied. This conditional logic, Deterministic Bayesian Logic, is constructed as a deterministic counterpart to the (probabilistic) Bayesian conditional. The logic is unrestricted, so that any logical operations are allowed. This logic is shown to be non-trivial and is not reduced to classical propositions. The Bayesian conditional of DBL implies a definition of logical independence. Interesting results are derived about the interactions between the logical independence and the proofs. A model is constructed for the logic. Completeness results a…
In this paper a conditional logic is defined and studied. This conditional logic, DmBL, is constructed as a deterministic counterpart to the Bayesian conditional. The logic is unrestricted, so that any logical operations are allowed. A notion of logical independence is also defined within the logic itself. This logic is shown to be non-trivial and is not reduced to classical propositions. A model is constructed for the logic. Completeness results are proved. It is shown that any unconditioned probability can be extended to the whole logic DmBL. The Bayesian conditional is then recovered from t…
Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful. We study how …
The Beth Definability Property holds for an algebraizable logic if and only if every epimorphism in the corresponding category of algebras is surjective. Using this technique, Urquhart in 1999 showed that the Beth Definability Property fails for a wide class of relevant logics, including T, E, and R. However, the counterexample for those logics does not extend to the algebraic counterpart of the super relevant logic KR, the so-called Boolean monoids. Following a suggestion of Urquhart, we use modular lattices constructed by Freese to show that epimorphisms need not be surjective in a wide clas…
From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this case 2. In the present work, we generalize the entire arrangement from propositional to first-order logic. Boole…
In this paper, a short survey about the concepts underlying general logics is given. In particular, a novel rigorous definition of a fuzzy negation as an operation acting on a lattice to render it into a fuzzy logic is presented. According to this definition, a fuzzy negation satisfies the weak double negation condition, requiring double negation to be expansive, the antitony condition, being equivalent to the disjunctive De Morgan law and thus warranting compatibility of negation with the lattice operations, and the Boolean boundary condition stating that the universal bounds of the lattice a…
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 ar…
In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition is represented by relations between…
