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.
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.
Some notes concerning the homogeneity of Boolean algebras and Boolean spaces
研究了具有可数生成的非主超滤子的布尔代数的齐次性，证明了一个布尔代数B是齐次的，如果它是可数生成的非主超滤子的并，并且有一个稠密子集D，使得对于D中的每一个a，相对代数B的限制a：= {b in B：b <= a}与B同构.特别地，原子布尔代数的可数个副本的自由积是齐次的。此外，一个布尔代数B是齐次的，如果它满足下列条件：（i）B有一个可数生成的超滤子，（ii）B不是c.c.c.，和（iii）对于B集减{0}中的每个a，有n个自同构h_1，.，h_n满足1=h_1（a）cup. cup h_n（a）.
In this paper we classify some special reducts of the countable atomless Boolean algebra which we call functional reducts. We prove that there are exactly $13$ such structures up to first order interdefinability.
Skew Boolean algebras (skew BA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed skew BAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use skew BAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed skew BAs in terms of nBAs of n-partitions.
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…
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…
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…
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…
