This article explores the connection between boolean-valued class models of set theory and the theory of arbitrary objects in roughly Kit Fine's sense of the word. In particular, it explores the hypothesis that the set theoretic universe as a whole can be seen as an arbitrary entity, which can in turn be taken to consist of arbitrary objects, viz. arbitrary sets.

2022-02-16math.LO

Stone and double Stone algebras: Boolean and Rough Set Representations, 3-valued and 4-valued Logics

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 …

2017-10-05math.LO

Leibnizian, Robinsonian, and Boolean Valued Monads

This is an overview of the present-day versions of monadology with some applications to vector lattices and linear inequalities.

2011-06-08math.LO

Keisler's Order and Full Boolean-Valued Models

We prove a compactness theorem for full Boolean-valued models. As an application, we show that if $T$ is a complete countable theory and $\mathcal{B}$ is a complete Boolean algebra, then $λ^+$-saturated $\mathcal{B}$-valued models of $T$ exist. Moreover, if $\mathcal{U}$ is an ultrafilter on $T$ and $\mathbf{M}$ is a $λ^+$-saturated $\mathcal{B}$-valued model of $T$, then whether or not $\mathbf{M}/\mathcal{U}$ is $λ^+$-saturated just depends on $\mathcal{U}$ and $T$; we say that $\mathcal{U}$ $λ^+$-saturates $T$ in this case. We show that Keisler's order can be formulated as follows: $T_0 \tr…

2018-10-12math.LO

Principal and Boolean congruences on θ-valued Lukasiewicz-Moisil algebras

The first system of many-valued logic was introduced by J. Lukasiewicz, his motivation was of philosophical nature as he was looking for an interpretation of the concepts of possibility and necessity. Since then, plenty of research has been developed in this area. In 1968, when Gr.C. Moisil came across Zadeh's fuzzy set theory he found the motivation he had been looking for in order to legitimate the introduction and study of infinitely-valued Lukasiewicz algebras, so he defined θ-valued Lukasiewicz algebras (or LMθ-algebras, for short) (without negation), where θis the order type of a chain. …

2012-04-19math.LO

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布尔值二阶逻辑再探

Boolean-valued second-order logic revisited

本文继续研究布尔值二阶逻辑与完全二阶逻辑的差异，证明前者紧致性数在真类Woodin基数下为ω₁（与完全二阶逻辑的最小可扩张基数不同），并引入内模型C²ᵇ，证明其为ZFC的最小内模型且具有良好性质。

2025-04-17math.LO

布尔值模型、预层和概形空间

Boolean valued models, presheaves, and étalé spaces

本文研究布尔值模型、预层和概形空间之间的关系，通过拓扑和范畴论方法，提出布尔值模型满足Loś定理的特征化条件，并区分完整性性质与混合性质。

2026-07-23math.CTmath.LO

无穷逻辑的布尔值语义

Boolean valued semantics for infinitary logics

本文利用力迫法改进Mansfield的布尔值完备性定理，针对无穷逻辑L∞ω建立更强完备性，并证明Craig插值性质与强省略类型定理，同时揭示L∞ω与力迫法的联系。

2023-05-12math.LO

Kleene Algebras and Logic: Boolean and Rough Set Representations, 3-valued, Rough and Perp Semantics

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.

2015-11-23math.LO

From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti

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…

2021-02-07math.LOquant-ph

Mutually exclusive nuances of truth in Moisil logic

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…

2014-12-09math.LO

On strictly nonzero integer-valued charges

A charge (finitely additive measure) defined on a Boolean algebra of sets taking values in a group $G$ is called a strictly nonzero (SNZ) charge if it takes the identity value in $G$ only for the zero element of the Boolean algebra. A study of such charges was initiated by Rudiger Göbel and K.P.S. Bhaskara Rao in 2002. Our main result is a solution to one of the questions posed in that paper: we show that for every cardinal $\aleph$, the Boolean algebra of clopen sets of $\{0,1\}^\aleph$ has a strictly nonzero integer-valued charge. The key lemma that we prove is that there exists a strictly n…

2016-08-03math.CAmath.GNmath.LO

A definability theorem for first order 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.

1997-06-12math.LO

Compound conditionals as random quantities and Boolean algebras

Conditionals play a key role in different areas of logic and probabilistic reasoning, and they have been studied and formalized from different angles. In this paper we focus on the de Finetti's notion of conditional as a three-valued object, with betting-based semantics, and its related approach as random quantity as mainly developed by two of the authors. Compound conditionals have been studied in the literature, but not in full generality. In this paper we provide a natural procedure to explicitly attach conditional random quantities to arbitrary compound conditionals that also allows us to …

2022-05-06math.LO

Twist-Valued Models for Three-valued Paraconsistent Set Theory

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent…

2019-12-01math.LO

A formalization of forcing and the unprovability of the continuum hypothesis

We describe a formalization of forcing using Boolean-valued models in the Lean 3 theorem prover, including the fundamental theorem of forcing and a deep embedding of first-order logic with a Boolean-valued soundness theorem. As an application of our framework, we specialize our construction to the Boolean algebra of regular opens of the Cantor space $2^{ω_2 \times ω}$ and formally verify the failure of the continuum hypothesis in the resulting model.

2019-04-23cs.LOmath.LO

Orthomodular-Valued Models for Quantum Set Theory

In 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo-Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti's model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which …

2017-05-14math-phmath.LOmath.MP

力迫的内部模态逻辑

The Internal Modal Logic of Forcing

该研究通过为模态公式构建内部克里普克语义，将模态集合论与布尔值模型相连，计算模态有效性并分析特殊代数行为，最终证明在特定语义下正规逻辑$\KTB$是相关模型中有效模态公式集。

2026-07-29math.LO

从量子视角看布尔函数的Walsh与Fourier-Hadamard支撑

On the Walsh and Fourier-Hadamard Supports of Boolean Functions From a Quantum Viewpoint

本文从量子计算视角研究布尔函数的Walsh与Fourier-Hadamard支撑，提出完全平衡函数和平衡集概念，补全经典结果并推广至伪布尔函数，以分析量子算法的能力。

2024-05-06math.COmath.LO