Logic With Verbs and its Mathematical Structure
The aim of this paper is to introduce the idea of Logic with Verbs and to show its mathematical structure.
math.LO↗
arXiv subjects
Jun Tanaka
Publications and source records attributed to Jun Tanaka.
Logic with Verbs
The aim of this paper is to introduce a logic in which nouns and verbs are handled together as a deductive reasoning, and also to observe the relationship between nouns and verbs as well as between logics and conversations.
cs.AI↗
Construction of A Lattice on the completion space of an algebra and an isomorphism to its Caratheodory Extension
In this paper, we will show how the Caratheodory Extension process is intimately related to the metric completion process. In particular, it will be shown how one is able to construct a lattice on the completion and to obtain an isomorphism to its Caratheodory Extension.
math.LO↗
Hahn Decomposition Theorem of Signed Lattice Measure
In this paper, we will define a signed Lattice measure on $σ$-algebras, as well as give the definition of positive and negative Lattice. Herein, we will show that the Hahn Decomposition Theorem decomposes any space X into a positive lattice A and a negative Lattice B such that $A \vee B$ =X and the signed Lattice measure of $A \wedge B $ is 0.
math.CA↗
Modern Set
In this paper, we intend to generalize the classical set theory as much as possible. we will do this by freeing sets from the regular properties of classical sets; e.g., the law of excluded middle, the law of non-contradiction, the distributive law, the commutative law,etc....
math.GM↗
Fuzzy signed measure
we will define a fuzzy signed measure on $σ$-algebras, as well as positive and negative sets. Herein, we will show that the Fuzzy Hahn Decomposition Theorem, which is a generalization of the classical Hahn Decomposition Theorem, decompose any space X into a positive set A and a negative set B such that A+B=X and the signed measure of $A \wedge B $ is 0.
math.LO↗
An isomorphism between the completion of an Algebra and its Caratheodory Extension
Let $Ω$ denote an algebra of sets and $μ$ a $σ$-finite measure. We then prove that the completion of $Ω$ under the pseudometric $d(A,B)$ = $μ^{\ast}(A \triangle B)$ is $σ$-algebra isomorphic and isometric to the Caratheodory Extension of $Ω$ under the equivalence relation $\sim$.
math.FA↗
A Realization of Measurable Sets as Limit Points
Starting with a sigma finite measure on an algebra, we define a pseudometric and show how measurable sets from the Caratheodory Extension Theorem can be thought of as limit points of Cauchy sequences in the algebra.
math.FA↗