Fuzzy Logic and Arithmetical Hierarchy III

Fuzzy logic is understood as a logic with a comparative and truth-functional notion of truth. Arithmetical complexity of sets of tautologies (identically true sentences) and satisfiable sentences (sentences true in at least one interpretation) as well of sets of provable formulas of the most important systems of fuzzy predicate logic is determined or at least estimated.     

On Retaining Classical Truths and Classical Deducibility in Many-Valued and Fuzzy Logics

In this paper, I identify the source of the differences between classical logic and many-valued logics (including fuzzy logics) with respect to the set of valid formulas and the set of inferences sanctioned. In the course of doing so, we find the conditions that are individually necessary and jointly sufficient for any many-valued semantics (again including fuzzy logics) to validate exactly the classically valid formulas, while sanctioning exactly the same set of inferences as classical logic. This in turn shows, contrary to what has sometimes been claimed, that at least one class of infinite-valued semantics is axiomatizable.     

Fuzzy Logic Programming and Fuzzy Control

We show that it is possible to base fuzzy control on fuzzy logic programming. Indeed, we observe that the class of fuzzy Herbrand interpretations gives a semantics for fuzzy programs and we show that the fuzzy function associated with a fuzzy system of IF-THEN rules is the fuzzy Herbrand interpretation associated with a suitable fuzzy program.     

概率逻辑与模糊逻辑在精细化学习诊断中的对比研究

精细化学习诊断有助于客观准确探究学生学习现状,为实施有针对性的补救教学提供理论和数据支持。本文对比研究概率逻辑与模糊逻辑在精细化学习诊断中的表现。首先,从“概念”视角介绍和对比两种逻辑。其次,介绍两个分别基于上述两逻辑的代表性模型：HO-PINC和Fuzzy-DINA。然后,对比两模型在五个实证数据上表现。最后,通过模拟研究进一步对比两模型的心理计量学性能。DIC等模型-数据拟合指标和RMSE等参数返真性指标的结果表明两模型对同一批数据有较一致的分析结果。建议实践者忽略模型选择对数据分析的影响,从概念或思辨视角入手选择使用概率逻辑或模糊逻辑来定义属性。     

Three Complexity Problems in Quantified Fuzzy Logic

We prove that the sets of standard tautologies of predicate Product Logic and of predicate Basic Logic, as well as the set of standard-satisfiable formulas of predicate Basic Logic are not arithmetical, thus finding a rather satisfactory solution to three problems proposed by Hájek in [H01].     

The Interpretability Logic of all Reasonable Arithmetical Theories

This paper is a presentation of astatus quæstionis, to wit of the problemof the interpretability logic of all reasonablearithmetical theories.We present both the arithmetical side and themodal side of the question.Dedicated to Dick de Jongh on the occasion of his 60th birthday     

An n-Player Semantic Game for an n + 1-Valued Logic

First we show that the classical two-player semantic game actually corresponds to a three-valued logic. Then we generalize this result and give an n-player semantic game for an n + 1-valued logic with n binary connectives, each associated with a player. We prove that player i has a winning strategy in game if and only if the truth value of is t _i in the model M, for 1 ≤ i ≤ n; and none of the players has a winning strategy in if and only if the truth value of is t ₀ in M.     

Mathematical Fuzzy Logic – What It Can Learn from Mostowski and Rasiowa

Important works of Mostowski and Rasiowa dealing with many-valued logic are analyzed from the point of view of contemporary mathematical fuzzy logic.A version of this paper has been presented during the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005.     

Game Logic and its Applications II

This paper provides a Genzten style formulation of the game logic framework GLm (0 m ), and proves the cut-elimination theorem for GLm. As its application, we prove the term existence theorem for GL used in Part I.     

多值逻辑与语义赋值博弈

文章在扩展博弈上,给出了多值逻辑的语义赋值博弈的一般框架,避免了博弈者在多值逻辑的语义博弈中声明无穷对象的问题;然后通过Eloise赢的策略定义博弈的语义概念——赋值,证明了多值逻辑的博弈语义与Tarski语义是等价的;最后,根据语义赋值博弈框架对经典逻辑进行了博弈化。     

Dialogue Games for Many-Valued Logics — an Overview

An overview of different versions and applications of Lorenzen’s dialogue game approach to the foundations of logic, here largely restricted to the realm of manyvalued logics, is presented. Among the reviewed concepts and results are Giles’s characterization of ?ukasiewicz logic and some of its generalizations to other fuzzy logics, including interval based logics, a parallel version of Lorenzen’s game for intuitionistic logic that is adequate for finite- and infinite-valued Gödel logics, and a truth comparison game for infinite-valued Gödel logic.     

Axiomatic Extensions of IMT3 Logic

In this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ³ ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x³) ∨ x ≈ ?, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of IMT3 is generated by their totally ordered members, we study the structure of all IMT3-chains in order to determine the lattice of all subvarieties of IMT3. Given a family of IMT3-chains the number of elements of the largest odd finite subalgebra in the family and the number of elements of the largest even finite subalgebra in the family turns out to be a complete classifier of the variety generated. We obtain a canonical set of generators and a finite equational axiomatization for each subvariety and, for each corresponding logic, a finite set of characteristic matrices and a finite set of axioms.     

Commodious Axiomatization of Quantifiers in Multiple-Valued Logic

We provide tools for a concise axiomatization of a broad class of quantifiers in many-valued logic, so-called distribution quantifiers. Although sound and complete axiomatizations for such quantifiers exist, their size renders them virtually useless for practical purposes. We show that for quantifiers based on finite distributive lattices compact axiomatizations can be obtained schematically. This is achieved by providing a link between skolemized signed formulas and filters/ideals in Boolean set lattices. Then lattice theoretic tools such as Birkhoff's representation theorem for finite distributive lattices are used to derive tableau-style axiomatizations of distribution quantifiers.     

Monadic Fuzzy Predicate Logics

Two variants of monadic fuzzy predicate logic are analyzed and compared with the full fuzzy predicate logic with respect to finite model property (properties) and arithmetical complexity of sets of tautologies, satisfiable formulas and of analogous notion restricted to finite models.     

The Undecidability of Iterated Modal Relativization

In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 ¹₁–complete. Two of these fragments do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is fi0 ⁰₁–complete. These results go via reduction to problems concerning domino systems.     

Modal Logic,Truth, and the Master Modality

In the paper (Braüner, 2001) we gave a minimal condition for the existence of a homophonic theory of truth for a modal or tense logic. In the present paper we generalise this result to arbitrary modal logics and we also show that a modal logic permits the existence of a homophonic theory of truth if and only if it permits the definition of a so-called master modality. Moreover, we explore a connection between the master modality and hybrid logic: We show that if attention is restricted to bidirectional frames, then the expressive power of the master modality is exactly what is needed to translate the bounded fragment of first-order logic into hybrid logic in a truth preserving way. We believe that this throws new light on Arthur Prior's fourth grade tense logic.