scholarly journals A Many-valued Logic for Lexicographic Preference Representation

2021 ◽  
Author(s):  
Angelos Charalambidis ◽  
George Papadimitriou ◽  
Panos Rondogiannis ◽  
Antonis Troumpoukis
Keyword(s):  
Propositional Logic ◽  
Rational Numbers ◽  
User Preferences ◽  
Satisfaction Level ◽  
Truth Values ◽  
Lexicographic Preferences ◽  
Preference Representation ◽  
Ranking Query

We introduce lexicographic logic, an extension of propositional logic that can represent a variety of preferences, most notably lexicographic ones. The proposed logic supports a simple new connective whose semantics can be defined in terms of finite lists of truth values. We demonstrate that, despite the well-known theoretical limitations that pose barriers to the quantitative representation of lexicographic preferences, there exists a subset of the rational numbers over which the proposed new connective can be naturally defined. Lexicographic logic can be used to define in a simple way some well-known preferential operators, like "A and if possible B", and "A or failing that B". We argue that the new logic is an effective formalism for ranking query results according to the satisfaction level of user preferences.

Personalisation in Highly Dynamic Grid Services Environments

2009 ◽  
pp. 284-313
Author(s):  
Edgar Jembere ◽  
Matthew O. Adigun ◽  
Sibusiso S. Xulu
Keyword(s):  
User Preferences ◽  
User Preference ◽  
Grid Services ◽  
Context Aware Computing ◽  
Preference Representation ◽  
Use Of Services ◽  
Order Preference ◽  
Computing Environments ◽  
Preference Models ◽  
Intuitive Interpretation

Human Computer Interaction (HCI) challenges in highly dynamic computing environments can be solved by tailoring the access and use of services to user preferences. In this era of emerging standards for open and collaborative computing environments, the major challenge that is being addressed in this chapter is how personalisation information can be managed in order to support cross-service personalisation. The authors’ investigation of state of the art work in personalisation and context-aware computing found that user preferences are assumed to be static across different context descriptions whilst in reality some user preferences are transient and vary with changes in context. Further more, the assumed preference models do not give an intuitive interpretation of a preference and lack user expressiveness. This chapter presents a user preference model for dynamic computing environments, based on an intuitive quantitative preference measure and a strict partial order preference representation, to address these issues. The authors present an approach for mining context-based user preferences and its evaluation in a synthetic m-commerce environment. This chapter also shows how the data needed for mining context-based preferences is gathered and managed in a Grid infrastructure for mobile devices.

A Mathematical Setting for Fuzzy Logics

1997 ◽  
Vol 05 (03) ◽  
pp. 223-238 ◽  
Author(s):  
Mai Gehrke ◽  
Carol Walker ◽  
Elbert Walker
Keyword(s):  
Fuzzy Logic ◽  
Propositional Logic ◽  
Normal Forms ◽  
Unit Interval ◽  
Fuzzy Logics ◽  
Real Numbers ◽  
Truth Values ◽  
Finite Algebras ◽  
Finite Algorithms ◽  
Interval Valued

The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and -x=1-x for conjunction, disjunction and negation, respectively, is the standard propositional fuzzy logic. This is shown to be the same as three-valued logic. The propositional logic obtained when the algebra of truth values is the set {(a, b)|a≤ b and a,b∈[0,1]} of subintervals of the unit interval with component-wise operations, is propositional interval-valued fuzzy logic. This is shown to be the same as the logic given by a certain four element lattice of truth values. Since both of these logics are equivalent to ones given by finite algebras, it follows that there are finite algorithms for determining when two statements are logically equivalent within either of these logics. On this topic, normal forms are discussed for both of these logics.

scholarly journals The Propositional Logic of Frege’s Grundgesetze: Semantics and Expressiveness

2017 ◽  
Vol 5 (6) ◽  
Author(s):  
Eric D. Berg ◽  
Roy T. Cook
Keyword(s):  
Propositional Logic ◽  
Truth Values ◽  
Logical Operators ◽  
The Way

In this paper we compare the propositional logic of Frege’s Grundgesetze der Arithmetik to modern propositional systems, and show that Frege does not have a separable propositional logic, definable in terms of primitives of Grundgesetze, that corresponds to modern formulations of the logic of “not”, “and”, “or”, and “if…then…”. Along the way we prove a number of novel results about the system of propositional logic found in Grundgesetze, and the broader system obtained by including identity. In particular, we show that the propositional connectives that are definable in terms of Frege’s horizontal, negation, and conditional are exactly the connectives that fuse with the horizontal, and we show that the logical operators that are definable in terms of the horizontal, negation, the conditional, and identity are exactly the operators that are invariant with respect to permutations on the domain that leave the truth-values fixed. We conclude with some general observations regarding how Frege understood his logic, and how this understanding differs from modern views.

scholarly journals Tableau-based Decision Procedure for Non-Fregean Logic of Sentential Identity

2021 ◽  
pp. 41-57
Author(s):  
Joanna Golińska-Pilarek ◽  
Taneli Huuskonen ◽  
Michał Zawidzki
Keyword(s):  
Propositional Logic ◽  
Decision Procedure ◽  
Classical Propositional Logic ◽  
Complexity Bound ◽  
Sentential Calculus ◽  
Truth Values

AbstractSentential Calculus with Identity ($$\mathsf {SCI}$$ SCI ) is an extension of classical propositional logic, featuring a new connective of identity between formulas. In $$\mathsf {SCI}$$ SCI two formulas are said to be identical if they share the same denotation. In the semantics of the logic, truth values are distinguished from denotations, hence the identity connective is strictly stronger than classical equivalence. In this paper we present a sound, complete, and terminating algorithm deciding the satisfiability of $$\mathsf {SCI}$$ SCI -formulas, based on labelled tableaux. To the best of our knowledge, it is the first implemented decision procedure for $$\mathsf {SCI}$$ SCI which runs in NP, i.e., is complexity-optimal. The obtained complexity bound is a result of dividing derivation rules in the algorithm into two sets: decomposition and equality rules, whose interplay yields derivation trees with branches of polynomial length with respect to the size of the investigated formula. We describe an implementation of the procedure and compare its performance with implementations of other calculi for $$\mathsf {SCI}$$ SCI (for which, however, the termination results were not established). We show possible refinements of our algorithm and discuss the possibility of extending it to other non-Fregean logics.

A formalization of an ℵ0-valued propositional calculus

1953 ◽  
Vol 49 (3) ◽  
pp. 367-376
Author(s):  
Alan Rose
Keyword(s):  
Propositional Calculus ◽  
Rational Numbers ◽  
Truth Values ◽  
Truth Value ◽  
Image Position

In 1930 Łukasiewicz (3) developed an ℵ0-valued prepositional calculus with two primitives called implication and negation. The truth-values were all rational numbers satisfying 0 ≤ x ≤ 1, 1 being the designated truth-value. If the truth-values of P, Q, NP, CPQ are x, y, n(x), c(x, y) respectively, then

scholarly journals Preferential factors and usability effects on vehicle modifications for the Malaysian independent disabled driver

2018 ◽  
Vol 7 (2.29) ◽  
pp. 781
Author(s):  
Mohd Khairul Anwar Mohd Dahuri ◽  
Mohd Nasir Hussain ◽  
Nor Fasiha Mohd Yusof ◽  
Mohamad Kasim Abdul Jalil
Keyword(s):  
Product Development ◽  
Body Part ◽  
User Preferences ◽  
Satisfaction Level ◽  
Kuala Lumpur ◽  
Secondary Control ◽  
Factors Influencing ◽  
Hand Control ◽  
Major Factors ◽  
Movement Limitation

The usability of driving modifications for a person with disabilities is known to be an important aspect in addressing the independent vehicle’s driving capabilities. The existence of assistive modifications such as hand control, secondary control, foot control, and also the wheelchair assisted vehicle, which were developed to accommodate limitations were found to be widely used. A survey was conducted on 202 Malaysian independent disabled drivers from Kuala Lumpur, Selangor, Perak, Johor Baharu, Terengganu, and also Melaka. The components evaluated in the survey include the origin of modification used for driving, factors influencing the decision on the type of vehicle modification, usability difficulties when modification is in use, the satisfaction level of product appearance, as well as to understand user preferences when making decisions between modifications. A Likert scale of 1 to 5 was used as measurement, in order to rate the score given by the responses for each question in the survey. As a result, it was discovered that aspects such as the modification colour, shape, the toggle, pedal grip, regulation, modification availability, and movement limitation on the involved body part are among the major factors influencing the driver to make modification for driving purposes. Major factors were also found to influence the driving modification preferences to suit the category of disabilities. It was also discovered that components such as the availability of modification information and emergency driving situations are some of the least important factors influencing the decision to make and select a suitable driving modification. However, these less important components must not be ignored as it also contributes to the improvement for independent disabled driving, and also for the purpose of assistive product development.  

scholarly journals Constructing illoyal algebra-valued models of set theory

2021 ◽  
Vol 82 (3) ◽  
Author(s):  
Benedikt Löwe ◽  
Robert Paßmann ◽  
Sourav Tarafder
Keyword(s):  
Set Theory ◽  
Propositional Logic ◽  
Truth Values ◽  
Models Of Set Theory

AbstractAn algebra-valued model of set theory is called loyal to its algebra if the model and its algebra have the same propositional logic; it is called faithful if all elements of the algebra are truth values of a sentence of the language of set theory in the model. We observe that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of illoyal models: tail stretches, transposition twists, and maximal twists.

Formalisations of further ℵ0-valued Łukasiewicz propositional calculi

1978 ◽  
Vol 43 (2) ◽  
pp. 207-210 ◽  
Author(s):  
Alan Rose
Keyword(s):  
Propositional Calculus ◽  
Rational Numbers ◽  
Modus Ponens ◽  
Truth Values ◽  
Rules Of Procedure ◽  
C And N ◽  
Image Position ◽  
Propositional Calculi

It has been shown that, for all rational numbers r such that 0≤ r ≤ 1, the ℵ0-valued Łukasiewicz propositional calculus whose designated truth-values are those truth-values x such that r ≤ x ≤ 1 may be formalised completely by means of finitely many axiom schemes and primitive rules of procedure. We shall consider now the case where r is rational, 0≥r≤1 and the designated truth-values are those truth-values x such that r≤x≤1.We note that, in the subcase of the previous case where r = 1, a complete formalisation is given by the following four axiom schemes together with the rule of modus ponens (with respect to C),the functor A being defined in the usual way. The functors B, K, L will also be considered to be defined in the usual way. Let us consider now the functor Dαβ such that if P, Dαβ take the truth-values x, dαβ(x) respectively, α, β are relatively prime integers and r = α/β thenIt follows at once from a theorem of McNaughton that the functor Dαβ is definable in terms of C and N in an effective way. If r = 0 we make the definitionWe note first that if x ≤ α/β then dαβ(x)≤(β + 1)α/β − α = α/β. HenceLet us now define the functions dnαβ(x) (n = 0,1,…) bySinceit follows easily thatand thatThus, if x is designated, x − α/β > 0 and, if n > − log(x − α/β)/log(β + 1), then (β + 1)n(x−α/β) > 1.

scholarly journals On evaluations of propositional formulas in countable structures

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 1-13
Author(s):  
Aleksandar Perovic ◽  
Dragan Doder ◽  
Zoran Ognjanovic ◽  
Miodrag Raskovic
Keyword(s):  
Propositional Logic ◽  
Rational Numbers ◽  
Infinite Subset ◽  
Inference Rules ◽  
First Order ◽  
Order Language ◽  
Finite Sequences

Let L be a countable first-order language such that its set of constant symbols Const(L) is countable. We provide a complete infinitary propositional logic (formulas remain finite sequences of symbols, but we use inference rules with countably many premises) for description of C-valued L-structures, where C is an infinite subset of Const(L). The purpose of such a formalism is to provide a general propositional framework for reasoning about F-valued evaluations of propositional formulas, where F is a C-valued L-structure. The prime examples of F are the field of rational numbers Q, its countable elementary extensions, its real and algebraic closures, the field of fractions Q(?), where ? is a positive infinitesimal and so on.

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