Search

The term "Weber-Fechner law" is arguably the most widely used misnomer in psychological science. The unification reflects a failure to appreciate the logical independence and disparate implications of Weber's law and Fechner's law as well as some closely aligned ones. The present statement, long overdue, is meant to rectify this situation. I discuss the roots and derivations of the relevant laws, eschewing formalism to bare essentials for sake of wider accessibility. Three of the most important conclusions are (a) Weber's law is not indispensable for deriving Fechner's law; (b) arguably, Fechner himself did not use Weber's law in his original derivations; and (c) many investigators mistake the principle that subjective distance is determined by physical ratio for Weber's law. In truth, the principle, here called the Weber principle, and Weber's law, are different and independent. I stress the importance of drawing the distinction and illustrate confusions in the literature coming from misapplications of Weber's law and the use of misnomer. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

In this paper we consider compound conditionals, Frechet-Hoeffding bounds and the probabilistic interpretation of Frank t-norms. By studying the solvability of suitable linear systems, we show under logical independence the sharpness of the Frechet-Hoeffding bounds for the prevision of conjunctions and disjunctions of n conditional events. In addition, we illustrate some details in the case of three conditional events. We study the set of all coherent prevision assessments on a family containing n conditional events and their conjunction, by verifying that it is convex. We discuss the case where the prevision of conjunctions is assessed by Lukasiewicz t-norms and we give explicit solutions for the linear systems; then, we analyze a selected example. We obtain a probabilistic interpretation of Frank t-norms and t-conorms as prevision of conjunctions and disjunctions of conditional events, respectively. Then, we characterize the sets of coherent prevision assessments on a family containing n conditional events and their conjunction, or their disjunction, by using Frank t-norms, or Frank t-conorms. By assuming logical independence, we show that any Frank t-norm (resp., t-conorm) of two conditional events A | H and B | K , T λ ( A | H , B | K ) (resp., S λ ( A | H , B | K ) ), is a conjunction ( A | H ) ∧ ( B | K ) (resp., a disjunction ( A | H ) ∨ ( B | K ) ). Then, we analyze the case of logical dependence where A = B and we obtain the set of coherent assessments on A | H , A | K , ( A | H ) ∧ ( A | K ) ; moreover we represent it in terms of the class of Frank t-norms T λ , with λ ∈ [ 0 , 1 ] . By considering a family F containing three conditional events, their conjunction, and all pairwise conjunctions, we give some results on Frank t-norms and coherence of the prevision assessments on F . By assuming logical independence, we show that it is coherent to assess the previsions of all the conjunctions by means of Minimum and Product t-norms. In this case all the conjunctions coincide with the t-norms of the corresponding conditional events. We verify by a counterexample that, when the previsions of conjunctions are assessed by the Lukasiewicz t-norm, coherence is not assured. Then, the Lukasiewicz t-norm of conditional events may not be interpreted as their conjunction. Finally, we give two sufficient conditions for coherence and incoherence when using the Lukasiewicz t-norm.

We present a short history of the notion of logical independence, its counterpart and gen-eralization in universal algebra, and their applications in different areas.

International audience; We investigate the relation between monotonicity and the no-show paradox in voting rules. Although the literature has established their logical independence, we show, by presenting logical dependency results, that the two conditions are closer than a general logical independency result would suggest. Our analysis is made both under variable and fixed-size electorates.

This paper deals with the upper and lower bounds of a class of uncertainty measures endowed with particular characteristics (decomposability, monotonicity, partial additivity and so on). We consider an initial partial assessment consistent with either probability or possibility or necessity, then we study the upper and lower envelopes of all possible extensions. By resorting to a notion of weak logical independence we get as lower or upper envelope a possibility or a necessity, respectively, starting either from a probability or from a possibility or from a necessity.

This paper is concerned with the notions of supervenience and mechanistic constitution as they have been discussed in the philosophy of neuroscience. Since both notions essentially involve specific dependence and determination relations among properties and sets of properties, the question arises whether the notions are systematically connected and how they connect to science. In a first step, some definitions of supervenience and mechanistic constitution are presented and tested for logical independence. Afterwards, certain assumptions fundamental to neuroscientific inquiry are made explicit in order to show that the presented definitions of supervenience are virtually uninteresting for theory construction in this field. In a third step, a new formulation of supervenience is developed that makes explicit reference to the notion of constitution and that bridges the gap between the philosophical concepts and explanatory practice in neuroscience.

This paper presents a unified account of bitstrings—i.e. sequences of bits (0/1) that serve as compact semantic representations—for the analysis of Aristotelian relations and provides an overview of their effectiveness in three key areas of the Logical Geometry research programme. As for logical effectiveness, bitstrings allow a precise and positive characterisation of the notion of logical independence or unconnectedness, as well as a straightforward computation—in terms of bitstring length and level—of the number and type of Aristotelian relations that a particular formula may enter into. As for diagrammatic effectiveness, bitstrings play a crucial role in studying the subdiagrams of the Aristotelian rhombic dodecahedron, and different types of Aristotelian hexagons turn out to require bitstrings of different lengths. The linguistic and cognitive effectiveness of bitstring analysis relates to the scalar structure underlying the bitstrings, and to the difference between linear and non-linear bitstrings.

This essay by Nishida Kitarō from 1927, translated into English here for the first time, is from the initial period of what has come to be called “Nishida philosophy” (Nishida tetsugaku), when Nishida was first developing his conception of “place” (basho). Nishida here inquires into the relationship between logic and consciousness in terms of place and implacement in order to overcome the shortcomings of previous philosophical attempts—from the ancient Greeks to the moderns—to dualistically conceive the relationship between being and knowing in terms of subject-object or form-matter. During the course of articulating his novel approach to consciousness and cognition, Nishida discusses what he takes to be the weaknesses of Greek hylomorphism, Kantian (and neo-Kantian) dualism, and Husserlian phenomenology. Dissatisfied with the attribution of mere passivity to placiality, and turning away from consciousness objectified as a subject of statement, Nishida imparts to consciousness qua place a certain logical independence as an active yet un-objectifiable “predicate.” This investigation of consciousness as the unobjectifiable place for objectification leads Nishida to the notion of what precedes consciousness itself, a “place of nothing” (mu no basho) that envelops the dichotomized structures of subject-predicate, being-nothing, subject-object, universal-particular, et cetera.

The notion of common cause closedness of a classical, Kolmogorovian probability space with respect to a causal independence relation between the random events is defined, and propositions are presented that characterize common cause closedness for specific probability spaces. It is proved in particular that no probability space with a finite number of random events can contain common causes of all the correlations it predicts; however, it is demonstrated that probability spaces even with a finite number of random events can be common cause closed with respect to a causal independence relation that is stronger than logical independence. Furthermore it is shown that infinite, atomless probability spaces are always common cause closed in the strongest possible sense. Open problems concerning common cause closedness are formulated and the results are interpreted from the perspective of Reichenbach's Common Cause Principle (RCCP).

Different conditional independence models have been proposed in literature; in this paper we consider models induced by conditional probabilities based on the definition of conditional cs-independence. These models need not comply with the symmetry property, so that they have not the graphoid structure. Hence, the well-known d-separation criterion for directed acyclic graphs may not be able to represent such independence models. Therefore, we introduce a new separation criterion called L-separation. We study its main properties and show how it allows to represent the above-mentioned independence models through directed acyclic graphs.

We show, by means of (counter)examples, that the axioms for fourgonal families (as used to construct elation generalized quadrangles) are independent.

The aim of this paper is that of studying a notion of independence for imprecise probabilities which is essentially based on the intuitive meaning of this concept. This is expressed, in the case of two events, by the reciprocal irrelevance of the knowledge of the value of each event for evaluating the other one, and has been termed epistemic independence. In order to consider more general situations in the framework of coherent imprecise probabilities, a definition of (epistemic) independence is introduced referring to arbitrary sets of logically independent partitions. Logical independence is viewed as a natural prerequisite for epistemic independence. It is then proved that the definition is always consistent, its relationship with the factorization rule is analysed, and some of its more relevant implications are discussed.

Various notions of independence of observables have been proposed within the algebraic framework of quantum field theory. We discuss relationships between these and the recently introduced notion of logical independence in a general operator-algebraic context. We show that C*-independence implies an analogue of classical independence.

For an arbitrary subselection of the identities defining the stochastic independence (of several events), there exist random events, with prescribed probabilities, satisfying only the identities from the subselection. A similar statement holds if the events are required to be exchangeable.

The logical independence of two statements is tantamount to their probabilistic independence, the latter understood in a sense that derives from stochastic independence. And analogous logical and probabilistic senses of having the same factual content similarly coincide. These results are extended to notions of non-symmetrical independence and independence among more than two statements.

The Shapley value is defined as the average marginal contribution of a player, taken over all possible ways to form the grand coalition N when one starts from the empty coalition and adds players one by one. In a previous paper, the authors have introduced an allocation scheme for a general model of coalition formation where the evolution of the coalition of active players is ruled by a Markov chain and need not finish with the grand coalition. This note provides an axiomatization which is weaker than the one in the original paper but allows a much more transparent correctness proof. Moreover, the logical independence of the axioms is proved., La valeur de Shapley est définie comme étant la moyenne de la contribution marginale d'un joueur pour toutes les façons possibles de former la grande coalition, en partant de la coalition vide et en ajoutant les joueurs un par un. Dans un précédent article, les auteurs ont introduit un schéma d'allocation pour un modèle général de formation de coalition, où l'évolution de la coalition des joueurs actifs est gouvernée par une chaîne de Markov, et qui ne finit pas nécessairement à la grande coalition. Cette note fournit une axiomatisation plus faible que celle proposée dans l'article original, et de plus permet une démonstration bien plus claire. De plus, nous montrons l'indépendance logique des axiomes.

A general probability space is defined to be causally complete if it contains common cause type variables for all correlations it predicts between compatible variables that are causally independent with respect to a causal independence relation defined between variables. The problem of causal completeness is formulated explicitly and several propositions are presented that spell out causal (in)completeness of certain classical and non-classical probability spaces with respect to a causal independence relation that is stronger than logical independence.

The connections between coherent probabilities and possibilities are studied from an inferential point of view. We introduce a condition of weak logical independence between two partitions L and L' assuring that the upper envelope of the extensions of a coherent probability on L is a possibility on the algebra generated by L'.

By means of a logical condition between two partitions \(\mathcal{L}\) and \({\mathcal{L}}\) (“weak logical independence”), we find connections between probabilities and possibilities. We show that the upper envelope of the extensions of a probability on \({\mathcal{L}}\) is a possibility on the algebra generated by \({\mathcal{L}^\prime}\). Moreover we characterize the set of possibilities obtained as extensions of a coherent probability on an arbitrary set: in particular, we find the two “extreme” (i.e., dominated and dominating) possibilities.

The subtle notion of conditioning is controversial in several contexts, for example in possibility theory where, in fact, different definitions have been introduced. We refer to a general axiomatic definition of conditional possibility and then we deal with ''partial assessments'' on (not necessarily structured) domains containing only elements of interest. We study a notion of coherence, which assures the extendability of an assessment as a conditional possibility and we introduce a procedure for checking coherence. Moreover, we propose a definition of independence for conditional possibility, which avoids some counterintuitive situations, and we study its main properties in order to compare it with other definitions introduced in literature. Then, we check which properties among the graphoid ones are satisfied: this allows to compare our definition with other independence notions given in the context of other uncertainty formalisms. This analysis is relevant for graphical models in order to single out and visualize dependence relations among random variables.

Starting from a general definition of conditional possibility introduced by Bouchon – Meunier , Coletti , Marsala in [ 2 ] we propose a notion of independence , which encompasses the critical situations presented by other independence definitions : for instance we obtain that , in any case , possibility independence implies logical independence . Moreover , we introduce a procedure for checking the consistency of a partial assessment with a conditional possibility .

In this paper we extend to upper and lower probabilities our approach to independence, based solely on conditional probability (in a coherent setting). Most difficulties arise (when this notion is put forward in the classical framework) either from the introduction of marginals for upper and lower probabilities (often improperly called, in the relevant literature, "imprecise" probabilities) when trying to extend to them the "product rule", or from the different ways of introducing conditioning for upper and lower probabilities. Our approach to conditioning in the context of "imprecise" probabilities is instead the most natural: in fact its starting point refers to a direct definition (through coherence) of the "enveloping" conditional "precise" probabilities. The discussion of some critical examples seems to suggest that the intuitive aspects of independence are better captured by referring to just one (precise) probability than to a family (such as that one singling-out a lower or upper probability).

International audience; We introduce the definition of a conditional possibility (and a conditional necessity by duality) as a primitive concept, ie a function whose domain is a set of conditional events. The starting point is a definition of conditional event E|H which differs from many seemingly “similar” ones adopted in the relevant literature, which makes the third value depending on E|H. It turns out that this function t(E|H) can be taken as a conditional possibility by requiring “natural” property of closure of truth-values of the conditional events with respect to max and min. We show that other definitions of conditional possibility measures, present in the literature, are particular cases of the one proposed here. Moreover, we introduce a concept of coherence for conditional possibility and a relevant characterization theorem, given in terms of a class of unconditional possibility measures.

One of the essential tasks in the teaching of a programming language is the introduction of the main constructs of the language—the syntax and semantics of statements, control, procedures, file handling, and so on. Most teachers present this topic by simulating the operation of the computer on the chalkboard by manual tracing of the execution of sample demonstration programs. Upon closer inspection it becomes clear that this task could be performed by the computer itself. A program implementing this idea would help the teacher in the class and the student (who could use the computer to repeat the instructor's performance at any time and on any problem of his choice) both in the class and in study. This article describes a collection of programs called PMS that performs exactly this function for the teaching of Pascal. PMS is organized as a collection of “mimlanguages” each of which demonstrates, and allows the user to experiment with, a certain category of Pascal features using a particular screen representation. Each of the subsystems contains a syntax-driven, screen-oriented translator associated with an interpreter that controls execution and the appropriate screen representation. This modular structure of PMS reflects the fact that programming is always taught by presenting certain essentially disjoint language features individually. In addition to their logical independence, these disjoint concepts also require different graphical approaches for their visual representation and this reinforces the need for modularization of the package. PMS is written in Pascal and runs on several computers including the IBM PC.

In describing the position of human ecology within the system of sciences, Robert Park was concerned with establishing the logical independence of ecology from geography. He used arguments of neo-Kantian philosophy, and to a lesser exten Comtean positivism, to accomplish his goal. The study of the philosophical context of Park's arguments on human science suggests that his views were more compatible with those of the regional geographers of his period than with those of the urban and social geographers who rediscovered his work.

The notion of absolute independence, considered in this paper has a clear algebraic meaning and is a strengthening of the usual notion of logical independence. We prove that any consistent and countable set in classical prepositional logic has an absolutely independent axiornatization.

For any binary operation, four alternatives exist. It could be (i) both commutative and associative; (ii) neither commutative nor associative; (iii) commutative but not associative; (iv) associative but not commutative. The basic arithmetical operations provide examples for the first two possibilities. This paper presents elementary examples for the last two possibilities. It is claimed that the study of these examples is extremely important in order to understand the logical independence of commutativity and associativity.

Consider a pipeline system, operating in a multitasking environment, with the following property: during the execution only one instruction of each program in the ready status is present in the pipeline stream. In this way we obtain the logical independence between the instructions being executed in the pipeline. A simplified schema of an architecture based on the above idea is presented here. The system has an high degree of modularity and is particularly suited for multitasking applications.

The FORCASE construct1 is a programming language control structure which specifies a number of independent tasks, called caselimbs, to be done in parallel. This construct has been implemented to run on a sequential computer by executing the tasks one after another. In order to maintain logical independence of the tasks, a memory management scheme is employed to guard against multiple reassignments of the same memory location by more than one caselimb. Whenever a FORCASE is begun, three copies of the run-time stack are created: the original, the update, and the caselimb copies. Each caselimb is handed a copy of the original stack and executed to completion. Then, each memory word in the stack is compared against the other two copies, and when all three copies disagree, a data integrity error is signaled. A FORCASE stack is used to store the original and update copies so that nested occurrences of FORCASE can operate successfully.

Publisher Summary This chapter focuses on large-scale parallel computers. The rapid evolution of microelectronic technology is making it possible to put hundreds of thousands, and eventually millions, of logically active circuit elements on a single silicon chip. The dispatching hardware examines upcoming items in the instruction stream for logical independence, and can dispatch multiple instructions simultaneously when they are found to be independent. Machines of this kind capable of dispatching as many as half a dozen instructions during a single clock cycle have been designed. If it proves possible to define a significant collection of computational primitives to which constrained data-motion patterns of this type can be applied, and if one can find smooth ways of integrating these primitives into general purpose computing environments, then the great economic advantages of special-purpose chip design may have a major impact on large-scale scientific computation.

Publisher Summary This chapter discusses independence versus logical independence in the countable case. The chapter explains that not every logically infinite countable set l is equivalent to an independent set l and presents a characterization and several results for those which are. The results are restricted to countable theories. The chapter describes a theorem that if l is a logically set of sentences of T and some l chain has the arbitrary interpolation property, T is equivalent to an independent set l over T and the proof explores that the idea is particularly concrete. Once proved, it is clear that it is the order properties of the endpoints and not the metric properties used in the proof that are essential to the proof of Lemma 2, which is true for an homeomorphic collection of finite-disjoint unions of intervals, D n , in which the endpoints stand in the same order relation as those of the C n.