A Logic for Multiple-source Approximation Systems with Distributed Knowledge Base

The theory of rough sets starts with the notion of an approximation space, which is a pair (U,R), U being the domain of discourse, and R an equivalence relation on U. R is taken to represent the knowledge base of an agent, and the induced partition reflects a granularity of U that is the result of a lack of complete information about the objects in U. The focus then is on approximations of concepts on the domain, in the context of the granularity. The present article studies the theory in the situation where information is obtained from different sources. The notion of approximation space is extended to define a multiple-source approximation system with distributed knowledge base, which is a tuple (U,R_P)_{Pßf N}(U,R_P)_{P\ss_f N}, where N is a set of sources and P ranges over all finite subsets of N. Each R _P is an equivalence relation on U satisfying some additional conditions, representing the knowledge base of the group P of sources. Thus each finite group of sources and hence individual source perceives the same domain differently (depending on what information the group/individual source has about the domain), and the same concept may then have approximations that differ with the groups. In order to express the notions and properties related with rough set theory in this multiple-source situation, a quantified modal logic LMSAS ^D is proposed. In LMSAS ^D , quantification ranges over modalities, making it different from modal predicate logic and modal logic with propositional quantifiers. Some fragments of LMSAS ^D are discussed and it is shown that the modal system KTB is embedded in LMSAS ^D . The epistemic logic S5^D_nS5^D_n is also embedded in LMSAS ^D , and cannot replace the latter to serve our purpose. The relationship of LMSAS ^D with first and second-order logics is presented. Issues of expressibility, axiomatization and decidability are addressed.     

The adequacy of the theories of deductive systems with respect to sentential calculi

Summary The sentential calculiR, under discussion, are axiomatizable and implication is among their primitive terms. The modus ponens and the rule of substitution are their primitive rules. ByS _r is denoted the set of sentences obtained from the formulae of the calculusR by substituting sentences of a given language for all variables. The variablesx, y, z ... represent the elements of the setS _r , the variablesX, Y, Z ... represent the subsets ofS _R . The formulacxy designates an implication withx as its antecedent andy as its consequent,cxy is always an element ofS _R δ(X) means, thatX is closed with respect to the modus ponens rule.A _R designates the class of allS _R -substitutions of the axioms of the sentential calculusR.     

Kant,the Third Antinomy and Transcendental Arguments

In this paper, I consider whether a reading of Kant's solution to the Third Antinomy can offer material for devising a new model of transcendental argument. The problem that this form of argument is meant to address is an antinomy between two apparently contradictory claims, q and ¬q, where we seem equally justified in holding both. The model has the following form: p; q is a necessary condition of p; the only justification we have for q is that it is a necessary condition of p; p is justified only in domain X (where X is a domain of objects of cognition); therefore, q is justified only in domain X. Because the argument shows that our justification for q is valid only in X, it also establishes that there is conceptual space to hold ¬q outside of X.     

A new puzzle about belief and credence

I present a puzzle about belief and credence, which takes the form of three independently supported views that are mutually inconsistent. The first is the view that S has a modal belief that p (e.g., S believes that probably-p) if and only if S has a corresponding credence that p. The second is the view that S believes that p only if S has some credence that p. The third is the view that, possibly, S believes that p without a modal belief that p. [Word Count: 85]     

Thermal stability and glass-forming ability of Se80− x Te20Ag x (x = 0, 3, 5, 7 and 9) chalcogenide glasses

The thermal stability and glass-forming ability (GFA) of Se_{80? x} Te₂₀Ag _x (x?=?0,?3,?5,?7 and 9) chalcogenide glasses have been investigated using differential scanning calorimetry (DSC). The DSC runs have been taken at five different heating rates (10, 20, 30, 40 and 50?K/min) under non-isothermal condition. The thermal stability and GFA are monitored through determination of the temperature difference ΔT?=?T _c???T _g, where T _c is the onset crystallization temperature, T _g is the glass transition temperature, H ^l is the stability parameter, ΔH _c is the enthalpy released during crystallization and F _i is the fragility index. The activation energy of crystallization E _c and crystallization rate factor K have also been determined as indicators of the thermal stability of the above-mentioned samples. It is found that Se₇₁Te₂₀Ag₉ is the most stable among all the samples of the series.     

Fractal growth of graphite nodules in iron

It has been found from a large number of statistical tests that graphite nodules in malleable iron grow with time according to a power law. The growth of the nodules and their fractal dimension have been investigated experimentally as a function of annealing time. Based on the assumption of carbon-diffusion-controlled growth in the initial stage, the growth equation is R _G = K ₁ t ^1/(D-1), where R _G is the radius of the graphite nodules, t the time, D the fractal dimension and K ₁ a constant. Assuming cementite-dissolution-controlled growth in the later stages of growth, the relation is R _G = K ₃ t ^3/D , where K ₃ is a constant. The fractal dimension, or the aggregate state of the graphite nodules, strongly influences the growth process.     

On the correlation of a naturally and an artificially dichotomized variable

A method is suggested for estimating the correlation of a naturally (X) and an artificially (Y) dichotomized variable. It is assumed that a normal random variable (L) underlies the artificially dichotomized variable. The proposed correlation coefficient recovers the product moment correlation coefficient between X and L from a fourfold table of X and Y. The suggested correlation coefficient ν is contrasted with the phi correlation and the biserial η. The biserial η was proposed by Karl Pearson and is conceptually related to the new correlation coefficient. However, in addition, Pearson's biserial η invokes the assumption that the marginal distribution of L is normal, which contradicts its basic assumptions and thus does not recover the true correlation of L and X. Finally, an approximation is provided to simplify the calculation of ν and its standard error.     

The Rāga Bhāva in the Sāṁkhya Kārikā: Rectifying an Age-Old Mistake

The last foot of the 23rd verse of the Sā?khya Kārikā (SK)—‘tāmasam asmāt viparyastam’—is in need of reinterpretation. Prevailing interpretations are generally based on the primary meaning of the verse. In that sense, it is understood as a declaration of the four tāmasika bhāva that are contrary to the sāttvika ones. Taking the primary meaning of the verse is problematic because it contradicts the gu?a-bhāva coherence required by the doctrine of satkārya. The doctrine of satkārya is one of the foundational principles of Sā?khya. The avirāga or rāga bhāva shows coherence to rajas rather than tamas. I show that the verse needs to be interpreted by taking the secondary meaning. Accordingly, avirāga or rāga is established as a bhāva of rajas rather than tamas. Further, I also show that the idea of bhāva in the Sā?khya Kārikā and the Bhagavadgītā may be closely related.     

The logic of ability,freedom and responsibility

The aim of this paper is to offer a rigorous explication of statements ascribing ability to agents and to develop the logic of such statements. A world is said to be feasible iff it is compatible with the actual past-and-present. W is a P-world iff W is feasible and P is true in W (where P is a proposition). P is a sufficient condition for Q iff every P world is a Q world. P is a necessary condition for Q iff Q is a sufficient condition forP. Each individual property S is shown to generate a rule for an agent X. X heeds S iff X makes all his future choices in accordance with S. (Note that X may heed S and yet fail to have it). S is a P-strategy for X iff X's heeding S together with P is a necessary and sufficient condition for X to have S. (P-strategies are thus rules which X is able to implement on the proviso P).Provisional opportunity: X has the opportunity to A provided P iff there is an S such that S is a P-strategy for X and X's implementing S is a sufficient condition for X's doing A. P is etiologically complete iff for every event E which P reports P also reports an etiological ancestry of E, and P is true. Categorical opportunity: X has the opportunity to A iff there is a P such that P is etiologically complete and X has the opportunity to A provided P. For X to have the ability to A there must not only be an appropriate strategy, but X must have a command of that strategy. X steadfastly intends A iff X intends A at every future moment at which his doing A is not yet inevitable. X has a command of S w.r.t. A and P iff X's steadfastly intending A together with P is a sufficient condition for X to implement S. Provisional ability: X can A provided P iff there is an S such that S is a P-strategy for X, X's implementing S is a sufficient condition for X's doing A, and X has a command of S w.r.t. A and P. Categorical ability: X can A iff there is a P such that P is etiologically complete and X can A provided P. X is free w.r.t. to A iff X can A and X can non- A. X is free iff there is an A such that X is free w.r.t. A.     

Knowledge despite falsehood

I examine the claim, made by some authors, that we sometimes acquire knowledge from falsehood. I focus on two representative cases in which a subject S infers a proposition q from a false proposition p. If S knows that q, I argue, S's false belief that p is not essential to S's cognition. S's knowledge is instead due to S's belief that p′, a proposition in the neighbourhood of p that S (dispositionally) believes (and knows). S thus knows despite her false belief. The widely accepted and plausible principle that inferential knowledge requires known premises is unscathed.     

Size effects and Hall–Petch relation in polycrystalline cobalt

The mechanical behaviour of polycrystalline hexagonal close-packed cobalt was investigated over a large range of grain size d in order to examine the occurrence of size effects. Crystallographic texture and amount of face centred cubic allotropic phase were maintained unchanged thanks to appropriate heat treatment procedures. The Hall–Petch (HP) relation exhibits two distinct behaviours from the very beginning of plastic strain levels. The conventional HP law is fulfilled for a number of grains across the thickness t higher than a critical value (t/d)_c = 14. For t/d lower than (t/d)_c, a multicrystalline regime is evidenced highlighting a strong reduction in flow stress. The high value of (t/d)_c is related to the low-stacking fault energy of cobalt in the basal plane. The size effect is predominant in the first work hardening stage where slip mechanisms and stacking faults predominate. In the second stage, driven by mechanical twinning processes, this effect is less sensitive. Finally, the size effect could also affect the end of the elastic stage, in link with nonlinear elasticity mechanisms.     

B-varieties with normal free algebras

The starting point for the investigation in this paper is the following McKinsey-Tarski's Theorem: if f and g are algebraic functions (of the same number of variables) in a topological Boolean algebra (TBA) and if C(f)C(g) vanishes identically, then either f or g vanishes identically. The present paper generalizes this theorem to B-algebras and shows that validity of that theorem in a variety of B-algebras (B-variety) generated by SCI _B -equations implies that its free Lindenbaum-Tarski's algebra is normal. This is important in the semantical analysis of SCI _B (the Boolean strengthening of the sentential calculus with identity, SCI) since normal B-algebras are just models of this logic. The rest part of the paper is concerned with relationships between some closure systems of filters, SCI _B -theories, B-varieties and closed sets of SCI _B -equations that have been derived both from the semantics of SCI _B and from the semantics of the usual equational logic.To the memory of Jerzy Supecki     

In situ simultaneous measurements of photodarkening and photoinduced defect creation in amorphous As2Se3 films

To understand the direct correlation between photodarkening (PD) and photoinduced defect creation (PDC) observed in amorphous chalcogenides, in situ simultaneous measurements of PD and photocurrent (PC) have been performed on amorphous As₂Se₃ films. The time evolution of PD and PDC during light excitations are empirically described by a stretched exponential function; 1 ??exp[?(t/τ) ^β ], where τ is the effective response time and β the dispersion parameter. The value of τ for the PDC is very much smaller than that for the PD, suggesting that there is no direct correlation between the two.     

Bradley — Terry — Luce models to incorporate within‐pair order effects: Representation and uniqueness theorems

The paper presents different representation theorems for the Bradley — Terry — Luce (BTL) models of Beaver and Gokhale and of Davidson and Beaver. In particular, algorithms that can be used in constructing BTL scales are provided. The uniqueness theorems show that the Davidson — Beaver model should be preferred to the Beaver — Gokhale model since the multiplicative order effect parameter is uniquely determined whereas the additive effect parameter is merely a ratio scale. Finally, a relationship to the simple BTL model is established. Let p(a, b) denote the probability that a is chosen when (a, b) is presented in a fixed order. Then the probabilities p(a, b) satisfy the Beaver — Gokhale model if and only if the balanced probabilities p_b(a, b):= ½ (p(a, b) + 1–p (b, a)) satisfy the simple BTL model.     

Axiomatic world theory: An overview the general theory of evolution in brief

The world, its many subsystems and all their theories, starting with logic, can be reduced to two related functions: a combinatorial system generator and a hamiltonian system organizer. These can be derived, in turn, from an Axiom of Lawfulness, the expansion being guided by pseudo‐category and pseudo‐functor analysis to produce an axiomatic theory of the world or general theory of evolution. Specifically, world evolution is generated by a constrained combinatorial world generator, F:G_(X), deduced from two related axioms: I. The Axiom of World Lawfulness and II. The Axiom of World Constraint Constants, c = c₁, c₂, of primordial physical combinatee (substance), c₁, and physical combinator (motion), c₂.

Axiom I postulates a lawful analysis by an analyzer adhering to appropriate coordinate systems, CS, of a lawful analysand obeying a conservation law, X = X. The analysand consists of a base combinatee (the set and elements), X = {x₁, x₂,… x_n}, and a base combinator, namely, the universal Boolean operator, NOR = NOT + OR. Base combinatee and combinator both have attributes of quantity combinatorially generated by NOR operating on the universal number, 1, and of quality generated by NOR operating on the universal dimensions, MLT (mass, length, time), including the null sets.

Axiom II fixes the base constants, c, = c₁, c₂, thereby converting X to material substance using c₁ and NOR to material motion using c₂. This comprehensive, quality and quantity‐competent foundational science is called Universal Combinatorics. Its elements comprise the logical alphabet or metavector, A = {c, 1, MLT; X, NOR}, where c is obtained from the remaining terms. These give: (1) the attributive pseudo‐functor, F = P_(c,1,MLT), where P is the power set of the indicated attributes, and (2) the logic generator, G_(X), where G = NOR_(NOR). F then maps G_(X) into world evolution, F:G_(X) → world evolution, as follows:

Expanding the abstract generator, F₁:G,_(x), with world constants eliminated, i.e., c = 0, generates Universal Grammar consisting of (1) the substantive content of the abstract science chain running from linguistic grammar to mathematics and logic and (2) a comprehensive epistemology equivalent to an explicit theory of the strategic aspects of the scientific method, including a universal hamiltonian theory structure informally related to a mathematical category. The four epistemological theorems are:

1. I. The Combinatorial System Generator, F:G_(X), (read as “The attributive functor, F, maps the logic generator, G_(X), into world theory” or “The world is an attributive combinatorial function of logic").

2. II. The Hamiltonian System Operating Theorem, h (an abstract theory‐category structure).

3. III. The System Stability Theorem, PI?, where PI is the extremal Performance Index or controlling law.

4. IV. The Intersystem Abstraction Ranking Theorem given by the Attributive Functor/ Function, F.

F₂ admits the world constants, c > 0, to materialize the grammar generator, G_(X), to an homologous concrete Euler combinatorial physical wave generator, namely, the superstring equation of quantum theory, E_(NI) = A(σ,τ), where E is the permutational function, NI, is the set of nonintegers and the solution is the dual amplitude, σ,τ. Expanding generates the elementary particles of nonadaptive physics and, by inference, the substantive content of Universal Physics consisting of three additional primary systems comprising the world, where a primary system is defined as one having a distinct but derivative extremal controlling law:

1. I. Nonadaptive physics and chemistry (harmonic hamiltonian wave systems) : Minimize Action, subject to conservation constraints.

2. II. Adaptive physics or biology (membrane bound duplicating polymer‐copolymer hamiltonian systems) : Maximize Survival, subject to energy constraints.

3. III. Sentient physics or sociopsychology (neuromatrix hamiltonian systems) : Maximize subjective Happiness, subject to survival constraints and

4. IV. Representational physics or language (a symbolic combinatorial routine): Maximizes the Information Gain, subject to happiness constraints.

The world can then be viewed as a perpetual superfluid computer implicitly using the epistemology of Axiom I as a world program to process the physical data base, c > 0, of Axiom II into world evolution. After evolving through Systems I and II, mankind, i.e., System III, evolves as an internal metacomputer which makes the combinatorial program explicit and uses it to put all four primary systems in standard hamiltonian theory (pseudo‐category) format and terminology. This can be viewed as a generalization of the Darwinian variation‐and‐selection theme in which combinatorial‐variation is recursively hamiltonian‐selected thereby incrementing world logic and logic constraints on successive primary systems. Because Universal Physics and Universal Grammar are functor‐related homologous concrete and abstract combinatorial pseudocategories, related by a pseudo‐functor, thus, differing only in the presence and absence, respectively, of the World Constants, c ≥ 0, they constitute, ipso facto, Universal Science (Formal Philosophy, World Evolution, World Unification, Explicit Theory of Everything, ETOE, or Axiomatic World Theory).

QED: Because intricate verified predictions, ranging from particles to personality types, mental disorders, political parties and the abstract sciences, result from a system which is merely expanding to fill its possibility set, it is concluded that the world is lawful and that this means it is an object deterministic but not fully analytically determinable combinatorial system. In the object domain, the world is system‐number complete at four. Dually, in the analytical codomain, understanding of it is approximately complete, as measured by a world information gain function. Hence, the dualistic, analysand‐analyzer world program is finite and has dualistic completion criteria, as required of an involuted program.     

Explaining after by before: Basic aspects of a dynamic systems approach to the study of development

The basic properties of a dynamic systems approach of development are illustrated by contrasting two simple equations. One, y_t+1 = f (y_t), is characteristic of dynamic systems models. The other, y_i = f (x_i), refers to what, for the sake of simplicity, is referred to as the standard developmental approach. We give illustrations from cognitive, language and social development to show the characteristic differences of these two types of models and show their complementarity. The article further compares the “Bloomington” with the “Groningen” approach to dynamic systems theorizing in developmental psychology. It continues with a discussion of two important questions. One involves the issue of measurement and the nature of developmental variables from the viewpoint of dynamic systems. The second concerns the question of short- and long-term time scales in developmental models, which is discussed on the basis of an example, namely dyadic interaction of young children in the context of different social statuses.     

Restricted ranking

This paper is a study of certain aspects of restricted ranking, a method intended for use by a panel ofm judges evaluating the relative merits ofN subjects, candidates for scholarships, awards, etc. Each judge divides theN subjects intoR classes so thatn _i individuals receive a gradei (i = 1, 2, ...,R; Σn _i =N) where theR numbersn _i are close toN/R (n _i =N/R whenN is divisible byR) and are preassigned and the same for all judges. This method is superior in several respects to other likely alternatives. Under the null hypothesis that allnR =N subjects are of equal merit, four tests of significance are developed. The effectiveness of the method is investigated both theoretically by means of the asymptotic relative efficiency and more generally by simulation studies. When the numbersn _i are not restricted to values close to or equal toN/R but instead are given values conforming to a normally distributed pattern, the resulting method is known as theQ-sort, so designated by certain investigators in psychotherapy. The simulation studies reveal that restricted ranking is only slightly inferior to complete ranking and generally superior in the cases considered to theQ-sort, although there are likely to be other situations when the latter is superior. The authors are indebted to Dr. Clyde Kramer of Virginia Polytechnic Institute for bringing theQ-sort ranking technique of psychotherapy to their attention.     

The mental representation of causal conditional reasoning: mental models or causal models

In this paper, two experiments are reported investigating the nature of the cognitive representations underlying causal conditional reasoning performance. The predictions of causal and logical interpretations of the conditional diverge sharply when inferences involving pairs of conditionals—such as if P₁then Q and if P₂then Q—are considered. From a causal perspective, the causal direction of these conditionals is critical: are the P_icauses of Q; or symptoms caused byQ. The rich variety of inference patterns can naturally be modelled by Bayesian networks. A pair of causal conditionals where Q is an effect corresponds to a “collider” structure where the two causes (P_i) converge on a common effect. In contrast, a pair of causal conditionals where Q is a cause corresponds to a network where two effects (P_i) diverge from a common cause. Very different predictions are made by fully explicit or initial mental models interpretations. These predictions were tested in two experiments, each of which yielded data most consistent with causal model theory, rather than with mental models.     

On a Contraction-Less Intuitionistic Propositional Logic with Conjunction and Fusion

In this paper we prove the equivalence between the Gentzen system G _LJ*\c , obtained by deleting the contraction rule from the sequent calculus LJ* (which is a redundant version of LJ), the deductive system IPC*\c and the equational system associated with the variety RL of residuated lattices. This means that the variety RL is the equivalent algebraic semantics for both systems G _LJ*\c in the sense of [18] and [4], respectively. The equivalence between G _LJ*\c and IPC*\c is a strengthening of a result obtained by H. Ono and Y. Komori [14, Corollary 2.8.1] and the equivalence between G _LJ*\c and the equational system associated with the variety RL of residuated lattices is a strengthening of a result obtained by P.M. Idziak [13, Theorem 1].An axiomatization of the restriction of IPC*\c to the formulas whose main connective is the implication connective is obtained by using an interpretation of G _LJ*\c in IPC*\c.