A system that looks for axioms

Besides the physicomathematical controversy concerning the ‘phenomenological justification’ and the specific formal structure of the calculus known as quantum logic (QL), there is a philosophical controversy concerning whether this ‘calculus of experimental propositions’ is properly speaking a logic rather than simply an algebraic structure only analogous to logic properly so called. Directly associated with this controversy is the issue of whether logic is an empirical science on the par with, e.g., physical geometry. This article is not concerned, however, with the general question of the empirical character of logic, either classical or quantal; it is rather concerned only with one seemingly decisive objection against regarding QL as logic, namely, the objection advanced by Jauch and Piron (1970, p. 176), who argue that since QL lacks an essential feature of logic — viz., a deduction scheme — it is `very questionable whether we may properly call the lattice of general quantum mechanics a logic.’ Their argument is based on what they regard to be the general failure of the lattice of QL propositions to admit a material implication connective or conditional operation by means of which the modus ponens deduction scheme can be incorporated into QL. The view of Jauch and Piron is also supported by Greechie and Gudder (1971, 1973), who examine a number of results which suggest that no reasonable material conditional exists in QL.

Quantum logic originated with the Birkhoff-von Neumann paper of 1936.2 The authors argued that the transition from classical to quantum mechanics involves the transition from a propositional calculus forming a Boolean algebra to a propositional calculus which constitutes “a sort of projective geometry”.* For von Neumann, the significance of this thesis for the foundational problems of the theory lay in the possibility of regarding the projective structure as the logic of a quantum set theory -- a non-Boolean theory of sets for properties of quantum mechanical systems in which dimensionality plays the role of cardinality and transition probabilities between events are induced by automorphisms of the logic.** Thus, probabilities are logical: they are not defined, as in the classical case, by measures over events represented as sets of maximal truth value assignments, parametrized by the points of a phase space. Indeed, the quantum sets have no “points” in the general case; there are no “hidden variables” underlying the statistics associated with a quantum logic or quantum set theory. This is a radically different account of the “irreducibility” of the quantum statistics from that provided by the Copenhagen interpretation.

To verify the correctness of Python programs, a novel approach for model checking Python programs with MSVL (Modeling, Simulation and Verification Language) is advocated. To this end, the rules for decoding the object-oriented semantics of Python with the process-oriented semantics of MSVL are defined, and the technique for automatically rewriting a Python program into its equivalent MSVL program is formalized, which in turn can be verified with the model checking tool MSV. In addition, an example is given to illustrate how the approach works. The approach fully utilizes the powerful expressiveness of MSVL to verify Python programs in a direct way, and helps to improve the quality of the software system.

An approach to quantum physics (QP) is proposed that is characterized by the attempt to give up the verificationist theory of truth underlying the standard interpretation of QP. As a first step, anobservatively minimal language L is constructed that is endowed with a Tarskian truth theory. Then, a set of axioms is stated by means of L that hold both in classical physics and in QP, and the further language Le of all properties is constructed. The concepts ofmeaning andtestability do not collapse in L and Le, hence quantum logic is interpreted as a theory of testability in QP, and QP turns out to be semantically incomplete. Furthermore, semantic and pragmatic compatibility of physical properties are distinguished in Le, and the concepts of testability and conjoint testability of statements are introduced. In this context some known quantum paradoxes can be avoided, and a new general principle (MGP) characterizes the truth mode of empirical physical laws. MGP invalidates the Bell theorem and, presumably, the Bell-Kochen-Specker theorem, and introduces apragmatic contextuality in QP in place of thesemantic contextuality that should occur because of these theorems.

BOOK REVIEWS Prior, A.N., Formal Logic. Oxford: The Clarendon Press, 1955. Pp. ix—329. $5.60. In the preface, the author indicates that this book is designed primarily as a textbook, though he hopes that it will prove to be of interest to others who are not Logic students. The content, which includes; Part I — The Classical Prepositional and Functional Calculi; Part II — The Traditional Logic of Terms; and Part HI — Modal, Three-valued and Extensional Systems ; is of such a nature that it should attract a variety of critico-analytical, perceptive readers other than those solely interested in Logic. Part I of the triparted arrangement commences with 'Truth-Functions and Tautologies' and is a systematic study of tautologous implications derived from fixed rules of transformation or inference and their relation to systems of tautologous alterations and conjunctions. In the axiomatizations of the Propositional Calculus to obtain a mechanical accuracy, a rigorously fixed symbolism is employed to express the laws laid down and established, as well as to represent the processes of proof. Using axioms and rules the author shows that these axioms are 'complete' and sufficient, not only to prove all tautologous formulae in the propositional calculus, but also to disprove all non-tautologous ones. Besides the fundamental axioms found in the Principia Mathematica of Whitehead and Russell, the newer and more economical ones derived by Lukasiewicz, and Hubert and Ackerman are employed. In answer to the question 'Why should the logician, whose primary interest is in inference, study other tautologous formulae beside those which are implications ?', the chapter entitled 'System with Propositional Constants and Functional Variables', is an attempt to show that all other tautologous formulae are either implications or definitional abbreviations for them. In defense of this thesis there are sections about The Implicational Calculus and Wajsberg's System in C and 0; Quine's Completeness Proof for the Full Calculus; Wajsberg's Completeness Proof for the Restricted Calculus; and Logic, Metalogic, and the Functorial Variables. Especially well expressed and explained in connection with the Theory of Quantification is the portion of the text on propositional operators, which form propositions, not out of other propositions, but out of names. The author illucidates the manner in which these new types of propositional formulae are introduced, and substituted for propositional variables in theses of the propositional calculus. With keen and astute logical acumen, he draws attention to the fact that there is a close resemblance (1) between the laws of equipollence and the principles now known as de Morgan Laws ; and (2) between universal quantification and conjunction, and existential quantification and alteration. 407 4o8Book-Reviews A. N. Prior's critical ability is especially pronounced in connection with 'Quantification and Equivalential Calculi: Definition Re-examined.' The work on extension, by the use of quantifiers and functional variables, of versions of the propositional calculus in which the only undefined truth-factor is E, 'If and only if,' includes the interesting note that Leániewski's special interest in a protothenic with an 'equivalential' basis arose from the personal view that he held about the nature of definition in formal systems. The author's account of definitions as assertions of equivalence include the contributions of Leániewski, Tarski and Sobocinski, who espoused the theory in its entirety, and of Lukasiewicz and Meredith, who partially adopted it. Despite the opinions of this distinguished company, the author clearly states that the conclusion of the account is wrong-headed and he shows that it makes it difficult to distinguish the definitions of a system from additional axioms. Indeed worthy of critical study, is Part II which is a truly remarkable synthesis of the Traditional Logic of Terms, that commences with the Aristotelian Syllogistic; is followed by a comprehensive treatment of Categorical Forms with Negative, Complex, and Quantified Terms; and concludes with Singular and Existential Propositions. When Professor Prior discusses Singulars and Generals in Traditional and Modern Logic, he explains with keen perception that there is a large element of anachronism in attributing to Aristotle any satisfactory way of interpreting his forms in the modern functional calculus and at the same time 'saving' all or most of his laws. The elaborate body of thought of...

Ten years ago Cheatham and Wegbreit [4] proposed a transformational program development methodology based on notions of top-down stepwise program refinement first expressed by Dijkstra [10] and Wirth [45]. A schema describing the process of this methodology is given in fig. 1. To develop a program by transformation, we first specify the program in as high a level of abstraction and as great a degree of clarity as our programming language admits. This high level problem statement program P is proved correct semimechanically according to some standard approach (see Flovd and Hoare [15, 21]), Next, using an interactive system equipped with a library of encoded transformations, each of which maps a correct program into another equivalent program, we select and apply transformations one at a time to successive versions of the program until we obtain a concrete, low level, effecient implementation version P'. The goals of transformational programming are to reduce programming labor, improve program reliability, and upgrade program performance. In order for labor to be reduced, the effort required to obtain P, prove it correct, and derive P' by transformation should be less than the effort required to code P from scratch, and also to debug it. Program reliability will be improved if P can be certified correct, and if each transformation preserves program meaning. Finally, program performance will be upgraded if transformations are directed towards increased efficiency.

High-level compiler transformations, especially loop transformations, are widely recognized as critical optimizations to restructure programs to improve data locality and expose parallelism. Guaranteeing the correctness of program transformations is essential, and to date three main approaches have been developed: proof of equivalence of affine programs, matching the execution traces of programs, and checking bit-by-bit equivalence of program outputs. Each technique suffers from limitations in the kind of transformations supported, space complexity, or the sensitivity to the testing dataset. In this paper, we take a novel approach that addresses all three limitations to provide an automatic bug checker to verify any iteration reordering transformations on affine programs, including non-affine transformations, with space consumption proportional to the original program data and robust to arbitrary datasets of a given size. We achieve this by exploiting the structure of affine program control- and data-flow to generate at compile-time lightweight checker code to be executed within the transformed program. Experimental results assess the correctness and effectiveness of our method and its increased coverage over previous approaches.

Composite steel and concrete columns have been used in the tall buildings due theirs high-resistance and the possibility to reduce cross sections when we compered composite columns with reinforced concrete columns. There are a lot of types of composite columns. We are concerned with columns, which are completely or partially concrete-encased steel members. In practice, a lot of composite columns are relatively slender and in design the second - order effects will usually need to be included. A partially concrete encased steel cross-section was selected for laboratory tests of composite columns. According to the results of the experiments (total of 18 columns were tested in two series), we analyzed the effects of the second - order theory. The experimental results were compared with theoretical results obtained from the model developed in the non-linear software. The evaluation of the results is also shown in comparison with the general design method according to Eurocode 4, Design of composite steel and concrete structures - Part 1.1 General rules and rules for buildings.

§1. The set Cn(R, A + X) is the set of all expressions deducible, on the ground of formulas A, from the class X of formulas of the implicational propositional calculus, by means of rules belonging to the set R. In other words, Cn(R, A + X) is the set of expressions obtained from the set X by the help of a two-parameter function Cn[R, A]. The formula is called the classical deduction theorem.The classical deduction theorem is true for the system 〈R, A〉 (where R is the set of primitive rules, A is the set of axioms of the propositional calculus) if it holds for the function Cn[R, A].

An application of the theory of categorical many-valued partial orders to sets with fuzzy equalities

The H + CO --> HCO reaction has been characterized with correlation consistent basis sets at five levels of theory in order to benchmark the sensitivities of the barrier height and reaction ergicity to the one-electron and n-electron expansions of the electronic wave function. Single and multireference methods are compared and contrasted. The coupled cluster method RCCSD(T) was found to be in very good agreement with Davidson-corrected internally-contracted multireference configuration interaction (MRCI+Q). Second-order Moller-Plesset perturbation theory (MP2) was also employed. The estimated complete basis set (CBS) limits for the barrier height (in kcal/mol) for the five methods, including harmonic zero-point energy corrections, are MP2, 4.66; RCCSD, 4.78; RCCSD(T), 4.15; MRCI, 5.10; and MRCI+Q, 4.07. Similarly, the estimated CBS limits for the ergicity of the reaction are: MP2, -17.99; RCCSD, -13.34; RCCSD(T), -13.79; MRCI, -11.46; and MRCI+Q, -13.70. Additional basis set explorations for the RCCSD(T) method demonstrate that aug-cc-pVTZ sets, even with some functions removed, are sufficient to reproduce the CBS limits to within 0.1-0.3 kcal/mol.

Martin-Lof's intuitionistic type theory (ITT) can be regarded as a programming logic. Yet its data type aspect has not been much addressed. In the present paper we show how to introduce the data types in ITT in a systematic way. The main tool is data logic. Data logics are introduced in [12], which are many-sorted second order theories. For a given data type we can establish its data logic, i.e. a set of non-logical axioms, then from the axioms its rules in ITT can be obtained.

Based on recent experimental results, a molecular model of the ferroelectric ordering in the induced smectic-${C}^{*}$ phase is proposed in terms of a coupling between dipole and quadrupole ordering of chiral molecules. The analysis of the correlations between the value and sign of the spontaneous polarization and the molecular structure of a broad variety of chiral dopants indicates that several new experimental results cannot be explained in the framework of the existing theory of ferroelectric ordering. In particular, the realistic theory is expected to account for the qualitative difference between the properties of the two different types of dopants that possess a chiral center either in the flexible chains or in the rigid core, respectively. The general statistical theory of ferroelectric ordering induced by a chiral dopant in the nonchiral smectic-$C$ host phase is developed and used to obtain the expression for the spontaneous polarization in terms of the dopant molar fraction, quadrupole order parameter, molecular chirality, and the geometrical parameters that characterize the relative orientation of molecular electric and steric dipoles with respect to the molecular rigid core. The different kinds of polar interactions between chiral molecules, which can be responsible for the appearance of the spontaneous polarization, are discussed in detail. In the context of this model it is possible to explain qualitatively the difference in properties between the two types of chiral dopants used in the experiment. The results of the theory enable one to understand why the spontaneous polarization is sensitive to the molecular structure of the host phase only if the dopant molecule possesses a chiral center in the rigid core. In the context of the same model one explains also the opposite signs of the polarization induced by the same chiral dopant in the smectic-$C$ host phases with the two different orientations of the molecular dipole. The latter result enables one also to understand the origin of the recently observed polarization sign inversion induced by a change of concentration of the chiral dopant. The theoretical conclusions are supported by some recent experimental data presented in this paper.

Ferrous objects distort magnetic fields and can significantly increase magneto-inductive positioning errors in indoor environments. In this work, we use image theory in order to formulate an analytical channel model for the magnetic field of a quasi-static magnetic dipole positioned above a perfectly conducting half-space. The proposed model can be used to compensate for the distorting effects that metallic reinforcement bars (rebars) impose on the magnetic field of a magneto-inductive transmitter node in an indoor environment. Good agreement is observed between the analytical solution and numerical solutions obtained from 2-D finite element simulations when the transmitter node is located more than 0.4 m above the distorters. Experimental results indicate that the image theory model shows significant improvement over the free space dipole model in estimating position along the normal to the plane of the rebars, typically reducing positioning errors by 36% in 90% of the cases.

Migration sous l'effet d'un champ electrique de 110Ag et de 124Sb dans l'argent