The impossibility of a bivalent truth-functional semantics for the non-Boolean propositional structures of quantum mechanics

Summary The general fact of the impossibility of a bivalent, truth-functional semantics for the propositional structures determined by quantum mechanics should be more subtly demarcated according to whether the structures are taken to be orthomodular latticesP _L or partial-Boolean algebrasP _A; according to whether the semantic mappings are required to be truth-functional or truth-functional ; and according to whether two-or-higher dimensional Hilbert spaceP structures or three-or-higher dimensional Hilbert spaceP structures are being considered. If the quantumP structures are taken to be orthomodular latticesP _L, then bivalent mappings which preserve the operations and relations of aP _L must be truth-functional . Then as suggested by von Neumann and Jauch-Piron and as proven in this paper, the mere presence of incompatible elements in aP _L is sufficient to rule out any semantical or hidden-variable proposal which imposes this strong condition, for anytwo-or-higher dimensional Hilbert spaceP _L structure. Thus from the orthomodular lattice perspective, the peculiarly non-classical feature of quantum mechanics and the peculiarly non-Boolean feature of the quantum propositional structures is the existence of incompatible magnitudes and propositions. However, the weaker truth-functionality condition can instead be imposed upon the semantic or hidden-variable mappings on theP _L structures, although such mappings ignore the lattice meets and joins of incompatibles and preserve only the partial-Boolean algebra structural features of theP _L structures. Or alternatively, the quantum propositional structures can be taken to be partial-Boolean algebrasP _A, where bivalent mappings which preserve the operations and relations of aP _A need only be truth-functional (c). In either case, the Gleason, Kochen-Specker proofs show that any semantical or hidden variable proposal which imposes this truth-functionality (c) condition is impossible for anythree-or-higher dimensional Hilbert spaceP _A orP _L structures. But such semantical or hidden-variable proposals are possible for any two dimensional Hilbert spaceP _A orP _L structures, in spite of the presence of incompatibles in these structures, in spite of the fact that Heisenberg's Uncertainty Principle applies to the incompatible elements in these structures, and in spite of the fact that these structures are non-Boolean in the Piron sense. The present paper is a sequel of the proceedings of theSociety of Exact Philosophy annual meeting, published in our issue 9:2, pp. 187–278. — Ed. I am indebted to my supervisor, Dr. Edwin Levy, for many hours of helpful discussion on the drafts leading to this paper.     

Fitting the factor analysis model in ℓ1 norm

The well‐known problem of fitting the exploratory factor analysis model is reconsidered where the usual least squares goodness‐of‐fit function is replaced by a more resistant discrepancy measure, based on a smooth approximation of the ?₁ norm. Fitting the factor analysis model to the sample correlation matrix is a complex matrix optimization problem which requires the structure preservation of the unknown parameters (e.g. positive definiteness). The projected gradient approach is a natural way of solving such data matching problems as especially designed to follow the geometry of the model parameters. Two reparameterizations of the factor analysis model are considered. The approach leads to globally convergent procedures for simultaneous estimation of the factor analysis matrix parameters. Numerical examples illustrate the algorithms and factor analysis solutions.     

Differential Method of Characterizing Gait Strategies From Step Lengths and Frequencies: Strategy of Velocity Modulation

The differential method consists of the analysis of the variation of gait parameters length, frequency, and velocity with respect to their mean values, respectively, ΔL = L — L_m , Δf = f — f_m , and Δv = v – v_m , where L_m , f_m , and v_m represent the mean values of those parameters. Assuming that the strategy of modulation of velocity implies that L and f are functions of v and that statistical analyses of ratios ΔL/Δv and Δf/Δv have established that there is a very significant linear correlation, close to 1, between those ratios, the mathematical procedure allows one to determine the equation of step length, L = a · f + b · v + K, where a and b are the slope and the intercept of the linear regression and K is close to L_m . The equation was experimentally tested on 140 gait sequences performed by 6 participants and for gait velocities ranging from 0.6 to 2.2 m/s and was found to be very representative of all individual values. The differential method provides another way of using the derivative of velocity, v = L·f, to characterize the strategy of velocity modulation, which then permits one to determine the linear equation of velocity, v = f · L_m + L · f_m — L_m · f_m , and to show that the respective parts played by each parameter in the progression velocity are approximately equal. The author establishes the uniqueness of the different linear adjustments and discusses the differential method's own modes of use, that is, interindividually or globally.     

Formation of FeNi with L10-ordered structure using high-pressure torsion

FeNi with the L1₀-ordered structure is formed over an astronomical timescale in meteorites. In this study, severe plastic deformation using high-pressure torsion (HPT) is employed for the production of the L1₀ structure in the laboratory. Its formation is confirmed by X-ray diffraction analysis and transmission electron microscopy. Processing of elemental nanopowders by HPT is an effective method for the formation of the L1₀ phase, which is enhanced by the addition of Co to FeNi or annealing after HPT. The formation of the phase must be associated with enhanced diffusion through HPT.     

On Reduction Systems Equivalent to The Lambek Calculus with the Empty String

The paper continues a series of results on cut-rule axiomatizability of the Lambek calculus. It provides a complete solution of a problem which was solved partially in one of the author's earlier papers. It is proved that the product-free Lambek Calculus with the empty string (L ₀) is not finitely axiomatizable if the only rule of inference admitted is Lambek's cut rule. The proof makes use of the (infinitely) cut-rule axiomatized calculus C designed by the author exactly for this purpose.     

On the size of refutation Kripke models for some linear modal and tense logics

LetL be any modal or tense logic with the finite model property. For eachm, definer _L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr _L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr _L (m) for some linear modal and tense logicsL.     

Further test of a subjective probability model of social inference

Subjects read paragraphs suggesting that a fictitious person (O) either did or did not possess an attribute X. Then, they reported their beliefs that they would like O (P_L), that O possessed X (P_X), that O was intelligent (P_I) and that O was sarcastic (P_s). Other beliefs necessary to test the predictive accuracy of Wyer's subjective probability model of cognitive organization were also reported. The model accurately predicted P_L on the basis of beliefs about X (the attribute to which information about O directly pertained); this accuracy was greater than that obtained in an earlier study in which real persons were evaluated, and did not depend upon the favorableness of X. However, predictions of P_L based upon beliefs about O's intelligence overestimated obtained values, while predictions based upon beliefs about O's sarcasm underestimated these values; moreover, P_I and P_S were underestimated and overestimated, respectively, by beliefs about X. It was speculated that these discrepancies were due to temporary inconsistencies among beliefs resulting from the information presented which did not have time to be resolved before beliefs were reported. Changes in P_L, P_I, and P_S resulting from additional information about X were accurately predicted in all cases. Implications of the model for other formulations of social evaluation processes are noted.     

Interpretation(s) of elasticity in operant demand

This brief report provides an account of varying interpretations of elasticity (η ) in the operant demand framework. General references to “demand elasticity” have existed since the Exponential model of operant demand was proposed by Hursh and Silberberg (2008). This term has been used interchangeably with Essential Value (EV), P_MAX, and the rate of change constant α . This report provides an in-depth account of η and the various ways in which this metric has been used to interpret fitted demand functions. A review of relevant mathematic terms, operations associated with differentiating parameters, and worked solutions for η are provided for linear and nonlinear demand functions. The relations between η and EV, P_MAX, and α are described and explained in terms of their mathematical bases and recommendations are provided regarding their individual interpretation. This report concludes with recommendations for providing additional mathematical detail in published works and emphasizing a consistent use of terms when describing aspects of operant demand.     

Norms,Rules and Steadfast Love: Towards an Inclusive Theology of Intimacy

Abstract

This article argues that defenders of the traditional view that marriage is the necessary condition for sexual experience generally make two important mistakes. First, they underestimate how open to emergent forms marriage actually is. And second, they confuse marriage as a norm for sexual expression with marriage as a rule. This article attempts a simple clarification of this basic distinction and, in so doing, is able to throw considerable fresh light on the place of marriage in the Christian understanding of sexuality. First, there will be some reminders about how adaptable the institution of marriage has actually been and may continue to be. Second, attempts will be made to explain the differences between a norm and a rule. Finally some possibilities and consequences arising from this distinction will be explored. These will provide some promptings towards what is called here ‘an inclusive theology of intimacy’. Radical Christians are cautioned regarding their peremptory search for alternatives to marriage. People inside and outside the churches who reject marriage may be rejecting versions of it which, while undoubtedly historical, simply do not bring to light the possibilities released by what might be called ‘the marital norm’.     

DEMAND EQUATIONS FOR QUALITATIVELY DIFFERENT FOODS UNDER FIXED‐RATIO SCHEDULES: A COMPARISON OF THREE DATA CONVERSIONS

Concurrent schedules were used to establish 6 hens' preferences for three foods. The resulting biases suggested wheat was preferred over honey‐puffed and puffed wheat, and puffed wheat was the least preferred food. The hens then responded under fixed‐ratio schedules for each food in 40‐min (excluding reinforcer time) sessions, with the response requirement doubling each session until no reinforcers were received. At the smaller ratios, the less preferred the food, the faster the hens' overall response rates (mainly as a result of shorter postreinforcement pauses) and the more reinforcers they received. The relations between the logarithms of the number of reinforcers obtained (consumption) and the response ratio (price) were well fitted by curvilinear demand functions. Wheat produced the smallest initial consumption (ln L), followed by honey‐puffed and puffed wheat, respectively. The response requirement at which the demand functions predicted maximal responding (P_max) were larger for wheat than for the other foods. Normalizing consumption and price, as suggested by Hursh and Winger (1995), moved the data for the three foods towards a single demand function; however, the P_max values were generally largest for puffed wheat. The results of normalization, as suggested by Hursh and Silberberg (2008), depended on the k value used. The parameter k is related to the range of the data, and the same k value needs to be used for all data sets that are compared. A k value of 8.0 gave significantly higher essential values (smaller α values) for puffed wheat as compared to honey‐puffed wheat and wheat, and the P_max values, in normalized standard price units, were largest for puffed wheat. Normalizing demand by converting the puffed and honey‐puffed wheat reinforcers to wheat equivalents (by applying the bias parameter from the concurrent‐schedules procedure) maintained separate demand functions for the foods. Those for wheat had the smallest rates of change in elasticity (a) and, in contrast to the other analyses, the largest P_max values. Normalizing demand in terms of concurrent‐schedule preference appears to have some advantages and to merit further investigation.     

Don’t Ever Do That! Long-term Duties in <Emphasis Type="Italic">PD</Emphasis><Subscript><Emphasis Type="Italic">e</Emphasis></Subscript><Emphasis Type="Italic">L</Emphasis>

This paper studies long-term norms concerning actions. In Meyer’s Propositional Deontic Logic (PD _e L), only immediate duties can be expressed, however, often one has duties of longer durations such as: “Never do that”, or “Do this someday”. In this paper, we will investigate how to amend PD _e L so that such long-term duties can be expressed. This leads to the interesting and suprising consequence that the long-term prohibition and obligation are not interdefinable in our semantics, while there is a duality between these two notions. As a consequence, we have provided a new analysis of the long-term obligation by introducing a new atomic proposition I (indebtedness) to represent the condition that an agent has some unfulfilled obligation. Presented by Jacek Malinowski     

Inductivism and probabilism

Synopsis I I set out my view that all inference is essentially deductive and pinpoint what I take to be the major shortcomings of the induction rule.II The import of data depends on the probability model of the experiment, a dependence ignored by the induction rule. Inductivists admit background knowledge must be taken into account but never spell out how this is to be done. As I see it, that is the problem of induction.III The induction rule, far from providing a method of discovery, does not even serve to detect pattern. Knowing that there is uniformity in the universe is no help to discovering laws. A critique of Reichenbach's justification of the straight rule is constructed along these lines.IV The induction rule, by itself, cannot account for the varying rates at which confidence in an hypothesis mounts with data. The mathematical analysis of this salient feature of inductive reasoning requires prior probabilities. We also argue, against orthodox statisticians, that prior probabilities make a substantive contribution to the objectivity of inductive methods, viz. to the design of experiments and the selection of decision rules.V Carnap's general criticisms of various estimation rules, like the straight rule and the impervious rule, are seen to be misguided when the prior densities to which they correspond are taken into account.VI Analysis of Hempel's definition of confirmation qua formalization of the enumerative (naive) conception of instancehood. We show that from the standpoint of the quantitative measure P(H/E):P(H) for the degree to which E confirms H, Hempel's classificatory concept yields correct results only for sampling at large from a finite population with a two-way classification all of whose compositions are equally probable. We extend the analysis to Goodman's paradox, finding cases in which grue-like hypotheses do receive as much confirmation as their opposite numbers. We argue, moreover, the irrelevancy of entrenchment, and maintain that Goodman's paradox is no more than a straightforward counter-example to the enumerative conception of instancehood embodied in Hempel's definition.VII We rebutt the objection that prior probabilities, qua inputs of Bayesian analysis, can only be obtained by enumerative induction (insofar as they are objective). The divergence in the prior densities of two rational agents is less a function of subjectivity, we maintain, than of vagueness.VIII Our concluding remarks stress that, for Bayesians, there is no problem of induction in the usual sense.     

Henkin Quantifiers and the Definability of Truth

Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension L _* ¹(H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close L _* ¹(H) with respect to Boolean operations, and obtain the language L ¹(H). At the next level, we consider an extension L _* ²(H) of L ¹(H) in which every sentence is an L ¹(H)-sentence prefixed with a Henkin quantifier. We repeat this construction to infinity. Using the (un)-definability of truth – in – N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996).     

A syntactical approach to modality

Conclusion The systems T _N and T _M show that necessity can be consistently construed as a predicate of syntactical objects, if the expressive/deductive power of the system is deliberately engineered to reflect the power of the original object language operator. The system T _N relies on salient limitations on the expressive power of the language L _N through the construction of a quotational hierarchy, while the system T _Mrelies on limiting the scope of the modal axioms schemas to the sublanguage L _infM ⁺, which corresponds exactly with the restrictive hierarchy of L _N. The fact that L _infM ⁺ is identical to the image of the metalinguistic mapping C ⁺ from the normal operator system into L _M reveals that iterated operator modality is implicitly hierarchical, and that inconsistency is produced by applying the principles of the modal logic to formulas which have no natural analogues in the operator development. Thus the contradiction discovered by Montague can be diagnosed as the result of instantiating the axiom schemas with modally ungrounded formulas, and thereby adding radically new modal axioms to the predicate system.The predicate treatment of necessity differs significantly from that of the operator in that the cumulative models for the predicate system are strictly first-order. Possible worlds are not used as model-theoretic primitives, but rather alternate models are appealed to in order to specify the extension of N, which is semantically construed as a first-order predicate. In this manner, the intensional aspects of modality are built into the mode of specifying the particular set of objects which the denotation function assigns to N, rather than in the specification of the basic truth conditions for modal formulas. Intensional phenomena are thereby localised to the special requirements for determining the extension of a particular predicate, and this does not constitute a structural modification of the first-order models, but rather limits the relevant class of models to those which possess an appropriate denotation function.     

Negative Equivalence of Extensions of Minimal Logic

Two logics L₁ and L₂ are negatively equivalent if for any set of formulas X and any negated formula ¬, ¬ can be deduced from the set of hypotheses X in L₁ if and only if it can be done in L₂. This article is devoted to the investigation of negative equivalence relation in the class of extensions of minimal logic.The author acknowledges support by the Alexander von Humboldt-StiftungPresented by Jacek Malinowski     

Understanding Public Reaction to Energy Proposals: An Application of the Fishbein Model1

The Fishbein attitude model was applied to voter decision-making on an energy ballot proposal. Questionnaires were sent to a random sample of potential voters in Oregon's 1976 general election and dealt with the Nuclear Safeguards Initiative, a measure that would place restrictions on future nuclear power plants. Questionnaire items probed the attitude toward the act of voting “Yes” on the measure (A_act), perceived likelihood of various consequences of voting “Yes” (B_i), evaluations of these consequences (e_i), the subjective norm (SN), normative beliefs (NB_i), motivation to comply with several referents (Mc_i), and voting intention (VI). A follow-up interview determined the actual voting behavior (VB) of persons responding to the questionnaire. The following model predictions were tested and strongly supported by the data: (a) VB=VI; and (e) external variables have a nonsignificant relation to VB once VI is partialled out. It was concluded that the Fishbein model should be extremely useful in understanding public reaction to future energy proposals.     

A model and algorithm for multidimensional scaling with external constraints on the distances

A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configurationX relative to the proximity dataP and the loss ofX relative to a pseudo-data matrixR. The matrixR is set up such that the side constraints to be imposed onX's distances are expressed by the relations amongR's numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solutionX. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matricesP andR. Further generalizations are introduced by substitutingR by a set ofR matrices,R _i ,i=1, ...r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time.     

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.