Rule Separation and Embedding Theorems for Logics Without Weakening

A full separation theorem for the derivable rules of intuitionistic linear logic without bounds, 0 and exponentials is proved. Several structural consequences of this theorem for subreducts of (commutative) residuated lattices are obtained. The theorem is then extended to the logic LR ⁺ and its proof is extended to obtain the finite embeddability property for the class of square increasing residuated lattices.     

The pseudodiagnosticity trap: Should participants consider alternative hypotheses?

The pseudodiagnosticity task has been used as an example of the tendency on the part of participants to incorrectly assess Bayesian constraints in assessing data, and as a failure to consider alternative hypotheses in a probabilistic inference task. In the task, participants are given one value, the anchor value, corresponding to P(D1|H) and may choose one other value, either P(D1|¬!H), P(D2|H), or P(D2|not;!H). Most participants select P(D2|H), or P(D2|¬!H) which have been considered inappropriate (and called pseudodiagnostic) because only P(D1|¬!H) allows use of Bayes' theorem. We present a new analysis based on probability intervals and show that selection of either P(D2|H), or P(D2|¬!H) is in fact pseudodiagnostic, whereas choice of P(D1|¬!H) is diagnostic. Our analysis shows that choice of the pseudodiagnostic values actually increases uncertainty regarding the posterior probability of H, supporting the original interpretation of the experimental findings on the pseudodiagnosticity task. The argument illuminates the general proposition that evolutionarily adaptive heuristics for Bayesian inference can be misled in some task situations.     

Admissibility of Cut in <Emphasis Type="Italic">LC</Emphasis> with Fixed Point Combinator

The fixed point combinator (Y) is an important non-proper combinator, which is defhable from a combinatorially complete base. This combinator guarantees that recursive equations have a solution. Structurally free logics (LC) turn combinators into formulas and replace structural rules by combinatory ones. This paper introduces the fixed point and the dual fixed point combinator into structurally free logics. The admissibility of (multiple) cut in the resulting calculus is not provable by a simple adaptation of the similar proof for LC with proper combinators. The novelty of our proof—beyond proving the cut for a newly extended calculus–is that we add a fourth induction to the by-and-large Gentzen-style proof. Presented by Robert Goldblatt     

A Cut-Free Sequent System for the Smallest Interpretability Logic

The idea of interpretability logics arose in Visser [Vis90]. He introduced the logics as extensions of the provability logic GLwith a binary modality . The arithmetic realization of A B in a theory T will be that T plus the realization of B is interpretable in T plus the realization of A (T + A interprets T + B). More precisely, there exists a function f (the relative interpretation) on the formulas of the language of T such that T + B C implies T + A f(C).The interpretability logics were considered in several papers. An arithmetic completeness of the interpretability logic ILM, obtained by adding Montagna's axiom to the smallest interpretability logic IL, was proved in Berarducci [Ber90] and Shavrukov [Sha88] (see also Hájek and Montagna [HM90] and Hájek and Montagna [HM92]). [Vis90] proved that the interpretability logic ILP, an extension of IL, is also complete for another arithmetic interpretation. The completeness with respect to Kripke semantics due to Veltman was, for IL, ILMand ILP, proved in de Jongh and Veltman [JV90]. The fixed point theorem of GLcan be extended to ILand hence ILMand ILP(cf. de Jongh and Visser [JV91]). The unary pendant "T interprets T + A" is much less expressive and was studied in de Rijke [Rij92]. For an overview of interpretability logic, see Visser [Vis97], and Japaridze and de Jongh [JJ98].In this paper, we give a cut-free sequent system for IL. To begin with, we give a cut-free system for the sublogic IL4of IL, whose -free fragment is the modal logic K4. A cut-elimination theorem for ILis proved using the system for IK4and a property of Löb's axiom.     

All Proper Normal Extensions of S5-square have the Polynomial Size Model Property

We show that every proper normal extension of the bi-modal system S5 ² has the poly-size model property. In fact, to every proper normal extension L of S5 ² corresponds a natural number b(L) - the bound of L. For every L, there exists a polynomial P(·) of degree b(L) + 1 such that every L-consistent formula is satisfiable on an L-frame whose universe is bounded by P(||), where || denotes the number of subformulas of . It is shown that this bound is optimal.     

Constructing a covariance matrix that yields a specified minimizer and a specified minimum discrepancy function value

A method is presented for constructing a covariance matrix Σ*₀ that is the sum of a matrix Σ(γ₀) that satisfies a specified model and a perturbation matrix,E, such that Σ*₀=Σ(γ0) +E. The perturbation matrix is chosen in such a manner that a class of discrepancy functionsF(Σ*₀, Σ(γ₀)), which includes normal theory maximum likelihood as a special case, has the prespecified parameter value γ₀ as minimizer and a prespecified minimum δ A matrix constructed in this way seems particularly valuable for Monte Carlo experiments as the covariance matrix for a population in which the model does not hold exactly. This may be a more realistic conceptualization in many instances. An example is presented in which this procedure is employed to generate a covariance matrix among nonnormal, ordered categorical variables which is then used to study the performance of a factor analysis estimator. We are grateful to Alexander Shapiro for suggesting the proof of the solution in section 2.     

Naming and Diagonalization, from Cantor to Godel to Kleene

We trace self-reference phenomena to the possibility of namingfunctions by names that belong to the domain over which thefunctions are defined. A naming system is a structure of theform (D, type( ),{ }), where D is a non-empty set; for everya D, which is a name of a k-ary function, {a}: D^k D is thefunction named by a, and type(a) is the type of a, which tellsus if a is a name and, if it is, the arity of the named function.Under quite general conditions we get a fixed point theorem,whose special cases include the fixed point theorem underlyingGödel's proof, Kleene's recursion theorem and many othertheorems of this nature, including the solution to simultaneousfixed point equations. Partial functions are accommodated byincluding "undefined" values; we investigate different systemsarising out of different ways of dealing with them. Many-sortednaming systems are suggested as a natural approach to generalcomputatability with many data types over arbitrary structures.The first part of the paper is a historical reconstruction ofthe way Gödel probably derived his proof from Cantor'sdiagonalization, through the semantic version of Richard. Theincompleteness proof–including the fixed point construction–resultfrom a natural line of thought, thereby dispelling the appearanceof a "magic trick". The analysis goes on to show how Kleene'srecursion theorem is obtained along the same lines.     

On a functional equation related to Thurstone models

If f and g are nonvanishing characteristic functions the functional equation g(s)g(t)g(?s ? t) = f(as)f(at)f(?as ? at) implies g(s) = e^ibsf(as), i.e., f and g corresponding to probability distributions of the same type. It is shown here that when f and g are allowed to vanish this equation also has solutions in which f and g correspond to distributions of different types. The practical implication is that there are nonequivalent. Thurstone models which cannot be discriminated by any choice experiment with three objects.     

A 2-categorial Generalization of the Concept of Institution

After defining, for each many-sorted signature Σ = (S, Σ), the category Ter(Σ), of generalized terms for Σ (which is the dual of the Kleisli category for \mathbb T_S{\mathbb {T}_{\bf \Sigma}}, the monad in Set ^S determined by the adjunction T_S \dashv G_S{{\bf T}_{\bf \Sigma} \dashv {\rm G}_{\bf \Sigma}} from Set ^S to Alg(Σ), the category of Σ-algebras), we assign, to a signature morphism d from Σ to Λ, the functor d_à{{\bf d}_\diamond} from Ter(Σ) to Ter(Λ). Once defined the mappings that assign, respectively, to a many-sorted signature the corresponding category of generalized terms and to a signature morphism the functor between the associated categories of generalized terms, we state that both mappings are actually the components of a pseudo-functor Ter from Sig to the 2-category Cat. Next we prove that there is a functor Tr^Σ, of realization of generalized terms as term operations, from Alg(Σ) × Ter(Σ) to Set, that simultaneously formalizes the procedure of realization of generalized terms and its naturalness (by taking into account the variation of the algebras through the homomorphisms between them). We remark that from this fact we will get the invariance of the relation of satisfaction under signature change. Moreover, we prove that, for each signature morphism d from Σ to Λ, there exists a natural isomorphism θ ^d from the functor Tr^L °(Id ×d_à){{{\rm Tr}^{\bf {\bf \Lambda}} \circ ({\rm Id} \times {\bf d}_\diamond)}} to the functor Tr^S °(d^* ×Id){{\rm Tr}^{\bf \Sigma} \circ ({\bf d}^* \times {\rm Id})}, both from the category Alg(Λ) × Ter(Σ) to the category Set, where d* is the value at d of the arrow mapping of a contravariant functor Alg from Sig to Cat, that shows the invariant character of the procedure of realization of generalized terms under signature change. Finally, we construct the many-sorted term institution by combining adequately the above components (and, in a derived way, the many-sorted specification institution), but for a strict generalization of the standard notion of institution.     

Nondecomposable conjoint measurement for bisymmetric structures

This paper discusses two “nondecomposable” conjoint measurement representations for an asymmetric binary relation ? on a product set A × X, namely (a, x) ? (b, y) iff f₁(a) + g₁(a)g₂(x) > f₁(b) + g₁(b)g₂(y), and (a, x) ? (b, y) iff f₁(a) + f₂(x) + g₁(a)g₂(x) > f₁(b) + f₂(y) + g₁(b)g₂(y). Difficulties in developing axioms for ? on A × X which imply these representations in a general formulation have led to their examination from the standpoint of bisymmetric structures based on applications of a binary operation to A × X. Depending on context, the binary operation may refer to concatenation, extensive or intensive averaging, gambles based on an uncertain chance event, or to some other interpretable process. Independence axioms which are necessary and sufficient for the special representations within the context of bisymmetric structures are presented.     

On two problems of Harvey Friedman

The paper considers certain properties of intermediate and moda propositional logics.The first part contains a proof of the theorem stating that each intermediate logic is closed under the Kreisel-Putnam rule xyz/(xy)(xz).The second part includes a proof of the theorem ensuring existence of a greatest structurally complete intermediate logic having the disjunction property. This theorem confirms H. Friedman's conjecture 41 (cf. [2], problem 41).In the third part the reader will find a criterion which allows us to obtain sets satisfying the conditions of Friedman's problem 42, on the basis of intermediate logics satisfying the conditions of problem 41.Finally, the fourth part contains a proof of a criterion which allows us to obtain modal logics endowed with Hallden's property on the basis of structurally complete intermediate logics having the disjunction property.Dedicated to Professor Roman SuszkoThe author would like to thank professors J. Perzanowski and A. Wroski for valuable suggestions.     

The Poznań school methodology of idealization and concretization from the point of view of a revised structuralist theory conception

My thesis is that some methodological ideas of the Pozna school, i.e., the principles of idealization and concretization (factualization), and the correspondence principle can be represented rather successfully using the relations of theoretization and specialization of revised structuralism.Let <n(i), t(j)> (i=1,...m, j=1,...k) denote the conceptual apparatus of a theory T, and a class M={} (i=1,...m, j=1,...k) the models of T. The n-components refer to the values of dependent variables and t-components to the values of independent variables of the theory. The n- and t-components in turn represent appropriate concepts. Consider T ^* as a conceptual enrichment of T with concepts <n(i ^*), t(j ^*)> (i<i ^* or j<j ^*) and models M ^*={<D ^*, n(i ^*), t(j ^*)>}. If the classes M and M ^* are suitably related, then the situation illustrates both the case of the theoretization-relation of (revised) structuralism and of the factualization-principle of the Pozna school.Assume now that the concepts n(i), t(j) of T for some i, j are operationalized using some special assumptions generating appropriate empirical values n and t for these concepts. Let M denote the class {<D,...n,...t,...>} which is formed by substituting n and t for values of concepts n(i), t(j) in the elements of M. If the classes M and M are related in a suitable way then the situation is an example of both the specialization-relation of (revised) structuralism and the concretization-principle of the Pozna school. The correspondence principle in turn can be represented as a limiting case of the theoretization-relation of (revised) structuralism.Many thanks to my anonymous referees for critical and fruitful comments and special thanks to Dr. Carol Norris for correcting the language of this paper.     

Continuous Orthogonal Complement Functions and Distribution-Free Goodness of Fit Tests in Moment Structure Analysis

It is shown that for any full column rank matrix X ₀ with more rows than columns there is a neighborhood $\mathcal{N}$ of X ₀ and a continuous function f on $\mathcal{N}$ such that f(X) is an orthogonal complement of X for all X in $\mathcal{N}$ . This is used to derive a distribution free goodness of fit test for covariance structure analysis. This test was proposed some time ago and is extensively used. Unfortunately, there is an error in the proof that the proposed test statistic has an asymptotic χ ² distribution. This is a potentially serious problem, without a proof the test statistic may not, in fact, be asymptoticly χ ². The proof, however, is easily fixed using a continuous orthogonal complement function. Similar problems arise in other applications where orthogonal complements are used. These can also be resolved by using continuous orthogonal complement functions.     

Vortex dynamics in La1.86Sr0. 14CuO4 studied by magnetoresistivity

Abstract

Vortex dynamics in La_1.86Sr_0.14CuO₄ have been studied by the measurement of ρ_c ^//i (T, H), where ρ_c ^//i is the c-axis resistivity for H//i (i = c or a-b). We argue that, at temperatures higher than the irreversibility temperature T _irr, the usual vortex picture breaks down owing to the thermal motion of vortices, resulting in a T- and T _in-dependent anisotropic parameter γ. After taking into account the dependence of γ on T and T _irr, we show that at each given temperature we can rescale the ρ_c ^//a-b (T, H) data onto the corresponding ρ_c ^//c (T, H) curves. This scaling property clearly indicates that the Lorentz-force-free mechanism is responsible for ρ_c ^//a-b (T, H). Furthermore, we also show that the measured ρ_c ^//a-b (T, H) data can be explained in terms of the recently developed extended Josephson coupling model which is verified by rescaling ρ_c ^//a-b (T) data for various fields onto a single curve.     

The kinetics of anodic oxide growth

Abstract

The process of anodic oxide growth has been investigated. It is shown that the obtained experimental results confirm the Fehlner-Mott oxidation model in the case of InSb, a typical compound of the 111-V group. We also present a kinetic theory of the anion migration in the formed oxide. The theory is based on the following assumptions. (a) The only species which moves through the formed oxide is the OH^? anion. (b) The chemical reactions of OH^? with In³⁺ and Sb³⁺ cations take place simultaneously and independently, in a thin layer of the oxide/InSb interface. The resulting solution of the kinetic equations describes the anion distribution in the formed oxide. A simple expression for the rate of oxidation is obtained. Another possible application of the presented theory is suggested.     

Lexicographic additive differences

Several authors have identified sets of axioms for a preference relation ? on a two-factor set A × X which imply that ? can be represented by specific types of numerical structures. Perhaps the two best-known of these are the additive representation, for which there are real valued functions f_A on A and f_X on X such that (a, x) ? (b, y) if and only if f_A(a) + f_X(x) > f_A(b) + f_X(y), and the lexicographic representation which, with A as the dominant factor, has (a, x) ? (b, y) if and only if f_A(a) > f_A(b) or {f_A(a) = f_A(b) and f_X(x) > f_X(y)}. Recently, Duncan Luce has combined the additive and lexicographic notions in a model for which A is the dominant factor if the difference between a and b is sufficiently large but which adheres to the additive representation when the difference between a and b lies within what might be referred to as a lexicographic threshold. The present paper specifies axioms for ? which lead to a numerical model which also has a lexicographic component but whose local tradeoff structure is governed by the additive-difference model instead of the additive model. Although the additive-difference model includes the additive model as a special case, the new lexicographic additive-difference model is not more general than Luce's model since the former has a “constant” lexicographic threshold whereas Luce's model has a “variable” lexicographic threshold. Realizations of the new model range from the completely lexicographic representation to the regular additive-difference model with no genuine lexicographic component. Axioms for the latter model are obtained from the general axioms with one slight modification.     

Psychosocial mediators of a theory-based resistance training maintenance intervention for prediabetic adults

Objective: Examine psychosocial mediators of the effects of high vs. low-dose resistance training (RT) maintenance interventions among older (ages 50–69), overweight and pre-diabetic adults.

Design: Participants (N = 123) completed a three-month supervised RT initiation phase and were subsequently randomised (time 1) to high or low-dose six-month unsupervised RT maintenance interventions (time 2), followed by a six-month no-contact phase (time 3).

Main Outcome Measures: Online measures of putative mediators and RT behaviour.

Results: RT intervention condition (high vs. low dose) had significant effects on change from time 1 to time 2 in behavioural expectation, self-regulation and perceived satisfaction (f² = .04–.08), but not outcome expectancies, RT strategies or behavioural intentions (f² ≤ .02). Change in each of the putative mediators, except for outcome expectancies (f² ≤ .02), had significant effects on RT behaviour at times 2 (f² = .12–.27) and 3 (f² = .23–.40). In a multiple mediation model, behavioural expectation (f² = .11) and self-regulation (f² = .06) mediated the effects of RT intervention condition on time 2 RT behaviour, whereas perceived satisfaction did not (f² = .01). Self-regulation was a significant mediator of intervention effects on time 3 RT behaviour (f² = .11), but behavioural expectation and perceived satisfaction were not (f² = .04).

Conclusions: Findings suggest that behavioural expectation and self-regulation are appropriate targets for RT maintenance interventions among at-risk older adults.     

Virtual Modality

Model-theoretic 1-types overa given first-order theory T may be construed as natural metalogical miniatures of G. W. Leibniz' ``complete individual notions', ``substances' or ``substantial forms'. This analogy prompts this essay's modal semantics for an essentiallyundecidable first-order theory T, in which one quantifies over such ``substances' in a boolean universe V(C), where C is the completion of the Lindenbaum-algebra of T.More precisely, one can define recursively a set-theoretic translate of formulae _N of formulae of a normal modal theory T_m based on T, such that the counterpart `<sub>i</sub>' of a the modal variable `x_i' of L(T_m) in this translation-scheme ranges over elements of V(C) that are 1-types of T with value 1 (sometimes called `definite' C-valued 1-types of T).The article's basic completeness-result (2.13) then establishes that varphi; is a theorem of T<sub>m</sub> iff [[ <sub>N</sub> () is aconsequenceof <sub>N</sub> (T<sub>m</sub>) for each extension N of T which is a subtheory of the canonical generic theory (ultrafilter) u]] = 1 – or equivalently, that T<sub>m</sub> is consistent iff[[there is anextension N of T such that N is a subtheory of the canonical generic theory u, and <sub>N</sub>() for all in T<sub>m</sub>]] > 0.The proof of thiscompleteness-result also shows that an N which provides a countermodel for a modally unprovable – or equivalently, a closed set in the Stone space St(T) in the sense of V(C) – is intertranslatable with an `accessibility'-relation of a closely related Kripke-semantics whose `worlds' are generic extensions of an initial universe V via C.This interrelation providesa fairly precise rationale for the semantics' recourse to C-valued structures, and exhibits a sense in which the boolean-valued context is sharp.