Modal Incompleteness Research Articles

Nonconservative extensions by propositional quantifiers and modal incompleteness

Abstract Propositional modal logics can be extended by propositional quantifiers, i.e. quantifiers binding proposition letters understood as variables. This paper investigates whether such an extension is always conservative. It is shown that the answer depends on the way in which propositional quantifiers are added. On a minimal approach, according to which propositional quantifiers only have to satisfy the classical principles of quantification, every classical modal logic has a conservative extension by propositional quantifiers. However, in the context of normal modal logics it is natural to require propositionally quantified extensions also to be closed under the rule of necessitation. It is shown that in this setting, there are normal modal logics whose propositionally quantified extensions are nonconservative. Nonconservativity is shown to be a special case of a number of model-theoretic notions of incompleteness. More tentatively, it is suggested that nonconservativity indicates incompleteness in a more substantial sense, concerning the intended target of capturing the logics of modalities.

A Two-Step Damage Identification Based on Cross-Model Modal Strain Energy and Simultaneous Optimization

Aiming at the problem of the effect on damage identification caused by noise and modal incompleteness, a two-stage damage identification method based on cross-model modal strain energy and simultaneous optimization was proposed. This paper proposed an improved damage index (Modified Index of Cross-model Modal Strain Energy, MICMSE) based on cross-model modal strain energy. This method first locates the damaged elements by MICMSE, and introduces simultaneous optimization to quantify the damage ratios of damaged elements. The two-step method avoids the calculation of the sensitivity of the modal parameters and the reanalysis of the characteristic equation, speeds up the convergence speed, and guarantees the recognition accuracy. The results of the numerical example show that the method proposed in this paper can effectively locate the damage elements and identify the damage ratios under different damage cases and noise levels.

Model Updating for Systems with General Proportional Damping Using Complex FRFs

In this paper, we propose a model updating method for systems with nonviscous proportional damping. In comparison to the traditional viscous damping model, the introduction of nonviscous damping will not only reduce the vibration of the system but also change the resonance frequencies. Therefore, most of the existing updating methods cannot be directly applied to systems with nonviscous damping. In many works, for simplicity, the Rayleigh damping model has been applied in the model updating procedure. However, the assumption of Rayleigh damping may result in large errors of damping for higher modes. To capture the variation of modal damping ratio with frequency in a more general way, the diagonal elements of the modal damping matrix and relaxation parameter are updated to characterize the damping energy dissipation of the structure by the proposed method. Spatial and modal incompleteness are both discussed for the updating procedure. Numerical simulations and experimental examples are adopted to validate the effectiveness of the proposed method. The results show that the systems with general proportional damping can be predicted more accurately by the proposed updating method.

COMPLETE ADDITIVITY AND MODAL INCOMPLETENESS

AbstractIn this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem (1979), “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class ofcompletely additivemodal algebras, or as we call them,${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to${\cal V}$-baos, namely the provability logic$GLB$(Japaridze, 1988; Boolos, 1993). We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is${\cal V}$-complete. After these results, we generalize the Blok Dichotomy (Blok, 1978) to degrees of${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.

Effects of modal incompleteness on quantification of mode shapes complexity in vibrating structures

In experimental modal analysis, the mode shapes turn out to be complex due to the non-proportionality of damping in actual structures. For identification purposes, several indices have been proposed to quantify this complexity. Unfortunately, the effectiveness of the above indices is affected by the problem of data incompleteness that arises when a limited set of modal data is available, as in the case of reduced eigenvector components or in the case of band limited eigenvalues range. In contexts such as model updating, this problem makes it difficult to pair experimental versus numerical data and methods to compare the two sets of data are based on the reduction of numerical models or, conversely, on the expansion of the experimental one. However none of them reveals completely satisfactory. The present work aims to analyze the effectiveness of the above modal complexity indices. To this end, the most sensitive index found from previous studies is used in order to detect the mode shapes complexity compared to the damping non-proportionality in vibrating structures for different levels of modal incompleteness.

Semi-Supervised Multi-Modal Clustering and Classification with Incomplete Modalities

In this paper, we propose a novel Semi-supervised Learning with Incomplete Modality (SLIM) method considering the modal consistency and complementarity simultaneously, and Kernel SLIM (SLIM-K) based on matrix completion for further solving the modal incompleteness. As is well known, most realistic data have multi-modal representations, multi-modal learning refers to the process of learning a precise model for complete modalities. However, due to the failures of data collection, self-deficiencies, or other various reasons, multi-modal examples are usually with incomplete modalities, which generate utility obstacle using previous methods. In this paper, SLIM integrates the intrinsic consistency and extrinsic complementary information for prediction and cluster simultaneously. In detail, SLIM forms different modal classifiers and clustering learner consistently in a unified framework, while using the extrinsic complementary information from unlabeled data against the insufficiencies brought by the incomplete modal issue. Moreover, in order to deal with missing modality in essence, we propose the SLIM-K, which takes the complemented kernel matrix into the classifiers and the cluster learner respectively. Thus, SLIM-K can solve the defects of missing modality in result. Finally, we give the discussion of generalization of incomplete modalities. Experiments on 13 benchmark multi-modal datasets and two real-world incomplete multi-modal datasets validate the effectiveness of our methods.

A new principle in the interpretability logic of all reasonable arithmetical theories

The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper from 2011 revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by ${\textbf{IL}}({\rm All})$. In this paper we present a new principle $\sf R$ in ${\textbf{IL}}({\rm All})$. We show that $\sf R$ does not follow from the logic ${\textbf{IL}}{\sf P_0W^*}$ that contains all previously known principles. This is done by providing a modal incompleteness proof of ${\textbf{IL}}{\sf P_0W^*}$: showing that $\sf R$ follows semantically but not syntactically from ${\textbf{IL}}{\sf P_0W^*}$. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle $\sf R$ is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories.

Modal Incompleteness Revisited

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some Σ\( \subseteq \)L and some \(\varphi \notin L,\varphi \) cannot be separated from Σ by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any class of complete BAOs. Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Lob logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind.

주파수 응답함수를 이용한 부분구조 합성에서 모드자름 오차 보정에 관한 수치적 연구

A FRF-based substructuring method attempts to predict the dynamic characteristics of a complex structure from predetermined FRFs of the comprising uncoupled substructures. Although this method has the advantage of being able to incorporate experimental component FRFs directly, it is prone to errors : measurement errors, coordinate incompleteness, modal incompleteness, etc. Among the various sources of errors, this paper deals with the problem of modal incompleteness (or residual problem) of which importance is underestimated compared to others. It is a well-known rule of thumb that such a problem can be overcome by including modes up to 2 or 3 times the upper frequency of interest. Using a simulated case study, it is demonstrated that even including modes up to 20 times the upper frequency of interest does not guarantee a satisfactory result. A method to compensate the residual errors is introduced. This method requires the whole FRF matrices of substructures which is practically impossible for a complex structure. An applicable alternative is suggested and applied successfully to the case study. Finally, the effects of measurement errors on the residual compensation are also discussed.

Incompleteness and Fixed Points

Our purpose is to present some connections between modal incompleteness andmodal logics related to the Gödel-Löb logic GL. One of our goals is to prove that for all m, n, k, l ∈ ℕ the logic K + □i() □jp ↔ p) → □ip is incomplete and does not have the fixed point property. As a consequence we shall obtain that the Boolos logic KH does not have the fixed point property.

Incompleteness and Fixed Points

Our purpose is to present some connections between modal incompleteness andmodal logics related to the Gödel-Löb logic GL. One of our goals is to prove that for all m, n, k, l ∈ ℕ the logic K + □i() □jp ↔ p) → □ip is incomplete and does not have the fixed point property. As a consequence we shall obtain that the Boolos logic KH does not have the fixed point property.

Syntactic aspects of modal incompleteness theorems

Syntactic aspects of modal incompleteness theorems