Search

We make the following contributions to modal logics with propositional quantifiers and modal logics with comparative operators in this dissertation:- We define a general notion of normal modal logics with propositional quantifiers. We call them normal Π-logics. Then, as was done by Scrogg's theorem on extensions of the modal logic S5, we study in general the normal Π-logics extending S5. We show that they are all complete with respect to their algebraic semantics based on complete simple monadic algebras. We also show that the lattice formed by these logics is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N with the natural order topology. Further, we show how to determine the computability of normal Pi-logics extending S5Π. A corollary is that they can be of arbitrarily high Turing-degree.- Regarding the normal Π-logics extending the modal logic KD45, we identify two important axioms: Immod: □∀p(□p → p) and 4∀: ∀p□φ→□∀p□φ. We argue that when □ is interpreted as the belief operator, we should not accept Immod in the logic of belief while 4∀ is desirable in settings with full introspection. Then we provide algebraic semantics based on complete well-connected pseudo monadic algebras for Box and ∀p and show that with respect to these algebras, normal Π-logics extending KD45 and 4∀ with a finite list of formulas are complete. We also give a general completeness theorem for atomic complete well-connected pseudo monadic algebras and a sufficient condition for the decidability of logics obtained by classes of these algebras, atomic or not. A special case of these general theorems is that the normal Π-logic of serial, transitive, and Euclidean Kripke frames, that is, the Kripke frames validating KD45, is axiomatized by KD45, 4∀, Immod, and ∃p(p∧∀q(q→□(p→q))) together with the usual axioms and rules for propositional quantifiers, and this logic is decidable.Other than completeness and decidability, we also show that 4∀ is not in the smallest normal Π-logic extending KD45, using a countermodel based on a possible-world frame with propositional contingency, and that 4∀ is valid in any complete Boolean algebra expansion validating KD45. - For modal logics with comparative operators, we first provide an axiomatization of the logic of comparing the cardinality of sets, as defined by Cantor. The main technical contribution is the observation that in a purely comparative language, we can define finiteness well enough so that an axiomatization can be done by combining the logic of comparative cardinality for finite sets and the logic of comparative cardinality for infinite sets. Note that these two logics are very different: the former contains the axiom of qualitative additivity: |A| ≥ |B| iff |A \ B| ≥ |B \ A| but not the axiom of absorption: if |A| ≥ |B| and |A| ≥ |C| then |A| ≥ |B ∪ C|, while the latter does the opposite.- Then we consider modal logics for comparative imprecise probability. In these logics, comparisons are made according to a set of probability measures and can be intuitively read as either "at least as likely as" (symbolized by ≥) or "more likely than" (symbolized by >). We first disambiguate two interpretations of "more likely than" based on a set of probability measures and show that the stronger interpretation is not definable from "at least as likely as" while the weaker sense is. Then, we go on to axiomatize the logic of imprecise probability in a sequence of languages obtained by adding to the language with just "at least as likely as", one by one, the comparative operator > for "more likely than" (in the stronger sense), a unary operator ♢ for "possibly", and a binary operator ψ for "possibly φ, and after learning the truth of φ, ψ''. We also comment on the expressivity of these languages and the decidability of the logics in these languages. In particular, we show that many distinctive features of the imprecise probability approach of representing the doxastic states of agents, such as the problem of dilation, are observable at this purely comparative level. Finally, we add a pair of operators Ipφ and Ip-φ, intuitively read, respectively, as "after learning the existence of an actually true new proposition, now named by p, φ", and "after learning the existence of an actually false new proposition, now named by p, φ". We show that this pair of operators allow us to formalize a common kind of information dynamics and will boost the expressivity of the language to a quantitative level.

The purpose of this dissertation is to develop and demonstrate methods appropriate for the quantification and propagation of uncertainty in large, high-consequence engineering projects. The term "rigorous uncertainty quantification" refers to methods equal to the proposed task. The motivating practical example is uncertainty in a Mars atmosphere model due to the incompletely characterized presence of dust. The contributions made in this dissertation, though primarily mathematical and philosophical, are driven by the immediate needs of engineers applying uncertainty quantification in the field. Arguments are provided to explain how the practical needs of engineering projects like Mars lander missions motivate the use of the objective probability bounds approach, as opposed to the subjectivist theories which dominate uncertainty quantification in many research communities. An expanded formalism for Dempster-Shafer structures is introduced, allowing for the representation of continuous random variables and fuzzy variables as Dempster-Shafer structures. Then, the correctness and incorrectness of probability bounds analysis and the Cartesian product propagation method for Dempster-Shafer structures under certain dependency conditions are proven. It is also conclusively demonstrated that there exist some probability bounds problems in which the best-possible bounds on probability can not be represented using Dempster-Shafer structures. Nevertheless, Dempster-Shafer theory is shown to provide a useful mathematical framework for a wide range of probability bounds problems. The dissertation concludes with the application of these new methods to the problem of propagating uncertainty from the dust parameters in a Mars atmosphere model to uncertainty in that model's prediction of atmospheric density. A thirty-day simulation of the weather at Holden Crater on Mars is conducted using a meso-scale atmosphere model, MRAMS. Although this analysis only addresses one component of Mars atmosphere uncertainty, it demonstrates the applicability of probability bounds methods in practical engineering work. More importantly, the Mars atmosphere uncertainty analysis provides a framework in which to conclusively establish the practical importance of epistemology in rigorous uncertainty quantification.

Due to the inherent uncertainties in ground and groundwater conditions, tunnel projects often have to face potential risks of cost overrun or schedule delay. Risk analysis has become a required tool (by insurers, Federal Transit Administration, etc.) to identify and quantify risk, as well as visualize causes and effects, and the course (chain) of events. Various efforts have been made to risk assessment and analysis by using conventional methodologies with precise probabilities. However, because of limited information or experience in similar tunnel projects, available evidence in risk assessment and analysis usually relies on judgments from experienced engineers and experts. As a result, imprecision is involved in probability evaluations. The intention of this study is to explore the use of the theory of imprecise probability as applied to risk analysis in tunneling. The goal of the methodologies proposed in this study is to deal with imprecise information without forcing the experts to commit to assessments that they do not feel comfortable with or the analyst to pick a single distribution when the available data does not warrant such precision. After a brief introduction to the theory of imprecise probability, different types of interaction between variables are studied, including unknown interaction, different types of independence, and correlated variables. Various algorithms aiming at achieving upper and lower bounds on previsions and conditional probabilities with assumed interaction type are proposed. Then, methodologies have been developed for risk registers, event trees, fault trees, and decision trees, i.e. the standard tools in risk assessment for underground projects. Corresponding algorithms are developed and illustrated by examples. Finally, several case histories of risk analysis in tunneling are revisited by using the methodologies developed in this study. All results obtained based on imprecise probabilities are compared with the results from precise probabilities.