Probability theory and probability logic

As a survey of many technical results in probability theory and probability logic, this monograph by two widely respected scholars offers a valuable compendium of the principal aspects of the formal study of probability.

Abstract. Surface flow on rilled hillslopes tends to produce sediment yields that scale nonlinearly with total hillslope length. The widespread observation lacks a single unifying theory for such a nonlinear relationship. We explore the contribution of rill network geometry to the observed yield–length scaling relationship. Relying on an idealized network geometry, we formally develop probability functions for geometric variables of contributing area and rill length. In doing so, we contribute towards a complete probabilistic foundation for the Hack distribution. Using deterministic and empirical functions, we then extend the probability theory to the hydraulic variables that are related to sediment detachment and transport. A Monte Carlo simulation samples hydraulic variables from hillslopes of different lengths to provide estimates of sediment yield. The results of this analysis demonstrate a nonlinear yield–length relationship as a result of the rill network geometry. Theory is supported by numerical modeling, wherein surface flow is routed over an idealized numerical surface and a natural surface from northern Arizona. Numerical flow routing demonstrates probability functions that resemble the theoretical ones. This work provides a unique application of the Scheidegger network to hillslope settings which, because of their finite lengths, result in unique probability functions. We have addressed sediment yields on rilled slopes and have contributed towards understanding Hack's law from a probabilistic reasoning.

Economic benefits of urban rail projects that improve travel-time reliability: Evidence from Tokyo, Japan

This paper aims to present a probability distribution function for voltage sags. A large number of real grid recordings, originally obtained by authors, measurements provided by Prof. Math Bollen or presented in the literature are analyzed. Two types of probability functions, Weibull and lognormal are considered for five cases of measurement results from different countries. Averaged probability distribution functions are proposed. Their applications for the optimization of the artificially generated voltage sags database can be used further for the training of neural networks for power quality characterization.

0. Requisites from Algebra, Logic and Probability. Part I: Logic. 1. Boole's Logic of Class Terms. 2. Formalization of Boole's Logic. 3. Boole's Propositional Logic. Part II: Probability. 4. Probability from Boole's Viewpoint. 5. Boole's Probability Made Rigorous. 6. Applications. Probability Logic. Bibliography. Index.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

: With better natural language semantic representations, computers can do more applications more efficiently as a result of better understanding of natural text. However, no single semantic representation at this time fulfills all requirements needed for a satisfactory representation. Logic-based representations like first-order logic capture many of the linguistic phenomena using logical constructs, and they come with standardized inference mechanisms, but standard first-order logic fails to capture the graded aspect of meaning in languages. Distributional models use contextual similarity to predict the graded semantic similarity of words and phrases but they do not adequately capture logical structure. In addition there are a few recent attempts to combine both representations either on the logic side (still, not a graded representation), or in the distribution side (not full logic). We propose using probabilistic logic to represent natural language semantics combining the expressivity and the automated inference of logic, and the gradedness of distributional representations. We evaluate this semantic representation on two tasks, Recognizing Textual Entailment (RTE) and Semantic Textual Similarity (STS). Doing RTE and STS better is an indication of a better semantic understanding. Our system has three main components, 1. Parsing and Task Representation, 2. Knowledge Base Construction, and 3. Inference. The input natural sentences of the RTE/STS task are mapped to logical form using Boxer which is a rule based system built on top of a CCG parser, then they are used to formulate the RTE/STS problem in probabilistic logic. Then, a knowledge base is represented as weighted inference rules collected from different sources like WordNet and on-the-fly lexical rules from distributional semantics.

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.

Probabilistic stit logic and its decomposition

On the expressive power of counting

In this paper we first introduce a new neighbourhood semantics for propositional intuitionistic logic. We then naturally extend this semantics to first-order intuitionistic logic. We also study bisimulation between neighbourhood models and prove some of their basic properties for both propositional and first-order intuitionistic logic.

Does probability extend classical logic? In a recent paper by Dubois and Prade [60], they seem to give a negative answer to this question. If we refer to a formal and strict meaning of the term “classical” logic, we cannot disagree with their conclusion. On the other hand, at the very beginning of a 1935's paper [49] by de Finetti (from which some excerpts are reported also in Dubois and Prade's paper) there is the following statement: • “it is beyond doubt that probability theory can be considered as a multi-valued logic (precisely: with a continuous range of values), and that this point of view is the most suitable to clarify the foundational aspects of the notion and the logic of probability”. KeywordsConditional ProbabilityClassical LogicNegative AnswerAxiom SystemConditional EventThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Probability and sampling in dentistry are two fundamentals which have great importance in clinical research. Many research works in dentistry shows lack of proper understanding and use of these two factors. The definition of probability is incredibly significant in daily life. Statistical analysis is based on this particularly useful definition. In fact, the function of probability in modern science is that of substituting for certainty. Probabilities are numbers that represent the probability that a specific occurrence will occur. We learn about the odds of many daily cases, ranging from weather predictions (probability of rain or snow) to lotteries (probability of winning a major jackpot). In biostatistical applications, probability theory underlies the statistical inference. Statistical inference means drawing generalizations or inferences on unknown population parameters. After selecting a sample from the population of interest, we calculate the characteristics under analysis, summarize the characteristics in our sample, and then draw inferences about the population based on what we find in the sample. Population and sampling are two critical aspects of study design. The population is a group of individuals who share common relations. A sample is a population subset. The size of the sample is the number of individuals in the sample. The more representative the sample of the population, the surer the researcher can be about the validity of the data. In this module, we will explore sampling methods, basic principles of probability, and applications of probability theory. The definition of probability is introduced, and the function of probability distributions is discussed in the statistical theory, with reference to the normal distribution and its characteristics. Sampling and sampling variations are defined, along with the sampling error, the standard error of the mean and the confidence intervals for determining the likely magnitude of the population mean. Medical study typically includes patients with an illness or disorder. The generalization of clinical research results is focused on several factors linked to the internal and external validity of the research methods. The sampling process is the key methodological problem that affects the generalizability of clinical research results. In this educational article, we also clarify the various methods of sampling in clinical research.

We consider various extensions of first-order logic. Informally, a logic 𝓛 is an extension of first-order logic if every sentence of first-order logic is also a sentence of 𝓛. We also require that 𝓛 is closed under conjunction and negation and has other basic properties of a logic. In Section 9.4, we list the properties that formally define the notion of an extension of first-order logic. Prior to Section 9.4, we provide various natural examples of such extensions. In Sections 9.1–9.3, we consider, respectively, second-order logic, infinitary logics, and logics with fixed-point operators. We do not provide a thorough treatment of any one of these logics. Indeed, we could easily devote an entire chapter to each. Rather, we define each logic and provide examples that demonstrate the expressive power of the logics. In particular, we show that none of these logics has compactness. In the final Section 9.4, we prove that if a proper extension of first-order logic has compactness, then the Downward Löwenhiem–Skolem theorem must fail for that logic. This is Lindstrom’s theorem. The Compactness theorem and Downward Löwenheim–Skolem theorem are two crucial results for model theory. Every property of first-order logic from Chapter 4 is a consequence of these two theorems. Lindström’s theorem implies that the only extension of first-order logic possessing these properties is first-order logic itself. Second-order logic is the extension of first-order logic that allows quantification of relations. The symbols of second-order logic are the same symbols used in first-order logic. The syntax of second-order logic is defined by adding one rule to the syntax of first-order logic. The additional rule makes second-order logic far more expressive than first-order logic. Specifically, the syntax of second-order logic is defined as follows. Any atomic first-order formula is a formula of second-order logic. Moreover, we have the following four rules: (R1) If φ is a formula then so is ￢φ. (R2) If φ and ψ are formulas then so is φ ∧ ψ. (R3) If φ is a formula, then so is ∃x φ for any variable x.