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From one of the founders of symbolic logic comes this collection of writings on logical subjects and related questions of probability. George Boole invented.
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- Studies in Logic and Probability
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- George Boole | SpringerLink
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Appropriate for upper-level undergraduates and graduate students, the contents range from The Mathematical Analysis of Logic to Boole's final works, including The Laws of Thought, the most systematic statement of his ideas on logic and probability. Boole had intended to create a follow-up volume but did not survive to fulfill his ambition; this volume features his further studies on the subject.
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Studies in Logic and Probability
This is evidenced by the several conferences on the history of logic, by a journal devoted to the subject, and by an accumulation of new results. This increased activity and the new results - the chief one being that Boole's work in probability is best viewed as a probability logic - were influential circumstances conducive to a new edition.
Chapter 1, presenting Boole's ideas on a mathematical treatment of logic, from their emergence in his early work on through to his immediate successors, has been considerably enlarged. Chapter 2 includes additional discussion of the uninterpretable'' notion, both semantically and syntactically.
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Chapter 3 now includes a revival of Boole's abandoned propositional logic and, also, a discussion of his hitherto unnoticed brush with ancient formal logic. Chapter 5 has an improved explanation of why Boole's probability method works. Chapter 6, Applications and Probability Logic, is a new addition. Changes from the first edition have brought about a three-fold increase in the bibliography. The first edition merited that status; the second deserves it even more Kyburg qu:This was an important and useful book in its first edition. It has now become even more important, since it now addresses a large class of contemporary problems in the handling and propagation of uncertainty.
We are always looking for ways to improve customer experience on Elsevier. We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit. If you decide to participate, a new browser tab will open so you can complete the survey after you have completed your visit to this website. Thanks in advance for your time. Skip to content. The valuation function, as in a Kripke model, allows us to assign properties to the worlds. The first generalization, which is most common in applications of modal probabilistic logic, is to allow the distributions to be indexed by two sets rather than one.
We depict this example with the following diagram. Inside each circle is a labeling of the truth of each proposition letter for the world whose name is labelled right outside the circle. The arrows indicate the probabilities. Probabilities of 0 are not labelled. In this case, pressing a button does not have a certain outcome. That is,.
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A significant feature of modal logics in general and this includes modal probabilistic logic is the ability to support higher-order reasoning , that is, the reasoning about probabilities of probabilities. Higher-order probability also occurs for instance in the Judy Benjamin Problem van Fraassen a where one conditionalizes on probabilistic information. Whether one agrees with the principles proposed in the literature on higher-order probabilities or not, the ability to represent them forces one to investigate the principles governing them. But the players randomize over their opponents.
Probabilities are generally defined as measures in a measure space. This is crucial for some probabilities to be defined on uncountably infinite sets; for example, a uniform distribution over a unit interval cannot be defined on all subsets of the interval while also maintaining the countable additivity condition for probability measures. The reason we may want entire spaces to differ from one world to another is to reflect uncertainty about what probability space is the right one.
Although probabilities reflect quantitative uncertainty at one level, there can also be qualitative uncertainty about probabilities. We might want to have qualitative and quantitative uncertainty because we may be so uncertain about some situations that we do not want to assign numbers to the probabilities of their events, while there are other situations where we do have a sense of the probabilities of their events; and these situations can interact. There are many situations in which we might not want to assign numerical values to uncertainties. One example is where a computer selects a bit 0 or 1, and we know nothing about how this bit is selected.
Results of coin flips, on the other hand, are often used examples of where we would assign probabilities to individual outcomes. One way to formalize the interaction between probability and qualitative uncertainty is by adding another relation to the model and a modal operator to the language as is done in Fagin and Halpern , We have discussed two views of modal probability logic. One is temporal or stochastic, where the probability distribution associated with each state determines the likelihood of transitioning into other states; another is concerned with subjective perspectives of agents, who may reason about probabilities of other agents.
A stochastic system is dynamic in that it represents probabilities of different transitions, and this can be conveyed by the modal probabilistic models themselves. But from a subjective view, the modal probabilistic models are static: the probabilities are concerned with what currently is the case. Although static in their interpretation, the modal probabilistic setting can be put in a dynamic context. Dynamics in a modal probabilistic setting is generally concerned with simultaneous changes to probabilities in potentially all possible worlds.
Intuitively, such a change may be caused by new information that invokes a probabilistic revision at each possible world. The dynamics of subjective probabilities is often modeled using conditional probabilities, such as in Kooi , Baltag and Smets , and van Benthem et al. Let us assume for the remainder of this dynamics subsection that every relevant set considered has positive probability.
George Boole | SpringerLink
For other overviews of modal probability logics and its dynamics, see Demey and Kooi , Demey and Sack , and appendix L on probabilistic update in dynamic epistemic logic of the entry on dynamic epistemic logic. In this section we will discuss first-order probability logics.
As was explained in Section 1 of this entry, there are many ways in which a logic can have probabilistic features. The models of the logic can have probabilistic aspects, the notion of consequence can have a probabilistic flavor, or the language of the logic can contain probabilistic operators. In this section we will focus on those logical operators that have a first-order flavor.
The first-order flavor is what distinguishes these operators from the probabilistic modal operators of the previous section. First-order probabilistic operators are needed to express these sort of statements. This sentence considers the probability that Tweety a particular bird can fly. These two types of sentences are addressed by two different types of semantics, where the former involves probabilities over a domain, while the latter involves probabilities over a set of possible worlds that is separate from the domain.
In this subsection we will have a closer look at a particular first-order probability logic, whose language is as simple as possible, in order to focus on the probabilistic quantifiers. The language is very much like the language of classical first-order logic, but rather than the familiar universal and existential quantifier, the language contains a probabilistic quantifier. The language contains two kinds of syntactical objects, namely terms and formulas.
The terms are defined inductively as follows:. As an example, consider a model of a vase containing nine marbles: five are black and four are white. The logic that we just presented is too simple to capture many forms of reasoning about probabilities. We will discuss three extensions here. First of all one would like to reason about cases where more than one object is selected from the domain.
Consider for example the probability of first picking a black marble, putting it back, and then picking a white marble from the vase. This approach is taken by Bacchus and Halpern , corresponding to the idea that selections are independent and with replacements.