In this course we will present and discuss some of the most important ideas and results covering these various contacts between logic and probability, including (1) the view of probability as extended logic, foundations based on comparative probability, (2) probabilities defined on expressive logical languages, (3) default reasoning, acceptance rules, and the quantitative/qualitative interface, (4) statistical relational models and foundations of probabilistic programming, (5) zero-one laws, random structures, and almost-sure theories. Logic and probability are related in many important ways: probability can in some sense be seen as a generalization of logic many logical systems have natural probabilistic semantics logic can be used to reason explicitly about probability and logic and probability can be combined into systems that capitalize on the advantages of each. We investigate the relationship between topological semantics and the more standard relational semantics, establish the foundational result that S4 is "the logic of space" (i.e., sound and complete with respect to the class of all topological spaces), and discuss richer epistemic systems in which topology can be used to capture the distinction between the known and the knowable, fact and measurement.Ĭourse materials Logic and Probability - Thomas Icard and Krzysztof Mierzewski Intuitively, the spatial notion of "nearness" can be co-opted as a means of representing uncertainty. We then develop the notion of a topological space using a variety of metaphors and intuitions, and introduce topological semantics for the basic modal language. We begin by motivating and reviewing the standard relational structures used as models for knowledge in epistemic logics. Some basic background in modal logic will be helpful, but is not essential no background in topology is assumed. Something TRUE in all worlds is said to be believed by the agent (It is TRUE that the agent has the Ace of Spades).This course is an introduction to topology and an exploration of some of its applications in epistemic logic. What is left over are the epistemic alternatives (worlds possible given ones beliefs). Then eliminate those worlds that are not possible given what the agent knows. ![]() First compute all the possible ways the cards could be dealt to the opponent. The ability to play is determined partially by the agent's belief in the opponents hand. Complete knowledge of the opponents hand is impossible to determine. From the point of view of an agent A (we will not represent A to keep notations simple), f will be read as: "A believes f".Įxample: An agent is playing poker. ex: temporal logic, deontic logic, epistemic logic.įor our purpose, we will use modal operators in the context of epistemic logic (logic of knowledge). The modal operators can be used in various contexts and take specific meanings. The two modal operators are duals of each other:.The formula f is TRUE if f is TRUE for in at least one world accessible from the current world. ![]()
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