This paper argues that a combined treatment of probabilities, time and actions is essential for an appropriate logical account of the notion of probability; and, based on this intuition, describes an expressive probabilistic temporal logic for reasoning about actions with uncertain outcomes. The logic is modal and higher-order: modalities annotated by actions are used to express possibility and necessity of propositions in the next states resulting from the actions, and a higher-order function is needed to express the probability operator. The proposed logic is shown to be an adequate extension of classical mathematical probability theory, and its expressiveness is illustrated through the formalization of the Monty Hall problem.
Many probabilistic logics are surveyed in @cite_2 . Among those logics, most depart from classical logic by adopting a probabilistic notion of validity and entailment. , on the other hand, remains strictly classical in this respect. The probabilistic modal logics described in Sections 4.1 and 4.2 of @cite_2 are probably the most similar to . However, they are propositional, lack the probabilistic diamond operator, and are atemporal.
@cite_2
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The use of conventional classical logic is misleading for characterizing the behavior of logic programs because a logic program, when queried, will do one of three things: succeed with the query, fail with it, or not respond because it has fallen into infinite backtracking. In [7] Kleene proposed a three-valued logic for use in recursive function theory. The so-called third truth value was really undefined: truth value not determined. This logic is a useful tool in logic-program specification, and in particular, for describing models. (See [11].) Tarski showed that formal languages, like arithmetic, cannot contain their own truth predicate because one could then construct a paradoxical sentence that effectively asserts its own falsehood. Natural languages do allow the use of "is true", so by Tarski's argument a semantics for natural language must leave truth-value gaps: some sentences must fail to have a truth value. In [8] Kripke showed how a model having truth-value gaps, using Kleene's three-valued logic, could be specified. The mechanism he used is a famiUar one in program semantics: consider the least fixed point of a certain monotone operator. But that operator must be defined on a space involving three-valued logic, and for Kripke's application it will not be continuous. We apply techniques similar to Kripke's to logic programs. We associate with each program a monotone operator on a space of three-valued logic interpretations, or better partial interpretations. This space is not a complete lattice, and the operators are not, in general, continuous. But least and other fixed points do exist. These fixed points are shown to provide suitable three-valued program models. They relate closely to the least and greatest fixed points of the operators used in [1]. Because of the extra machinery involved, our treatment allows for a natural consideration of negation, and indeed, of the other prepositional connectives as well. And because of the elaborate structure of fixed points available, we are able to Let P be an arbitrary theory and let X be any given logic. Let M be a set of atoms. We say that M is a X-stable model of P if M is a classical model of P and P∪¬M proves in logic X all atoms in M, this is denoted by P∪¬M ⊩xM. We prove that being an X-stable model is an invariant property for disjunctive programmes under a large class of logics. Two kinds of logics are mainly considered: paraconsistent logics and normal modal logics. For modal logics we use a translation proposed by Gelfond that replaces ¬a with ¬□a. As a consequence we prove that several semantics (recently introduced) for non-monotonic reasoning are equivalent for disjunctive programmes. In addition, we show that such semantics can be characterized by a fixed-point operator in terms of classical logic. We also present a simple translation of a disjunctive programme D into a normal programme N, such that the PStable model semantics of N corresponds to the stable semantics of D over the common language. We present the formal proof of this statement. This thesis presents a logical formalism for representing and reasoning with probabilistic knowledge. The formalism differs from previous efforts in this area in a number of ways. Most previous work has investigated ways of assigning probabilities to the sentences of a logical language. Such an assignment fails to capture an important class of probabilistic assertions, empirical generalizations. Such generalizations are particularly important for AI, since they can be accumulated through experience with the world. Thus, they offer the possibility of reasoning in very general domains, domains where no experts are available to gather subjective probabilities from. A logic is developed which can represent these empirical generalizations. Reasoning can be performed through a proof theory which is shown to be sound and complete. Furthermore, the logic can represent and reason with a very general set of assertions, including many non-numeric assertions. This also is important for AI as numbers are usually not available. The logic makes it clear that there is an essential difference between empirical, or statistical, probabilities and probabilities assigned to sentences, e.g., subjective probabilities. The second part of the formalism is an inductive mechanism for assigning degrees of belief to sentences based on the empirical generalizations expressed in the logic. these degrees of belief have a strong advantage over subjective probabilities: they are founded on objective statistical knowledge about the world. Furthermore, the mechanism of assigning degrees of belief gives a natural answer to the question "Where do the probabilities come from:" they come from our experience with the world. The two parts of the formalism offer combined, interacting, but still clearly separated, plausible inductive inference and sound deductive inference. Probabilistic abstract argumentation combines Dung’s abstract argumentation framework with probability theory in order to model uncertainty in argumentation. In this setting, we address the fundamental problem of computing the probability that a set of arguments is an extension according to a given semantics. We focus on the most popular semantics (i.e., admissible, stable, complete, grounded, preferred, ideal-set, ideal, stage, and semistable) and show the following dichotomy result: computing the probability that a set of arguments is an extension is either FP or FP#P-complete depending on the semantics adopted. Our polynomial-time results are particularly interesting, as they hold for some semantics for which no polynomial-time technique was known so far.