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1. Combining Logic and Probability Theory

  1. Prof. Jeff Paris - School of Mathematics
  2. Jeff Paris (mathematician)
  3. Why Probability?
  4. Continuum mechanics
  5. Intended learning outcomes

This theorem can be seen as a first, very partial clarification of the issue of probability preservation or uncertainty propagation. It says that if there is no uncertainty whatsoever about the premises, then there cannot be any uncertainty about the conclusion either. In the next two subsections we will consider more interesting cases, when there is non-zero uncertainty about the premises, and ask how it carries over to the conclusion. Finally, it should be noted that although this subsection only discussed probabilistic semantics for classical propositional logic, there are also probabilistic semantics for a variety of other logics, such as intuitionistic propositional logic van Fraassen b; Morgan and Leblanc , modal logics Morgan a,b, ; Cross , classical first-order logic Leblanc , ; van Fraassen b , relevant logic van Fraassen and nonmonotonic logic Pearl All of these systems share a key feature: the logic's semantics is probabilistic in nature, but probabilities are not explicitly represented in the object language; hence, they are much closer in nature to the propositional probability logics discussed here than to the systems presented in later sections.

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Goosens provides an overview of various axiomatizations of probability theory in terms of such primitive notions of conditional probability. In the previous subsection we discussed a first principle of probability preservation, which says that if all premises have probability 1, then the conclusion also has probability 1. Of course, more interesting cases arise when the premises are less than absolutely certain.

One can easily show that. We will now discuss Adams' methods to compute such bounds. Adams' results can be stated more easily in terms of uncertainty rather than certainty probability. If the probability function P is clear from the context, we will often simply write U instead of U P. In the remainder of this subsection and in the next one as well we will assume that all arguments have only finitely many premises which is not a significant restriction, given the compactness property of classical propositional logic.

Adams' first main result, which was originally established by Suppes , can now be stated as follows:. Theorem 2. If a valid argument has a small number of premises, each of which only has a small uncertainty i. Theorem 3. The upper bound provided by Theorem 2 can also be used to define a probabilistic notion of validity.

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Adams-probabilistic validity has an alternative, equivalent characterization in terms of probabilities rather than uncertainties. It can be shown that classical propositional logic is strongly sound and complete with respect to Adams' probabilistic semantics:. Adams , also defines another logic for which his probabilistic semantics is sound and complete. However, this system involves a non-truth-functional connective the probability conditional , and therefore falls outside the scope of this section.

Consider the following example. Then Theorem 2 says that. This upper bound on the uncertainty of the conclusion is rather disappointing, and it exposes the main weakness of Theorem 2.

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However, this premise is irrelevant, in the sense that the conclusion already follows from the other three premises. The weakness of Theorem 2 is thus that it takes into account the uncertainty of irrelevant or inessential premises. In argument A in the example above, premise s is absolutely irrelevant. Theorem 4. The proof of Theorem 4 is significantly more difficult than that of Theorem 2: Theorem 2 requires only basic probability theory, whereas Theorem 4 is proved using methods from linear programming Adams and Levine ; Goldman and Tucker Theorem 4 subsumes Theorem 2 as a special case: if all premises are relevant i.

Furthermore, Theorem 4 does not take into account irrelevant premises i. Hence Theorem 4 yields that. Given the uncertainties and degrees of essentialness of the premises of a valid argument, Adams' theorems allow us to compute an upper bound for the uncertainty of the conclusion. Of course these results can also be expressed in terms of probabilities rather than uncertainties; they then yield a lower bound for the probability of the conclusion.

For example, when expressed in terms of probabilities rather than uncertainties, Theorem 4 looks as follows:. They only provide a lower bound for the probability of the conclusion given the probabilities of the premises. However, in some applications it might also be informative to have an upper bound for the conclusion's probability. For example, if one knows that this probability has an upper bound of 0.

Why Probability?

They presuppose that the premises' exact probabilities are known. In such applications it would be useful to have a method to calculate optimal lower and upper bounds for the probability of the conclusion in terms of the upper and lower bounds of the probabilities of the premises. Hailperin , , , and Nilsson use methods from linear programming to show that these two restrictions can be overcome.

Their most important result is the following:. Theorem 5. This result can also be used to define yet another probabilistic notion of validity, which we will call Hailperin-probabilistic validity or simply h-validity. This notion is not defined with respect to formulas, but rather with respect to pairs consisting of a formula and a subinterval of [0,1]. Nilsson's work on probabilistic logic , has sparked a lot of research on probabilistic reasoning in artificial intelligence Hansen and Jaumard ; chapter 2 of Haenni et al.

Contemporary approaches based on probabilistic argumentation systems and probabilistic networks are better capable of handling these computational challenges. Furthermore, probabilistic argumentation systems are closely related to Dempster-Shafer theory Dempster ; Shafer ; Haenni and Lehmann However, an extended discussion of these approaches is beyond the scope of the current version of this entry; see Haenni et al.

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  4. In this section we will study probability logics that extend the propositional language L with rather basic probability operators. Subsection 2. There are several applications in which qualitative theories of probability might be useful, or even necessary. In some situations there are no frequencies available to use as estimates for the probabilities, or it might be practically impossible to obtain those frequencies.

    Continuum mechanics

    In such situations qualitative probability logics will be useful. One of the earliest qualitative probability logics is Hamblin's This means that it is not a normal modal operator, and cannot be given a Kripke relational semantics. Finally, it should be noted that with comparative probability a binary operator , one can also express some absolute probabilistic properties unary operators. The semantics of propositional probability logic involves a probability function P , satisfying certain properties.

    Here we consider P as an operator in the object language. It is thus natural to involve addition and more generally, linear combinations in a probability language with probability operators. But we will see that much can be expressed without linear combinations explicitly in the language.

    It is often desirable to have as few definitions as primitive and to generate further definitions from the primitive definitions.

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    7. This allows us to specify the language more concisely. Let us first look at what can be expressed using linear combinations of a basic primitive form. Here are some examples of what can be expressed. Note that we do not even consider coefficients of the probability term. In short, we very often face meta-uncertainty: uncertainty about the extent and nature of the uncertainty we face. There are various formal tools that can help with some of these uncertainties. These include imprecise probabilities and preference aggregation functions. But theses tools have their limits. In this subgroup we will investigate existing and new formal methods for quantifying meta-uncertainty as well as more qualitative approaches such as qualitative probability and utility assignments, precautionary reasoning, and maxi-min decision making.

      Bradley S Imprecise Probabilities. Zalta ed. Nature — Ecological Applications 12 2 : — Law, Probability and Risk 5 1 : 19— Cambridge University Press, Cambridge. Consequently, the ability to specify code and verify program correctness can be improved. Two developments are needed, the first of which introduces the binding time distinction into the lambda calculus in a manner analogous with the introduction of types into the untyped lambda calculus.

      Methods are also presented for introducing combinators for run-time. The second concerns the The authors describe here a framework in which the type notation of functional languages is extended to include a notation for binding times that is Extensions of First-Order Logic.

      Classical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic MSL provides a unifying framework in which to place, for example, second-order logic, type theory, modal and dynamic logics and MSL itself. The aim is two fold: only one theorem-prover is needed; proofs of the metaproperties of the different existing calculi can be avoided by borrowing them from MSL.

      To make the book accessible to readers from This is a Free Choice Petri Nets. Petri nets are a popular and powerful formal model for the analysis and modelling of concurrent systems, and a rich theory has developed around them. Petri nets are taught to undergraduates, and also used by industrial practitioners. This book focuses on a particular class of petri nets, free choice petri nets, which play a central role in the theory. The text is very clearly organised, with every notion carefully explained and every result proved.

      Intended learning outcomes

      Clear exposition is given for place invariants, siphons, traps and many other important analysis techniques. The material is organised along the This book develops the theory of typed feature structures, a data structure that generalizes both first-order terms and feature structures of unification-based grammars to include inheritance, typing, inequality, cycles, and intensionality. This book develops the theory of typed feature structures, a data structure that generalizes both first-order terms and feature structures of unificat Deductive and Declarative Programming.

      In this book, the author develops deduction-oriented methods for reasoning about functional and logic programs. The methods are based on the inductive theories of suitable data type specifications and exploit both classical theorem-proving and term rewriting. Detailed examples accompany the development of the methods, and their use is supported by a prototyping system that is documented at the end of the book.