Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications
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Sentential Probability Logic
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The next two chapters deal with the nineteenth-century work of Bolzano, De Morgan, and Boole, and the twentieth-century topic of probabilistic inference. The fourth chapter introduces an original form of logic in which probability values play a semantic role comparable to that of truth values in conventional verity logic. Serving to distinguish probability logic from verity logic, both of which nevertheless share a common formal syntax, is the key semantic notion of logical consequence. The fifth chapter covers an extension of probability logic that includes conventional probability.
Involved is a new logical notion, that of the suppositional B, supposing that A is the case replacing the ordinary conditional B, if A. Hailperin cites a number of valid logical consequences with conditional probabilities including an especially interesting example, Boole's Challenge Problem. Note that 0 is actually a kind of certainty, viz. According to this interpretation, the following theorem follows from the strong soundness and completeness of probabilistic semantics:.
Theorem 1. 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.
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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 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.
Inference in conditional probability logic
Of course, more interesting cases arise when the premises are less than absolutely certain. One can easily show that. 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. Theorem 2. If a valid argument has a small number of premises, each of which only has a small uncertainty i. Theorem 3.
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The upper bound provided by Theorem 2 can also be used to define a probabilistic notion of validity. Adams-probabilistic validity has an alternative, equivalent characterization in terms of probabilities rather than uncertainties. Adams , also defines another logic for which his probabilistic semantics is sound and complete.
2. Propositional Probability Logics
However, this system involves a non-truth-functional connective the probability conditional , and therefore falls outside the scope of this section. For more on probabilistic interpretations of conditionals, the reader can consult the entries on conditionals and the logic of conditionals of this encyclopedia. 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.
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. Essential premise set. Degree of essentialness. 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. Theorem 4 yields in general a tighter upper bound than Theorem 2. Hence Theorem 4 yields that. 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. For example, if one knows that this probability has an upper bound of 0. 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. 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. They differ from the logics in Section 2 in that the logics here involve probability operators in the object language. Section 3. 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.
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In such situations qualitative probability logics will be useful. This means that it is not a normal modal operator, and cannot be given a Kripke relational semantics.
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It should be noted that with comparative probability a binary operator , one can also express some absolute probabilistic properties unary operators. The resulting logic can be axiomatized completely, and is so expressive that it can even capture quantitative probabilistic logics, to which we turn now.
Some propositional probability logics include other types of formulas in the object language, such as those involving sums and products of probability terms.