- A Note on Binary Inductive Logic
- Hannes Leitgeb
- Probability and Induction
- Carnap's inductive probabilities as a contribution to decision theory
A Note on Binary Inductive Logic
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. 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.
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. 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. 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. We can define. Using this restricted probability language, we can reason about additivity in a less direct way. The formula.
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In Fagin et al. In Heifetz and Mongin , a sound and complete proof system is given for a logic without linear combinations, where additivity is broken up into implications. Many probability logics are interpreted over a single, but arbitrary probability space. Modal probability logic makes use of many probability spaces, each associated with a possible world or state. This can be viewed as a minor adjustment to the relational semantics of modal logic: rather than associate to every possible world a set of accessible worlds as is done in modal logic, modal probability logic associates to every possible world a probability distribution, a probability space, or a set of probability distributions.
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This modal setting involving multiple probabilities has generally been given a 1 stochastic interpretation, concerning different probabilities over the next states a system might transition into Larsen and Skou , and 2 a subjective interpretation, concerning different probabilities that different agents may have about a situation or each other's probabilities Fagin and Halpern Both interpretations can use exactly the same formal framework.
This general reading of the formula does not reflect any difference between modal probability logic and other probability logics with the same formula; where the difference lies is in the ability to embed probabilities in the arguments of probability terms and in the semantics.
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The following subsections provide an overview of the variations of how modal probability logic is modeled. In one case the language is altered slightly Subsection 4. The first two components of a basic modal probabilistic model are effectively the same as a Kripke frame whose relation is decorated with numbers probability values. Such a structure has different names, such as a directed graph with labelled edges in mathematics, or a probabilistic transition system in computer science.
The valuation function, as in a Kripke model, allows us to assign properties to the worlds. The semantics for formulas are given on pairs M,w , where M is a model and w is an element of the model. 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. The first set is the set W of worlds the base set of the model , but the other is an index set A often to be taken as a set of actions, agents, or players of a game.
Probability and Induction
Consider W , P , V , where. 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. Stochastic Interpretation: Consider the elements a and b of A to be actions, for example, pressing buttons on a machine.
In this case, pressing a button does not have a certain outcome. That is,. 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.
Carnap's inductive probabilities as a contribution to decision theory
Here, P 1 and P 2 are probability functions, which can have various interpretations, such as the probabilities of two agents, logical and statistical probability, or the probabilities of one agent at different moments in time Miller ; Lewis ; van Fraassen ; Halpern 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.
Subjective Interpretation: Suppose the elements a and b of A are players of a game. 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 rest of this dynamics section that every relevant set considered has positive probability.
In a modal setting, an operator [! Note that [! However, [! 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.