Probabilistic logic

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The aim of a probabilistic logic (or probability logic) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. The result is a richer and more expressive formalism with a broad range of possible application areas. The difficulty with probabilistic logics is that they tend to multiply the computational complexities of their probabilistic and logical components.

1 Proposals
2 Possible application areas
3 References
4 See also
5 External links

[edit] Proposals

There are numerous proposals for probabilistic logics:

The term "probabilistic logic" was first used in [N86], where the truth values of sentences are probabilities. The proposed semantical generalization induces a probabilistic logical entailment, which reduces to ordinary logical entailment when the probabilities of all sentences are either 0 or 1. This generalization applies to any logical system for which the consistency of a finite set of sentences can be established.

In the theory of probabilistic argumentation [KM95,H05], probabilities are not directly attached to logical sentences. Instead it is assumed that a particular subset $W$ of the variables $V$ involved in the sentences defines a probability space over the corresponding sub-σ-algebra. This induces two distinct probability measures w.r.t. $V$ , which are called degree of support and degree of possibility, respectively. Degrees of support can be regarded as non-additive probabilities of provability, which generalizes the concepts of ordinary logical entailment (for $V = {}$ ) and classical posterior probabilities (for $V = W$ ). Mathematically, this view is compatible with the Dempster-Shafer theory.

The theory of evidential reasoning [RLS90] also defines non-additive probabilities of probability (or epistemic probabilities) as a general notion for both logical entailment (provability) and probability. The idea is to augment standard propositional logic by considering an epistemic operator K that represents the state of knowledge that a rational agent has about the world. Probabilities are then defined over the resulting epistemic universe Kp of all propositional sentences p, and it is argued that this is the best information available to an analyst. From this view, Dempster-Shafer theory appears to be a generalized form of probabilistic reasoning.

Approximate reasoning formalism proposed by fuzzy logic is used in [G94] to obtain a logic in which the models are the probability distributions and the theories are the lower envelopes. The question of the consistency of the available information is strictly related with the one of the coherence of partial probabilistic assignment and therefore with Dutch book phenomenon.

The central concept in the theory of subjective logic [J01] are opinions about some of the propositional variables involved in the given logical sentences. An opinion is a two-dimensional extension of a single probabiliy value to express various degrees of ignorance. For the computation of overall opinions w.r.t. some query variables, the theory proposes respective operators for various logical connectives. Most of them are fully compatible with Dempster's rule of combination.

[edit] Possible application areas

[edit] References

[A98] E. W. Adams, 1998. A Primer of Probability Logic. CSLI Publications (Univ. of Chicago Press).
[C37] Rudolf Carnap, 1937. Logical Foundations of Probability. University of Chicago Press.
[C91] Chuaqui, R., 1991. Truth, Possibility and Probability: New Logical Foundations of Probability and Statistical Inference. Number 166 in Mathematics Studies. North-Holland.
[G94] Gerla, G., 1994, "Inferences in Probability Logic," Artificial Intelligence 70(1–2):33–52.
[H05] Haenni, R, 2005, "Towards a Unifying Theory of Logical and Probabilistic Reasoning," ISIPTA'05, 4th International Symposium on Imprecise Probabilities and Their Applications: 193-202. [1]
Hajek, Alan, 2001, "Probability, Logic, and Probability Logic," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic, Blackwell.
[J01] A. Jøsang, A., 2001, "A logic for uncertain probabilities," International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 9(3):279-311.
[KM95] Kohlas, J., and Monney, P.A., 1995. A Mathematical Theory of Hints. An Approach to the Dempster-Shafer Theory of Evidence. Vol. 425 in Lecture Notes in Economics and Mathematical Systems. Springer Verlag.
[K70] Henry Kyburg, 1970. Probability and Inductive Logic Macmillan.
[K74] H. E. Kyburg, 1974. The Logical Foundations of Statistical Inference, Dordrecht: Reidel.
[KT01] H. E. Kyburg and C. M. Teng, 2001. Uncertain Inference, Cambridge: Cambridge University Press.
[N86] Nilsson, N. J., 1986, "Probabilistic logic," Artificial Intelligence 28(1): 71-87.
[R05] Romeijn, J. W., 2005. Bayesian Inductive Logic. PhD thesis, Faculty of Philosophy, University of Groningen, Netherlands. [2]
[RLS92] Ruspini, E.H., Lowrance, J., and Strat, T., 1992, "Understanding evidential reasoning," International Journal of Approximate Reasoning, 6(3): 401-424.
[W02] Williamson, J., 2002, "Probability Logic," in D. Gabbay, R. Johnson, H. J. Ohlbach, and J. Woods, eds., Handbook of the Logic of Argument and Inference: the Turn Toward the Practical. Elsevier: 397-424.