Probabilistic logic

The aim of a probabilistic logic (also probability logic and probabilistic reasoning) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure of formal argument. The result is a richer and more expressive formalism with a broad range of possible application areas. Probabilistic logics attempt to find a natural extension of traditional logic truth tables: the results they define are derived through probabilistic expressions instead. A difficulty with probabilistic logics is that they tend to multiply the computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as those of Dempster-Shafer theory. The need to deal with a broad variety of contexts and issues has led to many different proposals.

Historical context

There are numerous proposals for probabilistic logics. Very roughly, they can be categorized into two different classes: those logics that attempt to make a probabilistic extension to logical entailment, such as Markov logic networks, and those that attempt to address the problems of uncertainty and lack of evidence (evidentiary logics).

That probability and uncertainty are not quite the same thing may be understood by noting that, despite the mathematization of probability in the Enlightenment, mathematical probability theory remains, to this very day, entirely unused in criminal courtrooms, when evaluating the "probability" of the guilt of a suspected criminal.[1]

More precisely, in evidentiary logic, there is a need to distinguish the truth of a statement from the confidence in its truth: thus, being uncertain of a suspect's guilt is not the same as assigning a numerical probability to the commission of the crime. A single suspect may be guilty or not guilty, just as a coin may be flipped heads or tails. Given a large collection of suspects, a certain percentage may be guilty, just as the probability of flipping "heads" is one-half. However, it is incorrect to take this law of averages with regard to a single criminal (or single coin-flip): the criminal is no more "a little bit guilty" than a single coin flip is "a little bit heads and a little bit tails": we are merely uncertain as to which it is. Conflating probability and uncertainty may be acceptable when making scientific measurements of physical quantities, but it is an error, in the context of "common sense" reasoning and logic. Just as in courtroom reasoning, the goal of employing uncertain inference is to gather evidence to strengthen the confidence of a proposition, as opposed to performing some sort of probabilistic entailment.

Historically, attempts to quantify probabilistic reasoning date back to antiquity. There was a particularly strong interest starting in the 12th century, with the work of the Scholastics, with the invention of the half-proof (so that two half-proofs are sufficient to prove guilt), the elucidation of moral certainty (sufficient certainty to act upon, but short of absolute certainty), the development of Catholic probabilism (the idea that it is always safe to follow the established rules of doctrine or the opinion of experts, even when they are less probable), the case-based reasoning of casuistry, and the scandal of Laxism (whereby probabilism was used to give support to almost any statement at all, it being possible to find an expert opinion in support of almost any proposition.).[1]

Modern proposals

Below is a list of proposals for probabilistic and evidentiary extensions to classical and predicate logic.

Possible application areas

See also


  1. 1 2 James Franklin, The Science of Conjecture: Evidence and Probability before Pascal, 2001 The Johns Hopkins Press, ISBN 0-8018-7109-3
  2. Nilsson, N. J., 1986, "Probabilistic logic," Artificial Intelligence 28(1): 71-87.
  3. Jøsang, A., 2001, "A logic for uncertain probabilities," International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 9(3):279-311.
  4. Jøsang, A. and McAnally, D., 2004, "Multiplication and Comultiplication of Beliefs," International Journal of Approximate Reasoning, 38(1), pp.19-51, 2004
  5. Jøsang, A., 2008, "Conditional Reasoning with Subjective Logic," Journal of Multiple-Valued Logic and Soft Computing, 15(1), pp.5-38, 2008.
  6. Gerla, G., 1994, "Inferences in Probability Logic," Artificial Intelligence 70(1–2):33–52.
  7. 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.
  8. 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.
  9. Ruspini, E.H., Lowrance, J., and Strat, T., 1992, "Understanding evidential reasoning," International Journal of Approximate Reasoning, 6(3): 401-424.

Further reading

External links

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