Probabilistic logic


The aim of a probabilistic logic 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 in evidence-based subjective logic. 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.
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.
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 : 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, the elucidation of moral certainty, the development of Catholic probabilism, the case-based reasoning of casuistry, and the scandal of Laxism.

Modern proposals

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