Probabilistic logic

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. Probabilistic logic makes traditional logic truth tables redundant because the results they define are dervied through probabilistic expressions instead. The difficulty with probabilistic logics is that they tend to multiply the computational complexities of their probabilistic and logical components.

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 with respect to $$V$$, 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 with repsect to some query variables, the theory proposes respective operators for various logical connectives.

Possible application areas

 * Argumentation theory
 * Artificial intelligence
 * Bioinformatics
 * Formal epistemology
 * Game theory
 * Philosophy of science
 * Psychology
 * Statistics