Probabilistic Logic - Modern Proposals

Modern Proposals

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

• The term "probabilistic logic" was first used in a paper by Nils Nilsson published in 1986, 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.

• The central concept in the theory of subjective logic are opinions about some of the propositional variables involved in the given logical sentences. A binomial opinion applies to a single proposition and is represented as a 3-dimensional extension of a single probability value to express various degrees of ignorance about the truth of the proposition. For the computation of derived opinions based on a structure of argument opinions, the theory proposes respective operators for various logical connectives, such as e.g. multiplication (AND), comultiplication (OR), division (UN-AND) and co-division (UN-OR) of opinions as well as conditional deduction (MP) and abduction (MT).

• Approximate reasoning formalism proposed by fuzzy logic can be used to obtain a logic in which the models are the probability distributions and the theories are the lower envelopes. In such a logic 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.

• Markov logic networks implement a form of uncertain inference based on the maximum entropy principle—the idea that probabilities should be assigned in such a way as to maximize entropy, in analogy with the way that Markov chains assign probabilities to finite state machine transitions.

• Systems such as Pei Wang's Non-Axiomatic Reasoning System (NARS) or Ben Goertzel's Probabilistic Logic Networks (PLN) add an explicit confidence ranking, as well as a probability to atoms and sentences. The rules of deduction and induction incorporate this uncertainty, thus side-stepping difficulties in purely Bayesian approaches to logic (including Markov logic), while also avoiding the paradoxes of Dempster-Shafer theory. The implementation of PLN attempts to use and generalize algorithms from logic programming, subject to these extensions.

• In the theory of probabilistic argumentation, probabilities are not directly attached to logical sentences. Instead it is assumed that a particular subset of the variables involved in the sentences defines a probability space over the corresponding sub-σ-algebra. This induces two distinct probability measures with respect to, 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 ) and classical posterior probabilities (for ). Mathematically, this view is compatible with the Dempster-Shafer theory.

• The theory of evidential reasoning 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.