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The term "probabilistic logic" was first published in Niels Nilsson's 1986 paper, in which the truth of any sentence / proposition is a probability. The proposed semantic generalization induces a probabilistic logical intension, which If the probability of all sentences is zero or one, they are summed up to ordinary logical implications. This generalization can not be applied to any logical system that can establish a logical relationship between a finite set of sentences.

The central concept in Subjective Logic is thought to be some logical propositions of some propositional variables. A binomial assumption is applied to a single proposition and expressed as a three dimensional extension of a probability value, to show different degrees of ignorance of the truth of the proposition. This theory suggests operators for each logical operator (such as AND and OR and XOR) to calculate derivatives based on a structure of the thought assumptions. As well as conditional derivative (MP) and conditional abstraction (MT) operators, .

The formalization of the reasoning suggested by fuzzy logic can be used to arrive at a rationale in which models are probabilistic distributions and theories are underlying envelopes. In such a logic, the issue of the coherence of the existing information is completely related to the coherence of one of the assignments Partial probabilistic, and consequently with the phenomenon of the Dutch book.

Markov Logic networks implement a form of non-coherent inference based on the principle of maximum entropy-the idea that probabilities should be assigned in such a way that the entropy is maximized, as compared with the way that the Markov chains assign probabilities to machine translations of finite terms.

Systems like the non-monetary reasoning system (by Pei Wang) or the probabilistic logic network (or PLG by Ben Goertzel) clearly add a confidence rating, in addition to a probability to each particle and every sentence. The rules of deduction and induction, this uncertainty, and as a result of the accompanying problems, entail pure Bayesian approaches (including Markov logic), while avoiding the contradictions of the Dempster-Shafer theory. The implementation of probabilistic logic network tries to use logic programming algorithms and generalize them according to these extensions.

In the theory of probabilistic debate , probabilities are not directly linked to sentences. Instead, it is assumed that a {\ displaystyle W} W subset of the variable {\ displaystyle V} V for sentences defines a probabilistic space on the corresponding Sigma sub-algebra. This creates two different probability probabilities for {\ displaystyle V} V, which is the degree of support and degree of possible readability. The degree of support can be considered as a non-additive probability of non-additivity, which implies the logical conclusion of the ordinary (with {\ displaystyle V = \ {\}} {\ displaystyle V = \ {\}} and the later possibilities (with {\ displaystyle V = W} {\ displaystyle V = W}). This view is mathematically compatible with the Dempster-Shafer theory.

The doctrinal reasoning theory defines probability probabilities (or cognitive probabilities) as non-addictive probabilities as a general concept for the logical consequence (verifiability) and probability. The goal is to upgrade the standard of propositional logic by considering a cognitive operator K, which describes the status of the knowledge of a rational representative of the world. Then probabilities on this universe Kp include all p propositions, and they are said to be the best information available to an analyst. In this view, Dempster-Shafer's theory appears as a generalization of probabilistic reasoning.

but fuzzy:

Certain sets (Crisp sets) are in fact the same as ordinary collections, which are introduced at the beginning of the classical theory of collections. Adding a definite attribute in fact creates a distinction that helps to easily create one of the most innovative and vital concepts in fuzzy logic called membership function.

In the case of definitive sets, the membership function has only two values ​​in its range (in mathematics, the board of a function is the same as the set of all outputs of the function).

Yes and No (one and zero), which are the same two possible values ​​in classical logical logic; therefore:

{\ displaystyle \ mathbf {\ mu} _ {A} {x} = \ left \ {{\ begin {matrix} 1 & {\ mbox {if}} \ x \ in A, \\ 0 & {\ mbox {if} } \ x \ notin A. \ end {matrix}} \ right.} {\ displaystyle \ mathbf {\ mu} _ {A} {x} = \ left \ {{\ begin {matrix} 1 & {\ mbox {if }} \ x \ in A, \\ 0 & {\ mbox {if}} \ x \ notin A. \ end {matrix}} \ right.}

Here is the {\ displaystyle \ mathbf {\ mu} _ {A} (x)} {\ displaystyle \ mathbf {\ mu} _ {A} (x)} element membership function {\ displaystyle x} x in the definite set { \ displaystyle A} A.

Fuzzy sets [edit]

Main article: Fuzzy sets

The range of the membership function from {\ displaystyle \ {0,1 \}} {\ displaystyle \ {0,1}} for definite sets to the closed interval {\ displaystyle [0,1]} {\ displaystyle [0,1] } For fuzzy sets.

in few words!

fuzzy logic extends probabilistic reasoning ... see Zadeh Works