Fuzzy Logic and Probability (original) (raw)

Fuzzy and Probability Uncertainty Logics

Readings in Fuzzy Sets for Intelligent Systems, 1993

Probability theory and fuzzy logic have been presented as quite distinct theoretical foundations for reasoning and decision making in situations of uncertainty. This paper establishes a common basis for both forms of logic of uncertainty in which a basic uncertainty logic is defined in terms of a valuation on a lattice of propositions. The (non-truth-functional) connectives for conjunction, disjunction, equivalence, implication, and negation are defined in terms which closely resemble those of probability theory. Addition of the axiom of the excluded middle tO the basic logic gives a standard probability logic. Alternatively, addition of a requirement for strong truth-functionality (truth-value of connective determined by truth-value of constituents) gives a fuzzy logic with connectives, including implication, as in Lukasiewicz' infinitely valued logic. A common semantics for all such variants is given in terms of binary responses from a population. The type of population, e.g., physical events, people, or neurons, determines whether the model is of physical probability, subjective belief, or human decision-making. The formal theory and the semantics together illustrate clearly the precise similarities and differences between fuzzy and probability logics.

A logic for reasoning about the probability of fuzzy events

Fuzzy Sets and Systems, 2007

In this paper we present the logic F P (L n , L) which allows to reason about the probability of fuzzy events formalized by means of the notion of state in a MValgebra. This logic is defined starting from a basic idea exposed by Hájek in [13]. Two kinds of semantics have been introduced, namely the class of weak and strong probabilistic models. The main result of this paper is a completeness theorem for the logic F P (L n , L) w.r.t. both weak and strong models. We also present two extensions of F P (L n , L): the first one is the logic F P (L n , RP L), obtained by expanding the F P (L n , L)-language with truth constants for the rationals in [0, 1], while the second extension is the logic F CP (L n , LΠ 1 2) allowing to reason about conditional states.

Fuzzy reasoning and the logics of uncertainty

International Symposium on Multiple-Valued Logic, 1976

This paper is concerned with the foundations of fuzzy reasoning and its relationships with other logics of uncertainty. The definitions of fuzzy logics are first examined and the role of fuzzification discussed. It is shown that fuzzification of PC gives a known multivalued logic but with inappropriate semantics of implication and various alternative forms of implication are discussed. In the main section the discussion is broadened to other logics of uncertainty and it is argued that there are close links, both formal and semantic, between fuzzy logic and probability logics. A basic multivalued logic is developed in terms of a truth function over a lattice of propositions that encompasses a wide range of logics of uncertainty. Various degrees of truth functionality are then defined and used to derive specific logics including probability logic and Lukasiewicz infinitely valued logic. Quantification and modal operators over the basic logic are introduced. Finally, a semantics for the basic logic is introduced in terms of a population (of events, or people, or neurons) and the semantic significance of the constraints giving rise to different logics is discussed.

A qualitative fuzzy possibilistic logic

International Journal of Approximate Reasoning, 1995

A formal logical system dealing with both uncertainty (possibility) and vagueness (fuzziness) is investigated. It is many-valued and modal. The system is related to a many-valued tense logic. A completeness theorem is exhibited.

Possible Semantics for a Common Framework of Probabilistic Logics

Advances in Soft Computing, 2008

This paper proposes a common framework for various probabilistic logics. It consists of a set of uncertain premises with probabilities attached to them. This raises the question of the strength of a conclusion, but without imposing a particular semantics, no general solution is possible. The paper discusses several possible semantics by looking at it from the perspective of probabilistic argumentation.

Fuzzy refutations for probability and multivalued logics

International Journal of Approximate Reasoning, 1994

The deductive apparatus of a fuzzy logic is usually proposed in Hilbert's style by fixing a fuzzy subset of logical axioms and a set of fuzzy inference rules. We sketch a "refutation approach" to fuzzy deduction. In particular, this enables us to face probability logic as a particular fuzzy refutation system in the framework of fuzzy logic. Namely, we propose a refutation system in which the probabilistic theories correspond to the lower envelopes and the complete probabilistic theories correspond to the probabilities. Finally, we apply the concept of fuzzy refutation system to multivalued logic.

A logic for reasoning about probabilities

Information and Computation, 1990

ABOUT PROBABILITIES 81 pendently of ous) can be extended in a straightforward way to the language of our first logic. The measurable case of our richer logic bears some similarities to the first-order logic of probabilities considered by Bacchus [Bac88]. There are also some significant technical differences; we compare our work with that of Bacchus and the more recent results on first-order logics of probability in [AH89, Ha1891 in more detail in Section 6.

Probability-Like Functionals and Fuzzy Logic

Journal of Mathematical Analysis and Applications, 1997

The aim of this paper is to show that fuzzy logic is a suitable tool to manage several types of probability-like functionals. Namely, we show that the superadditive functions, the necessities, the upper and lower probabilities, and the envelopes can be considered theories of suitable fuzzy logics. Some general results about the compactness in fuzzy logic are also obtained. ᮊ

A Fundamental Probabilistic Fuzzy Logic Framework Suitable for Causal Reasoning

2022

In this paper, we introduce a fundamental framework to create a bridge between Probability Theory and Fuzzy Logic. Indeed, our theory formulates a random experiment of selecting crisp elements with the criterion of having a certain fuzzy attribute. To do so, we associate some specific crisp random variables to the random experiment. Then, several formulas are presented, which make it easier to compute different conditional probabilities and expected values of these random variables. Also, we provide measure theoretical basis for our probabilistic fuzzy logic framework. Note that in our theory, the probability density functions of continuous distributions which come from the aforementioned random variables include the Dirac delta function as a term. Further, we introduce an application of our theory in Causal Inference.