Fuzzy reasoning

Formal symbolic methods generally assume one truth for any given query and do not admit uncertainty or multiple conclusions (without convoluted constructions). One common reaction to the inflexibility of symbolic logic is to directly encapsulate probability theory or Bayesian reasoning into a deductive framework. Such approaches extend traditional logics so that symbols not only denote fixed concepts but also can stand for rich probability distributions.

The direct translation of statistics to a deductive framework will necessarily result in certain paradoxes and counter-intuitive results that follow from probability theory (e.g., Simpson’ s Paradox, the Monty Hall Problem and the Two Envelopes Problem). Some systems, such as Wang’s Non-Axiomatic Reasoning System (NARS), avoid these problems by drawing inspiration from probabilistic reasoning but explicitly rejecting axioms of probability theory. NARS ranks the confidence of beliefs using measures that are intended to resemble probabilities but do not necessarily have any particular, strict interpretation. The values are manipulated by intuitive (but non-axiomatic) deduction rules that are designed to resemble human reasoning and biases in uncertain situations.