In quantitative logic, every atomic formula has the same truth degree. To overcome the shortcomings of truth degree, this paper gives randomized truth degree by introducing randomized mapping in Lukasiewicz three-valued propositional logic. Randomized truth degree is a generalization of truth degree, and the role of the different atomic formula is reflected in it. Based on randomized truth degree, randomized logic pseudo-metric is obtained and it is proved that randomized logic pseudo-metric is also an extension of metric space in quantified logic. Randomized truth degree is a combination of quantitative logic and probability logic and is also a generalization of probabilities of formulas in classical propositional logic.