A scientific theory must stand the verification by mans observation and experiments. It will be refuted by facts whenever its logical consequence contradicts some facts supported by observations and experiments. Thus, the process of scientific discovery is one of revising the theory according to the facts. The revision consists of the following steps:
discarding the laws of the theory which lead to the contradictions in such a way that the remaining part will be the maximum subset of the theory,
generating the facts supported by the experiments to create new laws for a new theory,
bringing forth the new scientific theory by merging the new laws with the remaining part of the old theory.
It is based on the scientists intuition and insight that the laws contradicting the experiments are deleted, while their intuition and insight are supported by implicit logical reasoning and analysis. Russells work shows us that for a study concerning logical analysis and reasoning, a formal inference system of logical connectives and quantifiers can be built up to do the study. R-calculus is such a formal inference system and it is capable of deriving and deleting the laws which are in conflict with the facts. Some examples are demonstrated to show how to use the calculus.