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A symbolic calculus named as the R-calculus is built to revise the defects of axiomatic systems mechanically when some counterexamples are found. The R-calculus consists of the rules of logical connective symbols and logical quantifier symbols of first order languages. The concept of reachability, soundness and completeness of the R-calculus are introduced. The basic theorem of software testing based on the R-calculus is also introduced to help mechanizing model checking.