The limitations of current logical systems in addressing vagueness and randomness restrict their application in the field of artificial intelligence (AI). To address this issue, we introduce an axiomatic mathematical system called statemental credibility logic (SCL). First, we jointly define logical operations and equivalence. Then, we introduce the concept of a truth measure. By extending the law of excluded middle and the law of non-contradiction, and using them in conjunction with other self-evident rules in the axiomization of SCL, we can incorporate classical propositional calculus and probability theory. Furthermore, SCL can handle statements that exhibit varying degrees of vagueness and randomness. We also extend SCL to deal with predicates that involve uncertainty. We believe that SCL and its extension have the potential to improve the reasoning capabilities of intelligent systems.
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