Propositional calculus is a method to study how propositions form more complex propositions and logical reasoning through some logical conjunction. Proposition refers to a sentence that has concrete meaning and can judge whether it is true or not.
If we regard propositions as the objects of operations like numbers, letters or algebraic expressions in algebra, and logical conjunctions as the symbols of operations like "addition, subtraction, multiplication and division" in algebra, then the process of composing complex propositions from simple propositions can be regarded as the process of logical operations, that is, the calculus of propositions.
This kind of logical operation also has certain properties like algebraic operation and satisfies certain operation rules. For example, it satisfies the laws of exchange, association and distribution, as well as logical identity, absorption, double negation, Dimmogan's law, syllogism and so on. Using these laws, we can make logical reasoning, simplify the complex and deduce whether two complex propositions are equivalent, that is, whether their truth tables are exactly the same, and so on.
A concrete model of propositional calculus is logical algebra. Logical algebra is also called commutative algebra. Its basic operations are logical addition, logical multiplication and logical negation, that is, OR, AND and NOT in propositional calculus. Operands are only 0 and 1, which are equivalent to true and false in propositional calculus.
Predicate calculus is also called propositional implication calculus. Predicate calculus is to analyze the internal structure of a proposition into a logical form with a subject and a predicate. A proposition consists of proposition connotation, logical connectives and quantifiers, and then study the logical reasoning relationship between such propositions.