What is calculus in mathematics

logical calculation

Quadruple \ ({\ mathcal {K}} \ text {= (} E, \, A, \, S, \, F \ text {)} \) with the following determinants:

  1. E. is a non-empty set, the elements of which serve as the basic symbols of the calculus (E. is also called the alphabet of \ ({\ mathcal {K}} \)).
  2. A. is a suitable subset of the free semigroup E.* above E. (E.* is the set of all words, i.e. the finite strings of basic characters in \ ({\ mathcal {K}} \)). A. is called the set of expressions of \ ({\ mathcal {K}} \); A. is usually defined inductively (see also Propositional calculus, Predicate calculus).
  3. S. is a special set of expressions, the set of sentences of \ ({\ mathcal {K}} \).
  4. F. is a mapping (it is called the derivative relation of \ ({\ mathcal {K}} \)), that of each subset XA. a part F.(X) ⊆ A. with the following properties:
    1. XF.(X),
    2. if X1X2so F.(X) ⊆ F.(X2),
    3. F.(F.(X)) ⊆ F.(X),
    4. to each aF.(X) there is a finite subset X0X, so that aF.(X0),
    5. F.(S.) ⊆ S.(⇒ F.(S.) = S.).
    (a) - (c) are the envelope properties of the derivative relation F., (d) symbolizes the finiteness law regarding F., and (e) means that only (valid) sentences can be derived from (valid) sentences.

The most important examples of logical calculi are the propositional and predicate calculus (see also elementary language). A distinction is often made between calculi with a syntactically defined set of sentences (and the corresponding derivation relation) and semantically defined set of sentences (and a corresponding inference relation).