A **propositional calculus** is a formal system, where:

- The
*alpha set*is a finite set of elements called*proposition symbols*or*propositional variables*. Syntactically speaking, these are the most basic elements of the formal language, otherwise referred to as*atomic formulæ*or*terminal elements*. In the examples to follow, the elements of are typically the letters, and so on.

- The
*omega set*is a finite set of elements called*operator symbols*or*logical connectives*. The set is partitioned into disjoint subsets as follows:

- In this partition, is the set of operator symbols of
*arity*.

- In the more familiar propositional calculi, is typically partitioned as follows:

- A frequently adopted convention treats the constant logical values as operators of arity zero, thus:

- Some writers use the tilde (~), or N, instead of ; and some use the ampersand (&), the prefixed K, or instead of . Notation varies even more for the set of logical values, with symbols like {false, true}, {F, T}, or all being seen in various contexts instead of {0, 1}.

- The
*zeta set*is a finite set of*transformation rules*that are called*inference rules*when they acquire logical applications.

- The
*iota set*is a finite set of*initial points*that are called*axioms*when they receive logical interpretations.

The *language* of, also known as its set of *formulæ*, *well-formed formulas* or *wffs*, is inductively defined by the following rules:

- Base: Any element of the alpha set is a formula of .
- If are formulæ and is in, then is a formula.
- Closed: Nothing else is a formula of .

Repeated applications of these rules permits the construction of complex formulæ. For example:

- By rule 1, is a formula.
- By rule 2, is a formula.
- By rule 1, is a formula.
- By rule 2, is a formula.

Read more about Propositional Calculus: Example 1. Simple Axiom System, Example 2. Natural Deduction System, Basic and Derived Argument Forms, Proofs in Propositional Calculus, Soundness and Completeness of The Rules, Interpretation of A Truth-functional Propositional Calculus, Alternative Calculus, Equivalence To Equational Logics, Graphical Calculi, Other Logical Calculi, Solvers

### Famous quotes containing the word calculus:

“I try to make a rough music, a dance of the mind, a *calculus* of the emotions, a driving beat of praise out of the pain and mystery that surround me and become me. My poems are meant to make your mind get up and shout.”

—Judith Johnson Sherwin (b. 1936)