Propositional Calculus

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:

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

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

1. By rule 1, is a formula.
2. By rule 2, is a formula.
3. By rule 1, is a formula.
4. By rule 2, is a formula.