Propositional Formula - An Algebra of Propositions, The Propositional Calculus

An Algebra of Propositions, The Propositional Calculus

An algebra (and there are many different ones), loosely defined, is a method by which a collection of symbols called variables together with some other symbols such as parentheses (, ) and some sub-set of symbols such as *, +, ~, &, V, =, ≡, ⋀, ￢ are manipulated within a system of rules. These symbols, and well-formed strings of them, are said to represent objects, but in a specific algebraic system these objects do not have meanings. Thus work inside the algebra becomes an exercise in obeying certain laws (rules) of the algebra's syntax (symbol-formation) rather than in semantics (meaning) of the symbols. The meanings are to be found outside the algebra.

For a well-formed sequence of symbols in the algebra—a formula -- to have some usefulness outside the algebra the symbols are assigned meanings and eventually the variables are assigned values; then by a series of rules the formula is evaluated.

When the values are restricted to just two and applied to the notion of simple sentences (e.g. spoken utterances or written assertions) linked by propositional connectives this whole algebraic system of symbols and rules and evaluation-methods is usually called the propositional calculus or the sentential calculus.

While some of the familiar rules of arithmetic algebra continue to hold in the algebra of propositions (e.g. the commutative and associative laws for AND and OR), some do not (e.g. the distributive laws for AND, OR and NOT).