## Recursive Definition of Proposition

A proposition (also called propositional form) here is a template for propositions. It is a 'legal' form for propositions. Every proposition must take one of these forms. It shows how larger more complex propositions can be generated from simpler ones.

Definition of Proposition (Propositional Form)

1. Basis Clause: The truth values 'true' and 'false', and all propositional variables such as P and Q are a proposition. Here a propositional variable is a variable that takes an individual specific proposition as its value.

2. Inductive Claus: If E and F are propositions, then , , , , and are propositions.
3. Extremal Clause: Nothing is a proposition unless it is obtained by 1. and 2.

Note : As you might have noticed, [ ] and ( ) are used interchangeably for propositions.

Example

, where P, Q, R, and S are propositional variables, is a proposition because it can be obtained by first generating , and by applying the inductive clause to the propositional variables P and Q, and R and S, respectively, then by combining them with agian applying the inductive clause.