Recursive Definition
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.