DHARMPROPOSITIONAL LOGIC 109(iv) Here, the LHS formula (A ↔ B), which read as “A if and only if B” is called a
biconditional formula. The formula (A ↔ B) return the truth value true if truth values of A
and B are identical ( that is either both true or both false).
A B (A ↔ B) (A → B) (B → A) (A → B) ∧ (B → A)
FFT T T T
FTF T F F
TFF F T F
TTT T T T
123 4 5 6 ← Column number
Fig. 5.10
Truth table shown in Fig. 5.10 proves the equivalence formula i.e.
(A ↔ B) ⇔ (A → B) ∧ (B → A)
Similarly verify the equivalence formula listed (v) and (vi).5.4 Propositional Logic..........................................................................................................
Continuation to the previous section 5.2 symbolization of the statements, we now define the
term propositional logic. Propositional logic has following character,
l It contains Atomic or Simple Statements called propositional variables.
viz. (A, B, C, .......) or (a, b, c, ........) or (A1, A2, A3, ......) or (B1, B2, .....) etc.
l It contains Operator Symbols or Connectors.
viz. ∧, ∨, ~, →
l It contains Parenthesis.
i.e ( , and )
l Nothing else is allowed.
Thus statements are represented by the propositional logic called statement formula. A
statement formula is an expression consisting of propositional variables, connectors and the
parenthesis. The scope of propositional variable/s is/are controlled by the parenthesis.
Let X is a set containing all statements and Y is another set consists of truth values
(true or false). Let we define the relation f i.e.,
f : X ⊗ X → Y
where, ⊗ is a boolean operator viz. ∧, ∨, ......, then relation f illustrates the mapping of com-
pound statements that are formed over set X using operator ⊗ to set Y that is either true (T) or
false (F). Assume statements A & B are in X then,
f (A, B) → {T, F}
or, ⊗ (A, B) → {T, F}
Now the question arises, how many different boolean operators are possible for ⊗. Since,
we are talking about dual-logic paradigm so each statement has two values T or F. For two
statements total numbers of possible different boolean operators are 222 = 2^4 = 16. These are
shown in Fig. 5.11.