Biconditional Statements in Propositional Logic

Biconditional Statements in Propositional Logic



Biconditional statements are compound propositions connected by the words “if and
only if.”

The symbol for “if and only if” is a ≡ (triple bar). Let’s consider the example below.

I will take a leave of absence if and only the administration allows me to. (p, q)

If we let p stand for “I will take a leave of absence” and q for “The administration allows
me to,” then the biconditional proposition “I will take a leave of absence if and only if
the administration allows me to” is symbolized as follows:

p≡q

Please note that the connective “if and only if” should not be confused with “only if.”
The connective “only if” is a connective of a conditional proposition. Let’s take the
example below:

I will take a leave of absence only if the administration allows me to. (p, q)

We have to take note that the proposition that comes after the connective “only if” is a
consequent. Thus, if we let p stand for “I will take a leave of absence” and q for “The
administration allows me to,” then the proposition is symbolized as follows: p ⊃ q.



Rules in Biconditional Propositions

1. A biconditional proposition is true if both components have the same truth value.
2. Thus, if one is true and the other is false, or if one is false and the other true,
then the biconditional proposition is false.

As we can see, the rules in biconditional propositions say that the only instance wherein
the biconditional proposition becomes true is when both component propositions have
the same truth value. This is because, in biconditional propositions, both component
propositions imply each other. Thus, the example above, that is, “I will take a leave of
absence if and only if the administration allows me to” can be restated as follows: