## Biconditional Propositions

**Biconditional Propositions**

Biconditional propositions are compound propositions connected by the words “**if and only if**.” As we learned in the previous discussion titled “Propositions and Symbols Used in Symbolic Logic,” 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

**for “The administration allows me to,” then the biconditional proposition “I will take a leave of absence**

*q***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

**for “The administration allows me to,” then the proposition is symbolized as follows:**

*q*

*p***⊃**

**. For a more detailed discussion on conditional propositions, see http://philonotes.com/index.php/2018/02/11/conditional-propositions/.**

*q***Rules in Biconditional Propositions**

- A biconditional proposition is
**true**if both components have the same truth value. - 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:

If I will take a leave of absence, then the administration allows me to; and if the administration allows me to, then I will take a leave of absence.

Thus, the symbol *p***≡ **** q** means

**is equal to**

*p***, and**

*q***is equal to**

*q***.**

*p*The truth table below illustrates this point.

The truth table above says:

- If
is*p***true**andis*q***true**, then*p**≡*is*q***true**. - If
is*p***true**andis*q***false**, then*p**≡*is*q***false**. - If
is*p***false**andis*q***true**, then*p**≡*is*q***false**. - If
is*p***false**andis*q***false**, then*p**≡*is*q***true**.

Now, suppose we have the example *~p **≡ **q*** .** How do we determine its truth value if

**is true and**

*p***is false?**

*q*Let me illustrate.

The illustration says that ** p** is

**true**and

**is**

*q***false**. Now, before we apply the rules in biconditional in the statement

*~*

*p***≡**

**, we need to simplify**

*q*

*~***first because the truth value “**

*p***true**” is assigned to

**and not to**

*p*

*~***. If we recall our discussion on the**

*p***rule in negation**, we learned that the negation of

**true**is

**false**. So, if

**is**

*p***true**, then

*~***is**

*p***false**. Thus, at the end of it all,

*~*

*p***≡**

**is**

*q***true.**

** **

We have provided a video of all our posts in Symbolic Logic. If you are interested, please visit the following:

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

See also “Propositions and Symbols Used in Symbolic Logic” http://philonotes.com/index.php/2018/02/02/symbolic-logic/