Biconditional Propositions

Symbolic Logic

 

 

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

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:

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 p is equal to q, and q is equal to p.

The truth table below illustrates this point.

The truth table above says:

  1. If p is true and q is true, then p q is true.
  2. If p is true and q is false, then p q is false.
  3. If p is false and q is true, then p q is false.
  4. If p is false and q is false, then p q is true.

Now, suppose we have the example ~p q. How do we determine its truth value if p is true and q is false?

Let me illustrate.

The illustration says that p is true and q is false. Now, before we apply the rules in biconditional in the statement ~p q, we need to simplify ~p first because the truth value “true” is assigned to p and not to ~p. If we recall our discussion on the rule in negation, we learned that the negation of true is false. So, if p is true, then ~p is false. Thus, at the end of it all, ~p q is 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/

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