Biconditional Statements in Propositional Logic

Biconditional statements are compound propositions related by the phrases “if and provided that.” 

The image for “if and provided that” is a ≡ (triple bar). Let’s contemplate the instance under.

I’ll take a depart of absence if and solely the administration permits me to. (p, q)

If we let p stand for “I’ll take a depart of absence” and q for “The administration permits me to,” then the biconditional proposition “I’ll take a depart of absence if and provided that the administration permits me to” is symbolized as follows:

p ≡ q

Please observe that the connective “if and provided that” shouldn’t be confused with “provided that.” The connective “provided that” is a connective of a conditional proposition. Let’s take the instance under:

I’ll take a depart of absence solely if the administration permits me to. (p, q)

We now have to take observe that the proposition that comes after the connective “provided that” is a consequent. Thus, if we let p stand for “I’ll take a depart of absence” and q for “The administration permits me to,” then the proposition is symbolized as follows: p ⊃ q. 

Guidelines in Biconditional Propositions

1. A biconditional proposition is true if each elements have the identical fact worth.
2. Thus, if one is true and the opposite is false, or if one is false and the opposite true, then the biconditional proposition is false.

As we will see, the principles in biconditional propositions say that the one occasion whereby the biconditional proposition turns into true is when each part propositions have the identical fact worth. It’s because, in biconditional propositions, each part propositions indicate one another. Thus, the instance above, that’s, “I’ll take a depart of absence if and provided that the administration permits me to” might be restated as follows:

If I’ll take a depart of absence, then the administration permits me to; and if the administration permits me to, then I’ll take a depart of absence.

Thus, the image p ≡ q means p is the same as q, and q is the same as p.

The reality desk under illustrates this level.

The reality desk 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’ve the instance ~p ≡ q. How will we decide its fact worth if p is true and q is fake?

Let me illustrate.

The illustration says that p is true and q is false. Now, earlier than we apply the principles in biconditional within the assertion ~p ≡ q, we have to simplify ~p first as a result of the reality worth “true” is assigned to p and to not ~p. If we recall our dialogue on the rule in negation, we realized that the negation of true is false. So, if p is true, then ~p is false. Thus, on the finish of all of it, ~p ≡ q is true.