The conditional – “p implies q” or “if p, then q”
Here p is the condition and q is the conclusion.
The logic behind the truth table can be illustrated with examples.
Example : “If you get an A, then I’ll give you a dollar.”
Let it consider as a promise, and this particular example will be true, if this promise is kept
Condition says: p is true and q is true
The conditional statement is saying that if p is true, then q will immediately follow and thus be true. So, the first row naturally follows this definition.
It’s true that you get an A and it’s true that I give you a dollar. Since I kept my promise, the implication is true.
Condition says: p is true and q is false
Similarly, the second row follows this because is we say “p implies q”, and then p is true but q is false, then the statement “p implies q” must be false, as q didn’t immediately follow p.
It’s true that you get an A but it’s false that I give you a dollar. Since I didn’t keep my promise, the implication is false.
Condition says: p is false, q is true.
If p is false, then p→q is true, no matter whether q is true or not. For instance:
if it’s false that you get an A? Whether or not I give you a dollar, I haven’t broken my promise. Thus, the implication can’t be false, so (since this is a two-valued logic) it must be true
Condition says: p is false, q is false.
If p is false i.e. you do not get A and q is false i.e. I do not give you a dollar. It directs that i haven’t broken the promise. So it must be true.
The biconditional uses a double arrow because it is really saying “p implies q” and also “q implies p”. Symbolically, it is equivalent to:
This form can be useful when writing proof or when showing logical equivalencies.
|p||Q||p→q||p→q||p ↔ q|
To help you remember the truth tables for these statements, you can think of the following:
The conditional, p implies q, is false only when the condition is true but the conclusion is false. Otherwise it is true.
The biconditional, p iff q, is true whenever the two statements i.e. condition as well as the conclusion have the same truth value. Otherwise it is false.