P => Q is only FALSE when the Premise(p) is TRUE AND Consequence(Q) is FALSE.
P => Q is always TRUE when the Premise(P) is FALSE OR the Consequence(Q) is TRUE.
P IMPLIES Q can be formulated in the following ways:
- If P then Q
- P only if Q
- P is a sufficient condition for Q
- Q is a necessary condition for P
- Q if P
- Q follows P
- Q provided P
- Q is a logical consequence of P
- Q whenever P
P = It is raining
Q = I get wet.
P => Q
- IF it is raining THEN I get wet
- I get wet ONLY IF it is raining
- It is raining IS A SUFFICIENT CONDITION FOR it is raining
- I get wet IS A NECESSARY CONDITION FOR it is raining
- I get wet IF it is raining
- I get wet FOLLOWS it is raining
- I get wet PROVIDED it is raining
- I get wet IS A LOGICAL CONSEQUENCE OF it is raining
- I get wet WHENEVER it is raining
The Premise(P) and the Consequence(Q) need not be related. The interpretation of the truth table depends on the formulation P and Q.
If the Moon is made of Cheese, then, the frog is blue.
The statement: The Moon is made of Cheese, and
The statement: The frog is blue.
are in no way related to each other, but nevertheless the truth table for P=>Q is valid. We would have to come up with a very magical Universe and a world for these two statements to be related and for them to make sense. That is just a question of imagination and interpretation not question of whether the true table is correct or not.
Even if we took these two statements for our world, then would both be false so:
False => False
So looking at the truth table where P is false and Q is false we see that P => Q is True.
So, If the The Moon is made of Cheese then the frog will be blue.
Usually, however P and Q will be related and everything will make sense.
The implies operator is ambigous so you need to use parenthesis when using it.
For example P => Q => R is ambigous. Does it mean (P=>Q) => R) or (P=>(Q => R))?
Here the truth table to show that they are not the same.
(p=>q) <=> (not p v q)