P implies Q


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:

  1. If P then Q
  2. P only if Q
  3. P is a sufficient condition for Q
  4. Q is a necessary condition for P
  5. Q if P
  6. Q follows P
  7. Q provided P
  8. Q is a logical consequence of P
  9. Q whenever P

P = It is raining

Q = I get wet.

P => Q

  1. IF it is raining THEN I get wet
  2. I get wet ONLY IF it is raining
  3. It is raining IS A SUFFICIENT CONDITION FOR it is raining
  4. I get wet IS A NECESSARY CONDITION FOR it is raining
  5. I get wet IF it is raining
  6. I get wet FOLLOWS it is raining
  7. I get wet PROVIDED it is raining
  8. I get wet IS A LOGICAL CONSEQUENCE OF it is raining
  9. 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.

For example.

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.

Picture 3

Logical Identity

(p=>q) <=> (not p v q)

Picture 4


P or Q


The compound statement P v Q is only true when P is true or Q is true.

P = The Man is running
Q = The Man is walking

The man is walking OR the man is running.

Although the Man cannot run and walk at the same, the truth table shows that if the Man could run and walk at the same time then P OR Q would be be true.

P= The Car Engine is running
Q=The Car Radio is on

Now the truth table seems to make more sense because we can experience both these things can happen at the same time.

So, the Car Engine is running OR the Car Radio is on, makes sense even if both things were happening at the same time (simply because they can happen at the same time).

see more explanations under P and Q


OR is associative so it can be used without any ambiguity.

For example:

((P v Q)  v R) <=> P v Q v R

P and Q

The compound statement P ^ Q is only true when P and Q are true.

P = The Car Engine is on
Q = The Car Radio is on

The truth table makes sense in all cases because the truths correlate with our real life experience.

P = The Cat is asleep
Q= The Cat is awake

P AND Q in the truth table is true, but a cat cannot be awake and asleep the same time – so how can it be true.

Well, the truth table knows nothing about what a cat can do or cannot do.
All the truth table shows, is all possible combination of truths using the AND operator.

A cat being asleep and awake at the same time is just a combination of truths that do not actually occur in reality.


AND is associative so it can be used without any ambiguity.

For example:

((P ^ Q)  ^ R) <=> P ^ Q ^ R