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.
((P ^ Q) ^ R) <=> P ^ Q ^ R