The first time I woke up this morning, I was dreaming about failing at ping pong. It was an informal group playing doubles table tennis according to the informal rules, all very pleasant until it was my turn to serve. And I couldn't. The muscle memory was gone! All of my attempts to hit the little white sphere were abject failures. In my old age, I was no longer capable of table tennis.
That was so depressing a dream that I went back to sleep and tried again.
The second time I woke up, I was dreaming about implication: If we
define assertion a to be "Mommy! Mommy! I just dropped a rotten
apple on the kitchen floor!", then we can define the implication
a implies b or
where b means "There is another mess in the kitchen that I have to clean up." Any time when a is true, b is also true.
Of course, knowing b doesn't tell us anything about a. Somebody else may have dropped a rotten apple or tipped over the milk carton, or the cat may have stuffed herself too full of the new cat food. The truth of b does not imply a and implication is not symmetric. One should always be careful of such things while waking up.
As I slowly awoke I began to recapitulate the propositional calculus of inference by developing and comparing truth tables. The key idea, which only returned to my mind at a certain level of consciousness, is that the inferential relationship (a ⇒ b) may also be thought of as an operator with its own truth table. Then one can see that the truth table for implication is the same as the truth table for the OR operator, ∨, except that the rows are interchanged. But we can remedy that by considering ¬a (NOT a) instead of a.
As I roused up completely from my state of semi-consciousness I was pleased to realize that in actual reality
Truly, this all makes sense even when one is fully awake.