"If-Then" Tables Explained
Sans Doute
(Part 3)

So far we've filled in one row of the "If-Then" Truth Table.

Truth Table: If-Then Statements
QH	RC	QH → RC
TRUE	TRUE	?
TRUE	FALSE	?
FALSE	TRUE	?
FALSE	FALSE	TRUE

So suppose I had picked another card instead. And it was the Queen of Diamonds.

And now what are the Truth Values?

Well, we didn't select the Queen of Hearts. So the antecedent is FALSE.

QH ≡ FALSE

But the Queen of Diamonds IS a red card. So in this case, the consequent is TRUE.

RC ≡ TRUE

So our entire statement:

QH → RC

... has the assignment:

FALSE → TRUE

Now that I've picked the wrong card of the right color, is my original statement:

QH → RC

... TRUE or FALSE?

Well, dang it, it's still TRUE. Why? Because it's still the case that picking the wrong card of the right color does not change the fact that IF I HAD picked the Queen of Hearts THEN it still WOULD HAVE BEEN red. Picking a wrong card of the right color doesn't change the truth of this statement.

In fact, we've seen that as long as the antecedent it FALSE, then the whole "If-Then" statement will always be TRUE.

At this point we'll take a pause - not that refreshes but educates. We just have understand that an IF-THEN statement simply does not require that the IF-part be TRUE.

Instead, it is asking what-if the IF-part is TRUE. That is, the question is IF the antecedent is TRUE, is the consequent also TRUE?

But what if we already know that the consequent is TRUE?

Well, a little reflection tells us it doesn't matter if the antecedent is TRUE or FALSE. Since the consequent is known to be TRUE, the whole IF-THEN statement is TRUE.

Look at it this way. Suppose you assert:

George Washington was the first President of the United States.

... which we represent as:

G

But after you've stated George was the first president, suppose some curmudgeon says:

But wait a minute! George was the first President, yes, but suppose the Eagles win the Superbowl?

In other words, he's constructed the following IF-THEN statement:

If the Eagles win the Superbowl, then George Washington was the first president of the United States.

... which we represent as:

W → G

Is this statement TRUE of FALSE?

Well, it's TRUE. After all, if the Eagles win the Superbowl, George was still the first President.

But now the curmudgeon says:

Hold on, then! George was the first President, but suppose the Eagles don't win the Superbowl?

¬W → G

Is this statement TRUE of FALSE?

Well, it's STILL TRUE! After all, even if the Eagles don't win the Superbowl, George was still the first President.

We see then that if we know the consequent is TRUE, then the whole IF-THEN statement is always TRUE regardless of the truth value of the antecenent.

So we can now fill in the blank, and the FALSE → TRUE part of the Truth Table is:

Truth Table: If-Then Statements
QH	RC	QH → RC
TRUE	TRUE	?
TRUE	FALSE	?
FALSE	TRUE	TRUE
FALSE	FALSE	TRUE

Hm. Things are starting to take shape. And so we will keep going clicking here.