"If-Then" Tables Explained
Sans Doute
(Part 2)
Let's look at the "If-Then" Truth Table again.
| Truth Table: If-Then Statements | ||
| A | B | A → B |
| TRUE | TRUE | TRUE |
| TRUE | FALSE | FALSE |
| FALSE | TRUE | TRUE |
| FALSE | FALSE | TRUE |
Now this is intended for logical arguments. That is, discussions that deal with statements that can unambiguously be called TRUE or FALSE.
And they must be TRUE or FALSE even if we don't know the answer - at least , we don't know the answer yet.
OK, let's begin by clearing the Truth Table. And then we'll figure out how to fill it back in.
| Truth Table: If-Then Statements | ||
| A | B | A → B |
| TRUE | TRUE | ? |
| TRUE | FALSE | ? |
| FALSE | TRUE | ? |
| FALSE | FALSE | ? |
What we will do is see how the TRUTH VALUES of the "If" part - called the antecedent - and of the "Then" part - called the consequent - affect the truth value of the whole "If-Then" statement.
Now, suppose I have a deck of cards. And I make the statement:
If I select the Queen of Hearts, then it will be a red card.
Which we can abbreviate as:
If QH then RC
... or
QH → RC
So my table is now:
| Truth Table: If-Then Statements | ||
| QH | RC | QH → RC |
| TRUE | TRUE | ? |
| TRUE | FALSE | ? |
| FALSE | TRUE | ? |
| FALSE | FALSE | ? |
Now I pick a card. But it is the Queen of Clubs.
And what are the Truth Values?
Well, we didn't select the Queen of Hearts. So the antecedent is FALSE.
QH ≡ FALSE
And the Queen of Clubs is a black card. So the consequent is FALSE as well.
RC ≡ FALSE
So our entire statement:
QH → RC
... has the assignment:
FALSE → FALSE
OK. I've picked the wrong card of the wrong color. But is my original statement:
QH → RC
... TRUE or FALSE?
Well, it's TRUE. Why? Because picking the wrong card of the wrong color does not change the fact that IF I pick the Queen of Hearts THEN it will be red. Picking the wrong card of the wrong color doesn't change the truth of this statement.
So we can fill in the blank, and the FALSE → FALSE part of the Truth Table is:
| Truth Table: If-Then Statements | ||
| QH | RC | QH → RC |
| TRUE | TRUE | ? |
| TRUE | FALSE | ? |
| FALSE | TRUE | ? |
| FALSE | FALSE | TRUE |
So now we'll continue - that is, if you click here.