CooperToons HomePage Merry History Dept. of Education CooperToons Books Previous Next

"If-Then" Tables Explained
Sans Doute
(Part 6)

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

Truth Table: If-Then Statements
A B A → B
TRUE TRUE TRUE
TRUE FALSE FALSE
FALSE TRUE TRUE
FALSE FALSE TRUE

But you might object that all we've done is by an empirical demonstrations using a deck of cards. So this table is fine for dealing blackjack, but is it valid for all mental reasoning in philosophy?

Well, the answer is yes.

But Captain Mephisto, I've often wondered how you could determine the Truth Values of the "If-Then" Table by reasoning alone.

No doubt you have. To do that we have to decide what If-Then really means.

Suppose you walk with some friends to a pub. You all are having a good convivial time. But as is sometimes the case, you start discussing philosophy such as Schleirmacher's view of the Last Supper compared with the New Testament conception and the faith of the Reformers. And so you get a bit loud. So the barmen - politely, of course - says:

Please don't speak louder or I'll have to ask you to leave

Now what is the gaffer actually saying?

Well, it's a two part statement based on two sentences:

Speak louder.
I'll ask you to leave ≡ AL

... where we added the "Not" symbol, ¬ before SL and then link the two parts by the "Or" symbol, (and of course we don't have to be polite if we are being logical).

¬SL ∨ AL

Now the "Or" statements have a Truth Table, too. And it's pretty easy to understand.

Truth Table: Or (∨)
A B A ∨ B
TRUE TRUE TRUE
TRUE FALSE TRUE
FALSE TRUE TRUE
FALSE FALSE FALSE

So an "Or" statement is only FALSE if both the individual parts are simultaneously FALSE.

Notice this is what we call the inclusive meaning of "Or". So if we say "I'm going to New York or Boston", then if you go to New York or to Boston the statement is TRUE. But if you actually go to both, the statement is still TRUE. Only if you go to Saskatoon or Toronto but neither New York or Boston is the statement FALSE.

[Note: There is a way to logically express "A OR B but NOT both", but we'll skip that for now.]

But back to the philosophical discussion in the pub.

Remember the barman asked us not to speak louder or he would ask us to leave. Now this is exactly the same thing as saying IF we do speak louder, then he will ask us to leave.

That is the sentence symbolically rendered as:

¬SL ∨ AL

... is the same as:

SL → AL

Of course the word "not" symbol, ¬, its own Truth Table and it's pretty simple.

Truth Table: Not (¬)
A ¬ A
TRUE FALSE
FALSE TRUE

We can now write the general sentence:

¬A ∨ B

... as the equivalent "If-Then" expression:

A → B

Or in other words:

"If-Then" (A → B) is the same as "Not-Or" (¬ A ∨ B).

We can now make a "Not-Or" Truth Table and prove that it is the same as our earlier "If-Then" Table.

We start off with just the assigned values of A and B.

A ¬ A B ¬A ∨ B
TRUE ? TRUE ?
TRUE ? FALSE ?
FALSE ? TRUE ?
FALSE ? FALSE ?

... and we fill in the ¬A column (which is easy)

A ¬ A B ¬A ∨ B
TRUE FALSE TRUE ?
TRUE FALSE FALSE ?
FALSE TRUE TRUE ?
FALSE TRUE FALSE ?

And now if you go back to the "Or" Truth Table, we can fill in the last column, the truth values for ¬A or B.

A ¬ A B ¬A ∨ B
TRUE FALSE TRUE TRUE
TRUE FALSE FALSE FALSE
FALSE TRUE TRUE TRUE
FALSE TRUE FALSE TRUE

And we can now compare the ¬A ∨ B with A → B

A B A → B ¬A ∨ B
TRUE TRUE TRUE TRUE
TRUE FALSE FALSE FALSE
FALSE TRUE TRUE TRUE
FALSE FALSE TRUE TRUE

So the two statements are completely the same:

How about that?

Socrates At Ease