IF-THEN vs. ONLY-IF
Logical Statements
The Conundrum of Equality

 

Other CooperToons Educational Essays have mentioned certain topics that drive students nuts. In physics there's the Clock Paradox of the Special Theory of Relativity. In chemistry, it's Pyrene and why this compound with 4n π-electrons (that is, 16 electrons) is aromatic. And in Set Theory there's the question as to why the Union of the Empty Class is Equal to the Universal Class.

Fortunately all of those topics are in (relatively) advanced areas. So study can be deferred until the basics have been mastered.

But there is a topic in philosophy - logic, actually - that causes students considerable difficulty. And this topic is at the basic, nay, even the introductory level. And that's:

Just why is the lousy

"IF A THEN B"

Statement

the same as the

"A ONLY-IF B"

Statement?

For instance, if the textbooks tell us that:

A IF B

is the same as

IF B THEN A

... shouldn't that mean that ...

A ONLY-IF B

is the same as

IF B THEN A

And yet, the textbooks tell us that:

A ONLY-IF B

is the same as

IF A THEN B

Just what the hey is going on?

It Is Not Conversational, Captain

We have to be honest. Despite their playing a fundamental role in logical deduction, IF-THEN statements are troublesome. For instance, by rigid logic, the following statements are all TRUE.

If George Washington was the first President of the United States, then he delivered the first State of the Union Address.

If George Washington was the King of France, then C. S. Lewis was the first President of the United States.

If 1 + 1 = 3, then 1 + 1 = 2.

The reason these oddball statements are TRUE is because we are talking about a specific type of IF-THEN Statement. This is called MATERIAL IMPLICATION.

Material Implication is what we call a TRUTH FUNCTIONAL statement. In a Truth Functional statement, the TRUTH or FALSITY of a complex sentence depends ONLY on whether the individual parts are TRUE or FALSE. What the individual parts actually state or mean is irrelevant.

As a result if we are talking about Truth Functional logic we can make TRUTH TABLES. And for MATERIAL IMPLICATION the table is:

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

For a detailed explanation of how you arrive at this table, just click here.

So to explain why IF A THEN B and the A ONLY IF B statements are the same, we have to delve into some basic logical concepts. And to this end you can't do better than to turn to one of the Titans of Philosophy. That's René Descartes and his famous Cogito.

IF I Think, THEN I Am (I Think).

The Cogito is the Latin statement

Cogito ergo sum.

... which everyone knows means:

I think, therefore I am.

So what exactly did René mean when he said, "I think, therefore I am"?

Remember that René was trying to arrive at absolute certainty. And his first goal was to prove his own existence.

After considerable thinking, René realized he was - yes - THINKING.

So in his mind popped the realization that IF he was thinking THEN he must actually exist!

Or as he said it:

I think, therefore I am.

We see then that:

I think, THEREFORE I am.

AND

IF I think THEN I exist.

... have the same meaning.

So far so good.

It Ain't Necessarily and Sufficiently So

But what exactly did René mean when he said IF he was thinking, THEN he existed?

Obviously, René said, the process of thinking requires the EXISTENCE of a thinker.

In other words, existence is NECESSARY for thoughts.

Now René didn't mean that everything that existed had thoughts. But IF something had thoughts, then you didn't need anything else to prove the existence of the thinker. That is, thoughts were SUFFICIENT to prove existence.

We see then that the relation between SUFFICIENT and NECESSARY conditions is easily shown by the IF-THEN statement:

IF I think, THEN I exist.

... generalizes to ...

IF [Sufficient Condition] THEN [Necessary Condition].

At this point we will digress and caution all new philosophers that the concepts of sufficiency and necessity are not always the same as in ordinary conversation.

For instance, it's common to think that the IF part is the CAUSE of the THEN part.

For instance, consider the sentence:

IF you give Arnold Schwarzenegger a boot in the tail, THEN he will be irritated.

Here the IF condition does indeed cause the THEN condition. And yes, such causality is often found in IF-THEN statements.

But the reader might think that the boot in the tail was necessary for the irritation. That, though, isn't the case. Arnold can get irritated for other reasons. But certainly a boot in the tail was sufficient to cause irritation.

And the irritation? How was that NECESSARY?

Again, not necessary in a colloquial sense. But we are asserting that as long as Arnold got a boot in the tail, then the statement that Arnold will be irritated MUST be TRUE.

This is, in fact, the logical definition of SUFFICIENT and NECESSARY conditions. If the TRUTH of one statement GUARANTEES that a second statement is ALWAYS TRUE, the first statement is the SUFFICIENT condition for second statement. And the second statement is the NECESSARY condition for the first.

If all this brouhaha still seems a bit esoteric, you can keep the concepts of SUFFICIENCY and NECESSITY straight simply by thinking about the IF-THEN sentence:

IF A THEN B

Just remember that the IF part A is the SUFFICIENT condition and the THEN part B is the NECESSARY condition.

I Think ONLY IF I Am

But getting back to our original problem of proving that A ONLY IF B is the same as IF A THEN B. We need to go back to square one and ask René to elaborate on his explanation.

Well, another way of explaining what René meant when he said:

I think, therefore I am.

... was that - as we said above - his existence ("I am") is NECESSARY for the process of thinking.

In other words, René could think only if he already existed.

Ha? (To quote Shakespeare.)

Would you repeat that?

The process of thinking is possible only if you are existing.

Did you say ONLY IF?

Yes. We said ONLY IF?

So we see that René's famous COGITO can be restated:

I think, therefore I am.

... is the same as:

I think ONLY IF I exist.

To repeat, we have just shown that the two sentences ...

I think, THEREFORE I am.

AND

I think ONLY IF I exist.

... have the same meaning.

But remember earlier we showed that:

I think, THEREFORE I am.

AND

IF I think THEN I exist.

... are also the same.

Since the IF-THEN statement and the ONLY-IF statement are equal to a third statement - the THEREFORE statement - we know that all statements are equal.

And so we must conclude:

IF I think, THEN I exist.

=

I think ONLY IF I exist.

... or in general terms ...

IF A THEN B.

=

A ONLY IF B.

... which is what the textbooks say.

A Little Formality, Please!

Now if things still seem a bit obscure, there is another approach to prove our point. We will use FORMAL LOGIC and provide a FORMAL PROOF that the two sentences are the same.

"Formal logic" does not mean you hold philosophical discussions wearing tuxedos or evening gowns. It means you represent natural language statements as formulas. And although there are tons of textbooks on formal logic, we'll just explain things as we go along.

First of all, we need to define some logical symbols, or as they are often called, the logical connectives.

Logical Connectives
Symbol	Official Name	Plain English
¬	Negation	"Not" "It is false that"
∧	Conjunction	"And"
∨	Disjunction	"Or"
→	Material Implication	"If-Then"

[Note: Although the symbol for NOT, ¬, doesn't really "connect" smaller or "atomic" sentences, it is still sometimes called a connective.]

Of course, you see there is no symbol for ONLY IF. So we'll invent one:

↑ ≡ ONLY IF

Now we can write the sentences symbolically:

IF A THEN B		=		A → B
A ONLY IF B		=		A ↑ B

At this point we need ONE MORE TIME to try to explain what we mean when we say:

I think ONLY IF I exist.

But we'll do things a little different. We'll think backwards and point out that:

I think ONLY IF I exist.

... means that ...

IF you don't exist then you don't think.

Symbolically then we can write these sentences as:

T ↑ E

=

¬E → ¬T

... and with these two statements in symbolic form and using the rules of logic, we now can provide our formal proof:

Statement	Explanation
T ↑ E	I think ONLY-IF I exist.
¬E → ¬T	If I don't exist, then I don't think (Equivalent Meaning)
¬(¬E) ∨ ¬T	IF-THEN (A → B) ≡ NOT-OR (¬A ∨ B)
E ∨ ¬T	Two NOTS cancel: ¬(¬A) ≡ A
¬T ∨ E	OR (∨) is commutative (A ∨ B ≡ B ∨ A)
T → E	NOT-OR ≡ IF-THEN (¬A ∨ B ≡ A → B)
"T ONLY IF E" EQUALS "IF T THEN E"	∴ QED

So we must conclude after all that:

A ONLY-IF B

is the same as

IF A THEN B

But ... (You Knew There's Be a "But")

What's that? Something still bothering you? Something still wrong?

Well, although we've proven that A IF-THEN B is the same as A ONLY-IF B, we have to admit this - quote - "proof" - unquote - doen't convince a lot of people. Often the lack of convincing is because the reader doesn't have the background needed to understand the details and nuances of the proof. But in this case, there's another reason why not everyone believes that A IF-THEN B is always the same as A ONLY-IF B.

A ONLY-IF B

is NOT ALWAYS the same as

IF A THEN B

And the reason?

Because.

Because what?

Just because.

And we literally mean just because.

Perhaps a word of explanation is in order.

True, we've proven that in TRUTH FUNCTIONAL PRPOSITIONAL LOGIC that

IF A THEN B

is the same as

A ONLY IF B

But that's not always the case in ORDINARY CONVERSATION.

Instead in everyday speech A ONLY-IF B very often mean A BECAUSE B.

For instance, take the sentence (please!):

IF Abraham Lincoln was elected President THEN Abe had more electoral votes than John Breckinridge.

We have no problem accepting this as TRUE and equating it with:

Abraham Lincoln was elected President ONLY IF Abe had more electoral votes than John Breckinridge.

But look at this sentence.

IF Abraham Lincoln was elected President, THEN Abe delivered the Gettysburg Address.

Now according to our Truth Table - that is, as long as IF-THEN is accepted as the Truth Functional Material Implication - we know that this statement is TRUE.

On the other hand, does that statement really mean the same as:

Abraham Lincoln was elected President ONLY IF Abe delivered the Gettysburg Address.

Clearly the ONLY-IF in ordinary conversation is not always the same as the Material Implication IF-THEN.

That is, in ordinary conversation ONLY-IF often has the same meaning as BECAUSE.

So we can accept that:

Abraham Lincoln was elected President ONLY-IF Abe had more electoral votes than John Breckinridge.

is TRUE and also means the same as:

Abraham Lincoln was elected President BECAUSE Abe had more electoral votes than John Breckinridge.

But we don't accept that:

Abraham Lincoln was elected President, ONLY-IF Abe delivered the Gettysburg Address.

... is TRUE because we interpret it to mean:

Abraham Lincoln was elected President BECAUSE Abe delivered the Gettysburg Address.

... which is also FALSE.

Furthermore, you should see by now that that BECAUSE is not Truth Functional. That is, the sentence:

Abraham Lincoln was elected President BECAUSE Abe delivered the Gettysburg Address.

... is of the form:

TRUE BECAUSE TRUE = FALSE.

... and yet:

Abraham Lincoln was elected President ONLY-IF Abe had more electoral votes than John Breckinridge.

... is of the form:

TRUE BECAUSE TRUE = TRUE.

We see then that a sentence like:

A BECAUSE B.

... is not TRUE or FALSE depending only on whether A or B are TRUE or FALSE. Therefore BECAUSE is not Truth Functional. So if you use ONLY IF in place of BECAUSE then that use cannot be the same as IF-THEN Material Implication.

Soddy, old chaps.

 

References

Introduction to Logic, Patrick Suppes, Van Nostrand, 1957.

Formal Logic: Its Scope and Limits, Richard Jeffrey, McGraw-Hill (1981).

Discourse on the Method, etc., René Descartes, E. P. Dutton, 1914.

Descartes: A Biography, Desmond Clarke, Cambridge University Press, 2006.

Stanford Encyclopedia of Philosophy, http://plato.stanford.edu

"René Descarte", http://plato.stanford.edu/entries

"Descartes' Life and Works", http://plato.stanford.edu/entries/descartes-works/

"Descartes' Epistemology", http://plato.stanford.edu/entries/descartes-epistemology/

Symbolic Logic: A First Course, Gary Hardegree, McGraw-Hill, 1994. This book has a chapter on equating ordinary conversational connectives with the logical equivalents and this includes the "If-Then" and "Only-If" problem. But when he first introduces the fact that A only if B is the same as If not B then not A, but he does not point out this is the same as If A then B until later. It would have been less confusing to introduce material implication - that is, if-then statements, before "only-if". But he does point out that certain connectives - such as "unless" - have more than one meaning in ordinary conversations.