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René Descartes
Have Watched

The Super Bowl?

"I watch, therefore I am!"

René Descartes and the Super Bowl

Should René have watched?


Everyone knows about (yawn) René Descartes. He's the guy that invented Cartesian geometry which used all those graphs with grids that tormented you ever since middle school. And if you know anything else about René, it was that he decided he existed because he could think.

We must give a mild warning to those unversed with René and his Life and Times. To avoid condescending smirks and sneers from his fans, remember René's last name is pronounced DAY-cart and his first name is re-NAY, the last syllable rhyming with "day".

Now one thing you read about René is he really never did any work. That is true in general, but we do know that when he graduated from the University of Poitiers in 1618, he got a job as a soldier for Prince Maurice of Nassau. This isn't Nassau in the Bahamas, but Nassau in Holland. Probably René's strong math background - from a natural ability and a rigorous training received when he was attending a Jesuit school before he went to Poitiers - landed him a job as a military engineer. He was stationed in Breda where the army also acted as a training school and no doubt René took advantage of whatever educational opportunities were available.

At Breda René also met Isaac Beeckman who was a leading philosopher and scientist of the time. The two men became good friends and René learned a lot of his sciences and philosophy from Isaac.

René left the army after two years. As far as getting the "ready" to live on, evidently some of the family's property was sold off, and René got a good chunk of cash. This kept him going for a number of years. He spent the rest of his life living in various places around Europe, gambling, going out to dinner with his friends, going to the theater, parties, and concerts, and given the fact that his maid had his daughter, having a good time.

It's not really odd that René turned to philosophy. At that time philosophy included what we think of as math and science. This was also the era before specialization. René studied biology, engineering, optics, astronomy, meterology, and math. He got the idea of modern analytic geometry (not fully developed until the 19th century) and worked with a number of famous mathematicians and scientists, particularly after he moved to Paris in the 1620's. Then around 1629 he headed back to Holland.

Sometime between 1629 and 1637 - when he published his famous book Discourse on the Method of Rightly Conducting One's Reason and of Seeking Truth in the Sciences - we know he had decided to work on the problem of how to obtain absolute certainty. He, like all intelligent and educated people, was aware that just believing something - no matter how fervently - meant nothing regarding its truth. So he decided to go back to basics.

He wondered. Did he, René Descartes, actually exist? How could he know since the senses could sometimes be deceiving? Then suddenly he realized he was thinking. If he was thinking, there must be a thinker. And if there was a thinker and it was he, the he existed.

As René later put it:

Accordingly, seeing that our senses sometimes deceive us, I was willing to suppose that there existed nothing really such as they presented to us; and because some men err in reasoning, and fall into paralogisms, even on the simplest matters of geometry, I, convinced that I was as open to error as any other, rejected as false all the reasonings I had hitherto taken for demonstrations; and finally, when I considered that the very same thoughts (pre-sentations) which we experience when awake may also be experienced when we are asleep, while there is at that time not one of them true, I supposed that all the objects (presentations) that had ever entered into my mind when awake, had in them no more truth than the illusions of my dreams. But immediately upon this I observed that, whilst I thus wished to think that all was false, it was absolutely necessary that I, who thus thought, should be somewhat; and as I observed that this truth, I think, therefore I am, was so certain and of such evidence, that no ground of doubt, however extravagant, could be alleged by the sceptics capable of shaking it, I concluded that I might, without scruple, accept it as the first principle of the philosophy of which I was in search.

To this day, you'll get a lot of head scratching about René's most famous quote. But there's no denying it's catchy: "I think, therefore I am" is probably the most famous philosophical soundbyte in history. Eggheads like to call it the Cogito because in Latin - the language of eggheads - René's statement is Cogito, ergo sum.

Now René and a lot of other people see his statement "I think, therefore I am" as what we call self-confirming or (to use Benjamin Franklin's correction of Thomas Jefferson) self-evident. The argument itself contains its proof. So why, we wonder, do people still debate about it?

For one thing, it seems a bit too glib and too easy. And something just doesn't seem to be right in self-confirming proofs. All too often they are simply disguised circular arguments.

Søren Kierkegaard, the famous 19th century Danish philosopher, certainly thought so. He pointed out that "I" appears in both the "I think" part of the sentence - called the antecedent - and the "therefore I am" part" - called the consequent. So the antecedent presupposes existence it claims to prove. Hence the argument is circular.

Linguist and etymologist John Ciardi (1916 - 1986) agreed. Ultimately if you believe "you are" because "you think", then you are presupposing that thoughts arise from a thinker.

So what to do?

Well, we will try what René himself said to do. If you are confronted by a difficult and complex problem, then break it down into small parts that you can handle. Solve each small part and when you put them back together, the whole problem will be solved.

When working at logical arguments, the best way to break them down is to use symbolic logic. Fortunately, the Cogito, - as we will call it, showing our solidarity with the eggheads - is actually about the easiest philosophical argument to render symbolically.

First of all, we write the Cogito in English:

I think, therefore I am.

... into a more "official" logical form:

If I think, then I exist.

Next, we now put René's rather windy argument (at least in the original paragraph) into the premise-conclusion form:

If I think, then I exist.
I think.
Therefore I exist.

Now we simplify the problem even further and break things down into "atomic sentences". These we represent as single letters:

I think T
I exist E

The "If-Then" nature of the sentence is represented by the arrow connective:

... and now can write the Cogito symbolically as:

T → E

... and the whole argument is rendered :

Premise 1 T → E If I think, then I exist.
Premise 2 T I think.
Conclusion E I exist.

Now comes a big surprise.

After all this effort we now admit are not really interested if what René said is actually TRUE. Instead, we want to know if his argument is VALID. That is, does the conclusion follow through proper reasoning using the premises?

First, we spill the beans and say, yes, the argument is valid. In fact, this form of an argument has it's own name, modus ponens which is Latin for the "affirming method". That is, if Something implies, Something Else, and we know the first Something is true, then we can be sure the Something Else is also true.

Generally modus ponens is expressed as:

If both A → B and A are TRUE, then B is TRUE.

But how do we know modus ponens is really valid?

Well, valid arguments are those that whenever the premises are TRUE, then the conclusion is TRUE. We don't just mean that you can find some cases where the premises are true and the conclusion is true. All cases where the premises are all true should give us a true conclusion.

On the other hand, if an argument is invalid, then you can find at least one counterexample. That is, you can find a case where the premises are TRUE, but the conclusion is FALSE. Finding just a single counterexample proves the argument is invalid.

Before proceeding we will review the Truth Table for the "If-Then" statements, known in philosophical circles as Material Implication. This lets us know if an "If-Then" sentence is TRUE based on whether the "If" part - the antecedent - and the "Then" part - the consequent - are TRUE or FALSE.

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

In other words, an "If-Then" statement is TRUE in all cases except when the "If" part is TRUE and the "Then" part is FALSE. To some people this makes perfect sense, but to other sit seems a bit strange. But you can see how we derive this Truth Table if you just click here.

So let's look at René's argument - and modus ponens - again:

Premise 1 T → E If I think, then I exist.
Premise 2 T I think.
Conclusion E I exist.

Reasoning from the table shows that René's argument is valid. For a valid argument, both Premise 1 and Premise 2 must be TRUE. But note that the antecedent of Premise 1 is identical to Premise 2 and so also must be TRUE. From the Truth Table we also see that a TRUE "If-Then" statement with a TRUE antecedent must also have a TRUE consequent.

Finally note that the consequent of Premise 1 - which we just said must be TRUE - is also the conclusion of the whole argument. So if the premises are TRUE, then the conclusion must be TRUE.

The argument is valid.

Note that a valid argument does not require that the Premises and Conclusions can only be TRUE. Validity only requires that if the premises are TRUE, then the conclusion must be TRUE.

Now if we don't want to try for this type of "direct proof", we could also look for a counterexample. If there is a counterexample then the conclusion must be FALSE but the premises are TRUE.

But the conclusion is also the consequent of Premise 1. Therefore the Truth Table tells us that Premise 1 can then be TRUE only if the antecedent is also FALSE. But the antecedent of Premise 1 is also Premise 2. So if our conclusion is FALSE, then we can't have both Premises TRUE, and so there are no cases where the premises are all TRUE and the conclusion false. Ergo, there are no counterexamples. René's argument - and modus ponens - is confirmed as valid.

So René was correct, n'est ce pas?

Weeeeelllllllll, there's just one thing.

No one disagrees that if "I think therefore I am" is TRUE and you do think, then you exist. That's not what we're asking about.

What we do wonder is whether the premise itself, "If I think, therefore I am", is TRUE. That's the question.

So how do we prove a premise?

Well, we have to change the Cogito from a premise to a conclusion. Then we have to find if there are new premises that make our (new) conclusion-and-former-premise TRUE. If there are no premises that we can accept, we can't accept the Cogito.

The first thing, then, is to create a new table with the Cogito as the conclusion:

Premise 1 [Something] [Something]
Premise 2 [Something Else] [Something Else]
Other Premises
(If Needed)
Conclusion T → E I think, therefore I exist.

In other words, what we see now is that the Cogito has hidden (or at least implicit) premises that are not expressed in the original argument. And now we have to find what those are.

So where do we go from here?

Now one thing to do if you are trying to prove something but getting nowhere is to try an indirect proof. Now an indirect proof means that you first assume the opposite of what you want to prove. Then you reason (properly, of course) until you derive a contradiction. That is, you find something that conflicts with a well-established fact. If you do manage to find such a contradiction, then the original non-negated conclusion was correct after all.

[Note: That a contradiction proves the opposite of a negated true statement is a well-known and relatively simple proof of elementary logic.]

Now a common mistake of beginners when they try an indirect proof is they don't realize that you must negate the conclusion. Instead they assume the conclusion is TRUE. Then they work though some steps until they come back to the original conclusion. So they conclude the theorem is correct.

Alas, as they find out when they get their paper back, this approach is fallacious. Proving an argument by assuming the argument is true is a circular argument. This is not a proper proof.

We, though, are doing something a little different. We will look at the contrapositive of the Cogito.

But first a little thought about - not existence - but non-existence.

Now what does non-existence imply? Well, for one thing, if you don't exist, then you can't do anything. And we mean anything. You can't watch television, have a beer, eat a bag of chips, or surf the net.

We repeat, if you don't exist, then you can't do anything.

We have, then, found one of our hidden premises. That is if we define:

E I exist.
D I do (something).

We render this inability to do anything as:

¬E → ¬D

... which with ¬ meaning "not" or "do not" simply means:

If I don't exist, then I don't do (anything).

OK. Now if you've studied logic you know this last sentence does not mean the same thing as, "If I don't do anything, then I don't exist." We know from the example of our own political leaders that this statement isn't correct. You certainly can do nothing and yet (and lamentably) exist.

On the other hand, the statement "If I don't exist, then I don't do (something)" is called the contrapositive of "If I do (something), then I do exist." The important point is that contrapositives are equivalent. Prove a contrapositive of any true statement and we know the statement is TRUE as well.

So symbolically this is represented as:

¬E → ¬D D → E

[Note: The proof of this is quite easy as long as you remember that "If-Then" is the same as "Not-Or", that is A → B ≡ ¬A ∨ B, and that ¬A ∨ B is the same as B ∨ ¬A].

We now return to René and ask. Just what did what he mean by "thinking". But most of all how did he know it was he who was thinking? Unless he knew the thoughts originated with him, he couldn't prove he existed.

Well, like a lot of people René used the word "think" in different ways. "Think" sometimes means processing information - maybe adding up numbers, multiplying terms, or dividing one number by another. That's why we can say that computers "think".

On the other hand, if we simply ponder something - that is just ruminate on various topics - we also say we're thinking. If you remember some past event or imagine something in the future, you're thinking.

But if we do a bit of thinking ourselves, we realize what René was talking about was the experience of perceiving thoughts. And here he was correct. He knew without doubt that he perceived or experienced thoughts.

And a crucial point for us to move forward is that when you perceive thoughts then clearly you are doing something. Or as we write symbolically:

P → D   If I perceive (thoughts), then I am doing (something).

... and we have our second new premise.

Finally, we can unambiguously recognize when we perceive thoughts. So our final new premise is about the simplest of all:

P   I perceive (thoughts and other things).

All right. Now let's put our premises into our table.

Premise 1     ¬E → ¬D   If I don't exist, then I don't do anything.
Premise 2     P → D   If I perceive (thoughts), then I am doing (something).
Premise 3     P   I perceive (thoughts).

Now before we begin with our proof we need to mention another type of reasoning. This is called modus tollens which is similar to modus ponens.

A → B Premise 1
¬B Premise 2
¬A Conclusion:

In other words, if you say "I think, therefore I exist", and you don't exist, then you can conclude "I don't think".

[Note: The proof of modus tollens is easy since we see from the Truth Table that if a true "If-Then" statement has a false consequent, the antecedent must also be false.]

OK. We list our premises once again:

Premise 1     ¬E → ¬D   If I don't exist, then I don't do anything.
Premise 2     P → D   If perceive (thoughts), then I am doing (something).
Premise 3     P   I perceive (thoughts).

And now we start the proof:

Step 1   D   Premise 2 and
Premise 3
  Modus Ponens (If A → B and A, then B.
Step 2   E   Step 1 and
Premise 1
  Modus Tollens (If A → B and ¬B, then ¬A) and also ¬¬A = A
Step 3   T → E   Step 2
"If I think, then I exist (I think, therefore I am)"

Now hold on there, pilgrim. Where did this new "I think" come from? We were talking about "perceiving" as a type of thinking. Now we're talking about any kind of thinking which doesn't even show up in any premise!

Well, go back and look at the "If-Then" Truth Table. If the consequent is TRUE, then it doesn't matter what the antecedent is. The antecedent can be TRUE of FALSE and the "If-Then" statement is still TRUE.

In other words, once we established Step 2 - that is, "I exist" - in our proof, then we can stick anything in as a new antecedent.

[Fill in the Blank], therefore I am.

So if we wish, we could have written:

I drink, therefore I am.

... or

I blink, therefore I am.

... or

I wink, therefore I am.

... and yes, even:

I watch the Super Bowl, therefore I am.

Go, team!



A simple proof of the deriviation of the Cogito is via a Truth Tree. A truth tree is a diagrammatic indirect proof. Details for making truth trees are found in the second reference (Jeffrey, 1981), and the tree itself for the Cogito can be seen if you click here.



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

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

Formal Logic: Its Scope and Limits, Richard Jeffrey, McGraw-Hill, 1981 (2nd Edition).

Stanford Encyclopedia of Philosophy,

"René Descarte",

"Descartes' Life and Works",

"Descartes' Epistemology",