Shantila's Inside Logic #9

Some Fallacies

You are still waiting at the airport, hoping to get Atlanta in time for your sister's birthday party.

You notice that the man across from you is scribbling on a notepad. He is the guy who won't confess. You wonder what he is writing. Maybe it is the confession. You are wondering what the confession is all about. Perhaps he is a cold-blooded murderer. Perhaps he is planning another murder right now. If so, he is dangerous man. --Anyway, who cares? You just wish the airplane would arrive from Denver so you can get to your sister's party on time.

You get up and walking around for awhile, completely bored. You see a tv and notice that the Jerry Springer show is on. According to the banner at the bottom of the screen, the topic today is

Jerry Springer is interviewing Sally Rose, a high school girl from Cleveland, Ohio.

Jerry Springer and the girl now begin slapping each other. Then the girl's boyfriend comes running on to the stage and begins beating up Jerry Springer. It is a big mess.

Good grief! you think. You decide you've had enough tv for awhile and walk back to your seat.

The man sitting across from you doesn't really look all that dangerous. His eyes are closed and he seems to be asleep. You lean forward slightly and peer at the sheet of paper on which he had been writing. It is filled with capital letters and symbols like > and ~ and some very messy marks where it looks like he was trying to make "&". Perhaps he is in the Mafia and this is some sort of code!

Suddenly you realize that he is looking straight at you. You lean back in your seat and nonchalantly look at your watch. You glance back at him. He is still staring at you.

He smiles. He gives you the creeps.

Joe walks up. He is holding his lottery ticket and a newspaper. And his fly is unzipped.

"I found the bathroom," he says, triumphantly.

"Great," you say.

"Would you mind checking this ticket for me?" he asks. Oh, wonderful. Now Joe thinks you're friends, just because you showed him to the bathroom.

"Hey, dude. Your pants are unzipped," you say, politely.

"Oh. Sorry," he says, zipping up his pants. "I always tell myself to zip them up when I am done. And when I am done I say, ok I am done. But nothing happens."

"Oh," you say.

"Can you check this ticket?"

"Ok, yes," you say.

He hands you the ticket and the paper. You read the numbers listed in the paper

And you read the numbers on the ticket.

The two sets of numbers are identical.

"Whoa! I can't believe this! Whoa!" you say. You are startled.

"Hey Joe -- you just won the lottery!"

"No!" Joe yells.

"Yes!" you jump up. "I can't believe it!"

"It's 8 million dollars!" Joe says.

"I can't believe it!" you say again, incredulously, staring back and forth at the ticket, then the paper, then the ticket again. The two sets of numbers remain identical.

"Maybe Britney will go out with me!" Joe exclaims.

"Who?" you ask.

"Britney," he says. "I mean, now that I'm rich?"

"Well I don't know," you say.

Suddenly, to your horror, you notice that the man who is going to confess has stood up and is now pointing a gun straight at your heart.

"Ok, then. Great. I'll take that ticket," he says. "I have always wanted to win the lottery and it looks like today is my lucky day!"

What? --He is dangerous! you think. Oh God! If he's a murderer, he's dangerous.-- He's a murderer!

****

Affirming the Consequent: Bad/AC

I hate to have to tell you this but you have just made a big mistake. In fact, you just committed a fallacy. You concluded that the man who is pointing the gun straight at your heart is a murderer. Your reasoning seems to be based on your thought that

if this man is a murderer, he is dangerous

and the thought that

he is dangerous.

That is bad reasoning; you flubbed up insofar as you were reasoning. Of course the thoughts might have simply flashed through your head due to your fear without your having moved from premises to a conclusion. In that case your thoughts might be be true or false but we might not call it reasoning.

In any case, as a piece of reasoning (which of course is our focus in logic) this is not valid, because the two premises here may be true even though the conclusion is false. It is true that murderers are dangerous. And obviously this man is dangerous, since he is using a gun to steal a lottery ticket. So both premises are true in this story. But that doesn't necessarily mean that the man is a murderer. The form of your reasoning is

M>D, D } M

We don't want a rule in our system of logic corresponding to this sequent! We don't want to be able to prove it! It is the "fallacy" (mistaken reasoning) called "affirming the consequent"--since it is invalid reasoning, we call it "Bad/AC".

Here is another example of the Bad/AC fallacy.

Think about it. This obviously is just plain bad thinking. The conditional premise is C>O -- and it clearly is true: anybody who lives in Cleveland lives in Ohio. The invalid reasoning assumes that O>C is true, that is, if Sally lives in Ohio, then she lives in Cleveland. But O>C may not be true, and obviously does not follow that C>O. There are many other places in Ohio other than Cleveland, such as Columbus and Bowling Green where she might live and still live in Ohio. Someone is making the Bad/AC mistake if they use C>O and think "Sally lives in Ohio, so she must live in Cleveland."

Look closely to see how this fallacious sequent differs from an application of MP, where we "affirm" the antecedent of the conditional and validly derive the consequent:

P>Q, P } Q [the valid rule MP]

P>Q, Q } P [the fallacy Bad/AC]

The main point here hinges on the fact that P>Q and Q>P have different meanings. The truth of one does not guarantee the truth of the other-- and it takes only one example (or counterexample) to show this. The C>O case is a good counterexample: C>O always is true, but O>C may not be true. The conditional does not necessarily go both ways.

Here is another example of the difference between MP and Bad/AC.

If Tom lives in Ohio, he lives in a mideastern state. Tom lives in Ohio. So, he lives in a mideastern state.

We can represent this with the sequent T>R, T} R. This is valid -- we can prove it easily using our valid rule MP.

Now consider this argument.

If Tom lives in Ohio, he lives in a mideastern state. Tom lives in mideastern state. So, he lives in Ohio.

We can represent this with the sequent T>R, R} T. Is it valid No, it is not valid. It is easy to see that arguments with this form are not valid (that is, they are invalid) because it is possible for the premises to be true but the conclusion false. The first premise is true because Ohio is a mideastern state --anybody who lives in Ohio lives in a mideastern state. Now suppose that Tom lives in Indiana. Since Indiana also is a mideastern state, the second premise also is true. But obviously the truth of the two premises does not guarantee the truth of the conclusion, so the sequent is invalid. There are in fact millions of actual people who live in a mideastern state but not in Ohio: so for each of them the premises are true but the conclusion is false. Since there are mideastern states other than Ohio, living in a mideastern state does not guarantee that one lives in Ohio.

The problem is that the conditional does not necessarily go both ways --the truth of T>R does not guarantee the truth of R>T, as the example makes very clear (since T>R is true but R>T is not true.) Notice that this feature of > differs from &, since a sentence P&Q guarantees Q&P, and vice versa --as we proved when we proved sequent 24 P&Q } Q&P.

Of course, in some cases the conditional does go both ways, unlike with T>R and R>T in the example. For instance, recall Britney's reasoning about her clothes. She said that she always wears the gold heels whenever she wears the gold blouse. This means both H>B and B>H are true. So the conditional can go both ways, but it most certainly does not always go both ways.

**

Denying the Antecedent: Bad/DA

There is another related fallacy we should consider. Suppose that for some reason you had come to believe (about the man who wouldn't confess) that ~M, he's not a murderer, and then on the basis of M>D (if he is a murderer, then he is dangerous) had concluded ~D. That also would be bad reasoning.

M>D, ~M } ~D

This fallacy is called "denying the antecedent"--we will call it Bad/DA. It is invalid for the same general reason that Bad/AC is invalid: the premises can all be true even though the conclusion is false. The man might be dangerous even though not a murderer. Likewise, recall the conditional C>O again and now consider this sequent

C>O, ~C } ~O

Is this valid? No, it is not! Certainly C>O is true and suppose ~C also is true: Sally does not live in Cleveland. Does this guarantee that ~O, she does not live in Ohio? No, of course not! She might very well live in Bowling Green; she might be your neighbor. Her not living in Cleveland does not necessarily mean that she does not live in Ohio. So this is a good example of the Bad/DA fallacy:

Notice how the invalid Bad/DA sequent differs from an application of MT, where we "deny" the consequent of a conditional, and validly derive the negation of the antecedent:

P>Q, ~Q } ~P [the valid rule MT]

P>Q, ~P } ~Q [the fallacy Bad/DA]

Whenever we reason using either of the two invalid forms, Bad/AC or Bad/DA, there is a danger of mistake. We may reach conclusions that are not supported by our reasons.

**

Non sequiter fallacy

Another simple fallacy is where the conclusion cannot be proved from the premises because the conclusion has nothing to do with the premises. For example,

P>Q, P } R

The conclusion R does not "follow" from the premises. The Latin term non sequiter ("does not follow") is used for this. Obviously we would not want to be able to prove such non sequiters as this in our system.