Shantila's Inside Logic #13

Disjunctions

Believe it or not, the man who is going to confess is still pointing his gun straight at your heart.

You just woke up after fainting. You were dreaming all sorts of things. You even dreamt you had a few days off.

You decide to hand the ticket back to Joe.

"Here, Joe," you say, handing him the ticket.

But the man keeps the gun pointed straight at you.

"I don't have the ticket," you say.

"Right," the man says.

"He has it now," you say. "His name is Joe."

"Right," the man says. He smiles his sick smile and keeps the gun pointed straight at you.

"Since he has the ticket, why don't you point the gun at him?" you ask.

"What good would that do?" the man asks. "He can't think straight."

Well, that's a good point.

But you have no choice but to try.

"Joe, listen," you say. "Listen to me. Just give him the ticket. Either you give it to him or he'll shoot us both. Suppose you don't give it to him."

"Then he'll shoot us both," Joe says.

"Wow! --That's right, Joe," you say. "So you understood it?"

"Yes."

"So you can reason with OR?"

"Yes, I can reason with or just fine," he says. "That is easy."

"You only have a partial logic block?" you ask.

"Yes," he says.

"Great," you say. "That's wonderful. Now give him the ticket."

"Why?" Joe says.

"Because he's stealing it!" you yell.

"But that's illegal," Joe says.

"Yes!" you say. "This man is a criminal! And he looks serious!"

A small crowd has gathered by now. The man with the gun pokes the gun back and forth at you. "Give me the ticket," he says.

"I hereby arrest you," Joe says loudly to the man with the gun.

"Oh, really? So just how are you going to do that?" the man asks.

"I work for the FBI," Joe says.

"Oh, really?" the man says sarcastically.

"Really?" you say with astonishment. You are dumbfounded.

"Isn't that sort of a bad job for a guy with a logic block?" the man asks.

"Its only partial logic block," Joe says.

"But don't you need to reason with and?" you ask. "And if? --How do you ever figure anything out?"

"Actually its been quite awhile--"

The man fires the gun into the air. It shatters lights and broken glass sprinkles down on you.

"Joe, if you're going to arrest him, it'd be a lot better if you had the gun," you say, momentarily forgetting that he does not understand if.

At that moment, the woman who's going to marry the man who kissed her nicely appears out of nowhere, double somersaulting over three rows of seats. In mid-flight she karate-kicks the man's hand with her right foot, so that the gun flies high into the air. She simultaneously knocks the man himself to the floor with her left knee and, after landing upright, she catches the gun and hands it to Joe.

Joe seizes the gun, whereupon the gun goes off, firing a bullet into the chest of the man who is going to confess.

The man now lies motionless on the floor.

"Joe, you killed him!" you scream.

"Oops," Joe says.

So much for the confession, you think.

"He's probably not dead," remarks the woman who was going to leave earlier but didn't. She stands up. "The last time this happened he only got knocked unconscious for awhile because he always wears a bulletproof vest."

She quickly gathers up her things and begins to walk away.

"Where are you going?" you ask.

"I am leaving," she says. "This time for good. Good-bye."

"Why?" you ask. "But he's going to confess!"

"Look," she turns around to face you. "It's going to be boring hanging out with him. Either he's dead now or he's still alive. If he's dead, he's going to be boring; if he's alive, it will be the same old thing, and if he's doing the same old thing, he's boring. Either way--"

"Either way he's going to be boring!" Joe says eagerly.

"Exactly," she says. "You got that right, Buster."

Joe beams. The woman walks off.

*

The astonishing thing here is that Joe proves that he is able to reason with "or"! He actually does it twice, first drawing the conclusion of your reasoning about what would happen if he didn't give up the ticket (both of you would get shot); and just now reaching the conclusion that, dead or alive, the man who got shot is going to be boring. Both of these arguments depend on or. Let us examine how it works.

We will use the symbol v to represent "or". Given any two statements P and Q, the statement "PvQ" is called the disjunction of P and Q (where P and Q each are called disjuncts).

The first rule associated with "or" is straightforward.

It does not matter what statement Q is. Given P, we can derive PvQ. We also can derive QvP. There is no question that this simple rule is valid. If P is true, then it also must be true that either P or Q.

For example, Sally lives in Cleveland. From this it follows that Sally lives in Cleveland or she lives in Los Angeles. Now it might be odd to reason in this way but clearly if the first sentence (Sally lives in Cleveland) is true, so also is the second one (she lives in Cleveland or LA). That is, from C we can reason validly to CvL (not matter what L is). And this is what our new vI rule says and it depends simply on the ordinary meaning of "or".

To consider another example, it is true that you are a human being. Therefore either you are a human being or you are a squirrel. From H we can derive HvS. This is simply how "or" works. This new rule is so easy to use that even a caveman could do it.

H } HvS

Notice that the new Rule vI is very different from reasoning from the disjunction HvS to S; indeed the sequent

HvS } S

is not valid and we do not want to be able to prove it, since the disjunction HvS may be true even though S is false. It is true that you are a human or you are a squirrel; but it is false that you are a squirrel (or so I assume).

It must be admitted that the new vI rule is, well, a bit boring. --But do not scoff at it! This rule is sometimes necessary and we do sometimes use this pattern in reasoning.

Suppose, for example, we have as a premise a conditional whose antecedent is a disjunction, such as (PvQ)>R, and suppose we know that one of the disjuncts P is true. We want to derive R. We cannot use MP directly because we don't have P>Q as a premise. Our new rule is useful here.

70 (PvQ)>R, P } R

We also can have logical truths that involve disjunctions. (Recall that a logical truth is a sentence that can be proved with no assumptions at all, as we discussed in an earlier chapter.) For example:

71 } P>(PvQ)

*

Inclusive and exclusive uses of "or".

The word "or" often isused so that PvQ is true just so long as at least one of P or Q is true, even if both P and Q are true. This is the "inclusive" of the word "or" and it is this meaning that we use the symbol v to represent.

For example, you are a human being (H). Since H is true, we know that HvQ will be true now matter what Q represents. And this is so even if Q happens to be true as well. Suppose Q represents "there are kangaroos in Australia" which is a true statement. Even though both H and Q are true, the disjunction HvQ still is true.

To consider another example of this type, suppose you want to meet with a TA to talk about your homework. Kevin says, "I will be in my office at 2p or Mark will be there." We can represent this as KvM. And notice that this promise does not turn out false even if it turns out that both Kevin and Mark happen to be the office at 2p. The truth of KvM is compatible with both K and M being true. KvM is true just so long as at least one of the two disjuncts are true even if both disjuncts turn out to be true. It is this inclusive use of "or" that is represented by our symbol v. (Otherwise our vI rule would be invalid, because it permits us to move from P to PvQ whether or not Q is true.)

But sometimes we do use the word "or" in a more restrictive way where in saying something of the form "P or Q" we intend to convey the idea that either P or Q is true, but not both. For example, Mom might say "you can have cake or you can have pie" where she intends to convey that you can have one or the other but not both. Similarly, one might say about a baseball series "either the Indians will win or the Yankees will win" meaning one but not both will win. This more restrictive use of "or" we call the "exclusive" use. One of the disjuncts is "excluded"-- meaning it isn't true (whereas, as noted above, the use that permits the truth of both disjuncts is the "inclusive" use).

Revising the earlier example, Kevin might have said, "I will be there or Mark will be there -- but not both of us will be there". This we can represent as (KvM)&~(K&M). And when the English word "or" actually is used by itself with this extra meaning (as can happen in its "exclusive" use), we can represent the meaning in this more complex way. So when Mom says, "You may have either cake or pie" she may very well intend to tell you that you can have one or the other --but not both; and we can represent what she means as (CvP)&~(C&P).

We normally assume that "P or Q" should be represented simply as PvQ unless the context makes it clear (perhaps by an emphasis on either) that "or" should be interpreted as being used with the extra "exclusive" content. If so, the additional content "and not both" is treated as part of the content of what is being said even though it is not explicitly uttered.

Once again we see that "interpretation" of ordinary English statements must be fluid and often it requires careful judgment.

Use of parentheses. Now consider the sentence "Either Kevin will be there, or Mark will be there, or Brad will be there." Notice this can be represented as (KvM)vB. It also can be represented correctly as Kv(MvB), since the bracketing by parentheses can be done either way without changing the meaning.