Shantila's Inside logic #7

Conjunction

Suppose you hear a commotion at the check-in desk. A man is trying to get directions to the bathroom. The person at the counter has pointed out a small handwritten sign on the counter, but apparently the man cannot read English, so the attendant reads it for him out loud.

When she is finished reading it, the attendant returns to her work shuffling old tickets. But the man does not leave. "Ma'am," he says. "I already read that sign."

Ok, so he can read English.

He then says, "Does that mean the closest men's bathroom is in B?"

"The closest women's is in C and the closest men's is in B," she says, without looking up. "Like I just told you. We apologize for the inconvenience."

"But about the closest men's bathroom itself?" he asks.

"Good Lord!" she says. "What is your name?"

"Joe," he says.

"Ok, Joe. Let's try to focus. I will say it again. The closest women's bathroom is in corridor C and the closest men's bathroom is in corridor B. We do apologize for the inconvenience."

"Ok," Joe says. "I understand that. Thank you. But--"

"Joe, come with me," you say.

With nothing better to do, you have walked over to the counter to help out. "I know where the closest men's bathroom is and I will show you where it is," you say to him.

"Do you know where the closest men's bathroom is?" he asks.

"Yes. I will show you where it is," you say.

"Thank you," Joe says. "I'd appreciate that."

You walk away with him.

He asks, "Did you hear the news about Britney?"

"Yes, I heard."

"I am trying to get a girlfriend like that." Joe says.

"Really?" you ask.

"She's making a comeback," he says.

"Really?" you ask.

"Of course, I'd need to have more money."

"That's true. You probably you would."

"Yah," he says.

Something falls out of his pocket.

"Wait, Joe," you say, picking up a small piece of paper. "You dropped something. Looks like a lottery ticket. Here."

"Thank you. I hope I win."

"Well, good luck. Ok, then. That's the bathroom down there on the right."

*

Looks like you have just met Joe, the guy who can't reason. Apparently he is going to be on your flight. And, astonishingly, it seems he cannot even reason with "and"! If he can't reason, how did he manage to get to the airport? How did he get dressed? Frankly, I don't really know. For in all of logic, there is nothing easier than reasoning with "and".

Let us use the symbol & to represent "and". Given any two statements P and Q, the statement "P&Q" is called the conjunction of P and Q (where P and Q each are called the conjuncts). From conjunction C&B as a premise it is obvious that drawing the conclusion C is good reasoning. (As in the story, letting C stand for "the closest womens's bathroom is in corridor C" and B stand for "the closest men's bathroom is in corridor B".) Likewise B follows from the conjunction C&B. This is valid reasoning. The test is as before: if the premise is true then the conclusion must be true. This gives us another rule.

The most difficult thing about using this rule is writing the symbol "&" correctly! It is quite obviously a good rule of reasoning. It might seem too trivial even to be mentioned. But in fact, it is essential, since a statement of the form P&Q is a different statement from P.

18 P&Q } P

19 P&Q } Q

There is an equally obvious counterpart rule in which we move from premises P and Q to the conjunction P&Q.

20 P,Q } P&Q

Notice that we could list the assumptions P and Q in a different order to prove this sequent, and the alternative proof is just as good as this one.

Even though the new rules are very simple to use, adding them to our system makes the system richer and allows us to write more complex statements and sequents.

Use of parentheses.

From now on we will use parentheses in an obvious way to prevent confusion. For example, the sentence ~(F&R) is the negation of the conjunction F&R. Notice how it differs from ~F&R, which is a conjunction of ~F and R. To see how these two sentences differ, let F stand for "The Falcons won" and R stands for "The Rockets won." Suppose the Falcons and Rockets played each other in football and suppose the Falcons won the game. So F is true, as is ~R.

Now consider the two sentences ~(F&R) and ~F&R.

~(F&R) is true (because it is not the case that both the Falcons and the Rockets won); but ~F&R is false (each of ~F and R is false).

This example shows that ~(F&R) and ~F&R have different meanings since the first is true but the second is not true. Also notice that ~(F&R) also does not guarantee ~F&~R, since this sentence also is false (due to ~F being false).

To consider another example of how to use parentheses, notice that the sentence P>(Q&R) is a conditional that has P as its antecedent and the conjunction Q&R as its consequent. An example of an English sentence with that form is the sentence on the handwritten sign:

Let L be "the bathrooms in this corridor are locked," C be "the closest women's bathroom is in corridor C," and B be "the closest men's bathroom is in corridor B." The sentence on the sign can be represented as L>(C&B). On the other hand, notice that (L>C)&B has a very different meaning, for it is a conjunction whose first conjunct is the conditional L>B and whose second conjunct is C. This sentence (L>C)&B means If the bathrooms in this corridor are locked, then the closest women's bathroom is in corridor C; and the closest men's bathroom is in corridor B. (The second sentence is telling us where the men's bathroom is, independent of whether or not the bathrooms in this corridor are locked.)

We now use one of our new rules in the following proof.

21 (P&R)>Q, P, R } Q

Notice that we use &I to derive a conjunction; and then we use MP to derive the consequent of the first premise.

In the following proof, we use MT to derive a negated conjunction.

22 (P&Q)>R, ~R } ~(P&Q)

In the following proof, we use both of our new rules &E and &I together with MP to derive a conjunction.

23 Q>R, P&Q } P&R

Notice that there are alternative proofs for this sequent that are just as good as this proof, since some of the steps could have been done in a different order.