Shantila's Inside Logic #10

Interpretation

Let us review your situation. You want to get to Atlanta but the plane is very late. You hear an announcement.

"The plane to Atlanta is in the Detroit airport but the weather is not very good. It will leave within an hour unless tornadoes start up again."

You are frustrated. Actually, you are really pissed! You will be able to surprise your sister at her party only if your plane leaves within an hour. You should call your sister if you are not going to be able to surprise her. So --you should call your sister if the plane is not going to leave within an hour. It was going to be a lot of fun, but it isn't working out right.

"Hey! Wake up!" Joe is yelling at you.

What? You can't focus. You must have fainted.

"I must have fainted," you think.

The man with the gun is still standing there. He is still pointing the gun at you.

Joe is holding your head. You are still holding the lottery ticket.

You feel like crying.

Joe pats you on the back.

"It'll be ok," he says.

"Thanks, Joe," you say.

You try to stand up.

"Give me the ticket right now!" the man yells.

*

Obviously you were having some thoughts about your situation (even while having passed out, apparently). Is there any reasoning here? If we can interpret this passage as containing reasoning (that is, containing an "argument"), we need first to identify a conclusion. It is good to see that there can be varying interpretations of a passage like this. But, pretty clearly, one "conclusion" that you reach here is that you should call your sister if the plane is not going to leave within an hour. Let us represent this as ~H>C (where H stands for "the plane is going to leave within an hour" and C stands for "you should call your sister").

Now we can consider whether or not there is a valid argument here for this conclusion. There are two main points to consider.

First, there obviously are some statements in the passage that should not be interpreted as relevant to this conclusion. For example, the fact that you are upset does not provide any support for the conclusion ~H>C. Neither does the fact that you feel like crying, or that Joe is being nice to you. To interpret the passage as providing reasons in support of the conclusion ~H>C we need, first, to be able to focus on the relevant statements and ignore the statements that are not relevant to the reasoning for the conclusion.

Secondly, there are a number of words in these sentences that can be interpreted properly in terms of the simple special language that we have developed to symbolize reasoning. Look at the first sentence. "You want to get to Atlanta but the plane is very late." An English statement of this form "A but L" guarantees at least that A&L, even though the English "but" statement contrasts A with L in some way (in this case L contrasts with what you want) and this extra information is not represented in A&L. Nonetheless A&L is a good enough interpretation for basic logic because if "A but L" is true, then that means at least that A&L is true.

There are other English words that can be interpreted using our symbols, just as "but" is properly interpreted using &. Here are some other phrases whose logical content can accurately be represented using &:

P although Q: P&Q, very similar to "but"

P yet Q: P&Q, also similar to "but"

neither P nor Q: ~P&~Q, means "both ~P and ~Q"

There also are phrases, in addition to the usual "if/then" that can be represented using our conditional operator >. Two such phrases, and their representation, are:

P unless Q: ~Q>P represents this best

P only if Q: this is best represented as ~Q>~P

Notice how P only if Q (that is, ~Q>~P) differs from P if Q (that is, Q>P). So in English the use of "only if" gives us a very different meaning from the simple use of "if".

Now let us symbolize the passage using these interpretations.

You want to get to Atlanta but the plane is very late. A&L

Your plane now is in Detroit but the weather is not very good. D&~W

It will leave within an hour unless it tornadoes start up again. ~T>H

That is what the announcement said. You are frustrated. Actually, you are really pissed! You begin to talk with Joe about it. (These are all statements, but let us just ignore them, at least for now. None of them seem relevant to your conclusion. If later we notice that one of them is relevant we could come back and reconsider.)

You will be able to surprise your sister at her party only if your plane leaves within an hour. ~H>~S

You should call your sister if you are not going to be able to surprise her. ~S>C

So you should call your sister if the plane is not going to leave within an hour. ~H>C

Ok, so now we want to see if there is a valid argument for the conclusion ~H>C. Here are the sentences we have symbolized.

A&L

D&~W

~T>H

~H>~S

~S>C

So let us try to prove ~H>C on the basis of these assumptions. Indeed, we can do so, as follows. We have five premises, which as usual we list on the first five lines.

We assumed ~H on line 6 (for a CP strategy). Then we can get ~S using the Rule MP, because we also have ~H>~S. Then we get C using MP. So ~H>C, via CP.

Notice the first three sentences here played no role at all in our proof! We didn't have to use them to get the conclusion we were seeking to establish. Even the 3d premise, ~T>H did not turn out relevant (even though it contains H and the conclusion contains ~H). One might initially have assumed that it would turn out relevant, but in fact it was not relevant to the reasoning in this particular argument.

So we have shown that indeed there is a valid argument in the passage for the conclusion ~H>C. This passage was, of course, specially constructed to make sure we could find the argument! But when we look for arguments in a newspaper or magazine or book, it may not be so easy to find an argument. Of course, not all communication or thought involves the construction of arguments or the presentation of reasons. And even when people are trying to develop an argument, what they say may be difficult to interpret. Rational and fair communication requires that we do our best to make sense of what others say and to try to interpret what they say charitably.

One additional problem is that sometimes we may fail to mention premises that are necessary because we assume the premise is common knowledge or so obvious that is does not even need to be mentioned.

Suppose, for example, you find out that ~H, and you've already reasoned to ~H>C (as described above). Suppose then you say to Joe right away, "That means I am going to need a phone." Why? Well, you haven't said it explicitly, of course, but it would obviously be reasonable to interpret you as assuming the premise C>P --if you should call your sister, then you are going to need a phone. This is an example of a premise that you might not mention since it is so obvious. When we interpret what others say or write, we sometimes need to add premises that they fail to mention in order to be able to interpret the reasoning as valid -- and in many cases it is not difficult to do this.

* Practice

10.1 Symbolize each of the following sentences using the special symbolic language that we have developed. Use these symbols. D: I will drive. T: I am too tired to drive. B: There are going to be back-seat drivers. W: I want to drive. G: I will go. J: Joe is going. S: Sally is going.

(a) I will drive unless I am too tired to do it.

(b) I will drive only if Sally goes.

(c) I will drive but I do not want to do it.

(d) Neither Joe nor Sally is going.

(e) I will neither drive nor go.

(f) I will drive although I do not want to do it.

(g) I will drive if I am not too tired to do it.

(h) I will go unless there are going to be some back-seat drivers.

(i) I will go unless Joe goes.

(j) I will go only if Joe goes.

(k) I will go if Joe goes.

(l) I will not drive unless Joe goes.

(m) I will drive although I am too tired to do it.

(n) If there are going to be back-seat drivers, I will not drive.

(o) I will not drive unless there are going to be back-seat drivers.

(p) I will drive only if there are not going to be any back-seat drivers.

(q) I will drive only if I am not too tired to do it.

(r) Joe is going although Sally is going.

10.2 Each of the following arguments is valid. Interpret each of them by symbolizing each of the sentences (using the same symbols as in the preceding exercise), identify the conclusion and premises, and construct a derivation to show that the reasoning is valid.

(a) I will drive unless Joe goes. I want to drive but I will not drive. Sally is going if Joe goes. So Sally is going.

(b) Neither Sally nor Joe is going. I will go only if Sally goes. So I am not going.

(c) Sally is not going and neither will I. Joe is going only if I go. So Joe is not going either.

(d) I will drive only if there are not going to be back-seat drivers. I will go although there are going to be backseat drivers. So I will go but I will not drive.

(e) I will go only if Sally goes; and if she goes I will go. I will drive unless Joe goes; and if he goes I will not drive. Both Sally and Joe are going. So I will go but I will not drive.

10.3 Both of the following arguments are valid. Interpret each of them by symbolizing the sentences and identifying the premises and conclusion. Then construct a proof to show that the reasoning is valid. Use these symbols: M: Joe will marry Britney. A: Joe is an alien from outer space. W: Joe wins the lottery.

(a) Joe will marry Britney unless he is an alien from outer space. Joe is not an alien. So he will marry Britney.

(b) Joe will marry Britney only if he wins the lottery. He is not going to win the lottery. So he will not marry Britney.

10.4 Suppose a thief were to reason as follows.

I am going to steal the lottery ticket right away if the girl has it. The girl has the ticket unless the boy has it. The boy has the ticket only if he held on to it; but he did not hold on to it. Therefore I am going to steal the lottery ticket right away.

This thief’s reasoning is valid. (a) Symbolize the reasoning using the following symbols; and (b) construct a proof to show that the reasoning is valid.

S: I am going to steal the lottery ticket right away.

G: The girl has the ticket.

B: The boy has the ticket.

H: The boy held on to the ticket.

10.5 Suppose a professor were to reason as follows.

I am going to try to steal the students’ money if their money is in their backpacks. The students’ money is in their backpacks unless they spent it on new clothes. They spent their money on new clothes only if they look spiffy; but they actually do not look spiffy. So I am going to try to steal their money.

This professor’s reasoning is valid. (a) Symbolize the reasoning using the following symbols; and (b) construct a proof to show that the reasoning is valid.

T: I am going to try to steal the students’ money.

B: The students’ money is in their backpacks.

C: The students spent their money on new clothes.

S: The students look spiffy.

10.6 March will soon be here. This means at least two things. First, the Bowling Green State University Philosophy 103 (Introduction to Logic) class will have their second test (March 4), and the logic students are working hard. Secondly, the BGSU basketball teams also are working hard. They want to win the league championships and go to the national tournaments. The following examples of reasoning are about some games on a certain Saturday.

Some BUT NOT ALL (!!) of the following pieces of reasoning are valid. (So this is a new type of exercise!)

(a) Symbolize each of them. (b) If valid, construct a derivation to show that it is valid; and (c) if it is not valid, explain why it is not valid.

Use these symbols. F: The Falcons win. M: Miami wins. T: Toledo wins. K: Kent State wins. A: Akron wins. Interpret "losing" as "not winning" (for example, represent "Miami loses" as ~M).

(a) If the Falcons lose, Miami wins. So the Falcons will win if Miami loses.

(b) If Toledo wins, so will Kent St. If Kent St wins, so will the Falcons. Therefore, the Falcons will win if Toledo does.

(c) If Miami wins, so will Toledo. But Miami won't win. So Toledo won't either.

(d) If Kent State wins, Toledo will lose. So Kent State will lose if Toledo wins.

(e) The Falcons will win if Miami wins and Toledo loses. So if Miami wins, then the Falcons will win if Toledo doesn't win.

(f) Akron will lose. Toledo will win if Akron does not lose. Therefore, Toledo also will lose.

(g) If Akron wins, Miami will win. Miami will win. So Akron will win.

(h) If Akron wins, Miami will lose. Miami will win. So Toledo will lose.

(i) If Miami wins, Toledo will lose. Miami is going to win. If Toledo loses, Akron will lose. Akron will win if the Falcons lose. Therefore, the Falcons will win! *Yay!