MODULE: Basic logic

TUTORIAL L03: Basic concepts

Here are a few basic concepts in logic that you ought to be familar with, whether you are studying symbolic logic or not.

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L03.1 Negation

The negation of a statement α is a statement whose truth-value is necessarily opposite to that of α. So for example, for any English sentence α, you can form its negation by appending "it is not the case that" to α to form the longer statement "it is not the case that α".

In formal logic, the negation of α can be written as "not-α", "~α" or "¬α".

Here are some concrete examples:

Statement (α)	Negation (¬α)
It is raining	It is not the case that it is raining (i.e. It is not raining.)
1+1=2	It is not the case that 1+1=2 (i.e. 1+1 is not 2.)
Spiderman loves Mary	It is not the case that Spiderman loves Mary.

There are two points about negation which should be obvious to you:

• A statement and its negation can never be true together. They are logically inconsistent with each other.
• A statement and its negation exhaust all logical possibilities - in any situation, one and only one of them must be true.

L03.2 Exercises

? Question 1 - Some exercises for you :

• What is the negation of "God exists"?
[Show answer]
"God does not exist.", "It is not the case that God exists."
• Is "I must not leave" the negation of "I must leave"?
[Show answer]
No! It should be "It is not the case that I must leave." This includes the possibility that it does not matter whether I should leave or not. Note that these two statements do not exhaust all possibilities.
• Is "Tom is very happy" the negation of "Tom is very depressed"?
[Show answer]
These statements are inconsistent but they are not negations of each other. The negation of the first is "It is not the case that Tom is very happy." This includes situations where Tom is neither happy nor depressed, or where Tom is a little depressed but not very depressed. Remember that a statement and its negation must exhaust all logical possibilities.

L03.3 Consistency

Suppose S is a set that contains one or more statement. S is consistent when it is logically possible for all of the statments in the set to be true at the same time. Otherwise S is inconsistent. Some examples:

• Consistent: Peter is three years old. Jane is four years old.
• Consistent: Peter is three years old. Peter is a fat rabbit.
• Inconsistent: Peter is three years old. Peter is a fat rabbit. Peter is five years old.
• Inconsistent: Peter is three years old. It is not the case that Peter is three years old.
• Inconsistent: Peter is a rabbit. All rabbits are three years old. Peter is one year old.
• Inconsistent: Peter is a completely white rabbit that is completely black.

Here are a few important points about consistency:

• If you have two statements that are both true, they are certainly consistent with each other.
• If you have two statements that are both false, they might or might not be consistent with each other. See if you can give your own examples.
• In the last example above, we have just one single inconsistent statement. An inconsistent statement must be false. But if you have a set of statements { P, Q, R, S }, the whole set is said to be inconsistent even if R and S are both true, and inconsistency is only due to inconsistency between P and Q.
• Although statements that are inconsistent with each other cannot all be true at the same time, it might be possible for them to be false at the same time.
• Every statement is inconsistent with its negation.

Inconsistency and self-defeating statements

Notice that there is a difference between making self-defeating statements and inconsistent statements. Suppose a tourist from a non-English speaking country says: "I cannot speak any English." Since what is being spoken is an English sentence, the tourist is obviously saying something false. However, strictly speaking the sentence is not logically inconsistent because it actually describes a logically possible situation. It is quite possible for the speaker not to be able to speak any English. What is impossible is to say the sentence truly. In these situations, it is more appropriate to say that the utterance is self-defeating rather than inconsistent.

Here are some funny actual examples of self-defeating / inconsistent statements:

During the installation of the Microsoft Wireless Optical Desktop for Bluetooth, you may receive an error message similar to the following when you try to restart your computer:

Keyboard Error or No Keyboard Present
Press F1 to continue, DEL to enter setup

An actual webpage: (Why is this strictly speaking NOT a case of inconsistency?)

L03.4 Exercises

? Question 1 - Which of the following sets of statements are consistent? Why?

1. Vegetables are good for your health. Vegetables are bad for your health. [Show answer]
   Inconsistent.
2. Joseph likes steak. Sharifa does not like steak. [Show answer]
   Consistent.
3. I knew I would pass the final exam. It is just bad luck that I didn't. [Show answer]
   Inconsistent. If you know something, what you know must be true.
4. Marilyn has never played basketball. But if she were to play basketball today, she will become the world's best basketball player tomorrow. [Show answer]
   Unlikely to be true, but consistent.
5. World War II actually did not happen. It is a lie cooked up by historians and politicians. People who said they remember what happened are actually part of the whole conspiracy. [Show answer]
   Again, unlikely to be true, but consistent.
6. No matter what is going to happen in the future, I shall still love you. [Show answer]
   Strictly speaking, not consistent. If I do not exist tomorrow, then I would not be able to continue to love you.
7. Tom can only fly very slowly. No human being can fly. [Show answer]
   Consistent, since it is not claimed that Tom is a human being. Maybe Tom is a bird. Notice that it is wrong to say that the sentences are consistent if Tom is not a human, but inconsistent if he is a human. Whether or not they are consistent depends only on what the sentences mean, and the sentence "Tom can fly in the sky without assistance" means the same thing whether Tom is human or something else. This is because the meaning of the name "Tom" does not tell you whether Tom is a human or something else.
8. Ah Kee is the best restaurant in Hong Kong. There are no good restaurants in Hong Kong. [Show answer]
   Consistent, since the best just has to be better than others, and might not be very good.
9. If God exists then God loves human beings. If God loves human beings then he would not want them to suffer at all. Many human beings experience a lot of suffering. God exists. [Show answer]
   The sentences are consistent. The sentences entail that people suffer, and God does not want people to suffer -- but that is not a contradiction. The sentences would only be inconsistent if you added the premise, "If God does not want human beings to suffer, then human beings do not suffer". Without that premise, it is easy to imagine that people suffer even though God exists and does not want us to suffer. Perhaps God does not have the power to prevent suffering, or maybe he has other desires that he can satisfy only if people suffer (for example, he might want us to have free will).

? Question 2 - Are these statements true or false?

1. If A is inconsistent with B, then if B is false, A must be true. [Show answer]
   False, since they might both be false. Example : "There is only one book on the table" is inconsistent with "There are only two books on the table". Both are false when there is no book on the table.
2. If statement A is inconsistent with statement B, and statement A is also inconsistent with statement C, then B is inconsistent with C. [Show answer]
   False. For a start, statement B might be the same as statement C. Every statement is of course consistent with itself (unless the statement itself is inconsistent).

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L03.5 Entailment

A sentence X entails Y if Y follows logically from X. In other words, if X is true then Y must also be true, e.g. "30 people have died in the riots" entails "more than 20 people died in the riots", but not vice-versa.

• If X entails Y and we find out that Y is false, then we should conclude that X is also false. But of course, if X entails Y and we find out that X is false, it does not follow that Y is also false.

• If X entails Y but Y does not entail X, then we say that X is a stronger claim than Y (or "Y is weaker than X"). For example, "all birds can fly" is stronger than "most birds can fly", which is still stronger than "some birds can fly".

  A stronger claim is of course more likely to be wrong. To use a typical example, suppose we want to praise X but are not sure whether X is the best or not, we might use the weaker claim "X is one of the best" rather than the stronger "X is the best". So we need not be accused of speaking falsely even if it turns out that X is not the best.

L03.6 Exercises

? Question 1 -

• If a statement A entails a statement B, does it follow that B entails A? [Show answer]
  No. For example, "Peter and Amie are happy" entails "Peter is happy" but not the other way round.
• If a statement A entails a statement B, and B entails another statement C, does A entail C? [Show answer]
  Yes. Since if A is true, then B must be true, and if B is true, C must be true. So if A is true, C must be true.
• Suppose a statement A entails a statement B, and B is false. Can we tell whether A is true or false? [Show answer]
  A cannot be true. If A were true, then B must also be true, which it is not.
• Consider these two statements : "Peter loves Beth or Peter loves April.", "If Peter loves Beth then Peter loves April." Do they entail that Peter loves Beth? Do they entail that Peter loves April? [Show answer]
  They entail that Peter loves April. They do not entail that Peter loves Beth.

? Question 2 - What do these statements entail which they do not entail on their own?

• Either it is raining or it is cloudy. It is not raining. [Show answer]
  It is cloudy.
• If Peter is upstairs, then someone is in the basement. Nobody is in the basement. [Show answer]
  Peter is not upstairs.

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L03.7 Logical Equivalence

If we have two statements that entail each other then they are logically equivalent. For example, "everyone is happy" is equivalent to "nobody is not happy", and "the glass is half full" is equivalent to "the glass is half empty".

• If two statements are logically equivalent, then they must always have the same truth value.

L03.8 Exercises

? Question 1 -

• Is the statement "good things are not cheap" logically equivalent to the statement "cheap things are not good"? [Show answer]
  Yes. They are both true when there is nothing that is both good and cheap, and they are false when there is at least one thing that is both good and cheap.
• See if you can form a different statement that also starts with "some" that is logically equivalent to "some animals are birds". [Show answer]
  Some birds are animals.
• Which of these statements are logically equivalent? (a) It is not true that there is no life on Mars. (b) There may be life on Mars. (c) It is rather unlikely that there is life on Mars. (d) There is life on Mars. (e) There may not be life on Mars. (f) It is the case that there is life on Mars. (g) There is no life on Mars. [Show answer]
  (a), (d) and (f).
• Which of these statements are logically equivalent? (a) It is not true that I have money. (b) It is false that I have no money. (c) I have lots and lots of money. (d) I have no money. [Show answer]
  (a) and (d).
• What about the statements in this photo?

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