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ENG101A: Freshman EnglishLecture 2: Basic Logic and Making Inferences

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Learning objectives:1. What is logic?2. What is a statement?3. Basic logic (negation, consistency,

entailment, logical equivalence) (Adapted from

http://philosophy.hku.hk/think/logic/statements.php)

4. Making inferences

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What is logic?The term "logic" is often used in many different ways. It is sometimes understood broadly as the systematic study of the principles of good reasoning. Sometimes, "logic" is understood more narrowly as what we might call "deductive logic". Roughly speaking, deductive logic is mainly about the consistency of statements and beliefs, as well as the validity of arguments.

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What is a statement?In logic we often talk about the logical properties of statements and how one statement is related to another.

So what is a statement?

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What is a statement? There are three main sentence types in

English:1. Declarative sentences are used for

assertions, e.g. "He is here."2. Interrogative sentences are used to

ask questions, e.g. "Is he here?"3. Imperative sentences are used for

making requests or issuing commands, e.g. "Come here!"

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What is a statement?For present purposes, we shall take a statement to be any declarative sentence. A declarative sentence is a complete and grammatical sentence that makes a claim.

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Here are some examples of statements in English:

Snow is white.The moon is made of green cheese.Everyone is here.Whatever will be, will be.

Note: As you can see, statements can be true or false, and they can be simple or complex. But they must be grammatical and complete sentences.

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These are not statements:1. The United Nations [ A proper name, but

not a sentence ]2. Bridge over troubled waters. [ Not a

complete sentence ]3. Come here right now! [ A command that is

not a complete sentence making a claim]4. Will you be available on Tuesday or

Wednesday? [ A question ]5. HJGAS&*^@#JHGKJAS*&^*!@GJHGAA*&S

[ Ungrammatical ]

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Is it a statement?There is an easy test to decide whether something is a statement in English. Suppose you have a sentence and you add "it is true that" to the front. If the resulting expression is grammatical, then it is a statement. Otherwise it is not.

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Exercise: Are these statements?1. One plus one equals three.2. Can you come to the party please?3. AJH$%^#@!4. If it rains then the street will be wet.5. We all feel very sorry for you.6. Come here!7. A chicken is a song that weighs ten tons.8. All statements are true.9. It is true that it is raining.

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Basic logic: Negation The negation of a statement (α) is a statement whose truth-value is necessarily opposite to that of α. 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".

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Some examples:Statement (α):

1. It is raining.

2. 1+1=2

3. Spiderman loves Mary.

Negation (¬α)

1. It is not the case that it is raining (i.e. It is not raining.)

2. It is not the case that 1+1=2 (i.e. 1+1 is not 2.)

3. It is not the case that Spiderman loves Mary.

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Note: 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.

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Exercises: 1. What is the negation of "God exists"?

2. Is "I must not leave" the negation of "I must leave"?

3. Is "Tom is very happy" the negation of "Tom is very depressed"?

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

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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.

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Are these statements consistent?1. Vegetables are good for your health.

Vegetables are bad for your health.2. Joseph likes steak. Sally does not like

steak.3. I knew I would pass the final exam. It is

just bad luck that I didn't. 4. Marilyn has never played basketball.

But if she were to play basketball today, she would become the world's best basketball player tomorrow.

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Are these statements 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.

6. No matter what is going to happen in the future, I shall still love you.

7. Tom can only fly very slowly. No human being can fly.

8. Ah Kee is the best restaurant in Hong Kong. There are no good restaurants in Hong Kong.

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EntailmentA 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.

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What do the statement entail?

Either it is raining or it is cloudy. It is not raining.

If Peter is upstairs, then someone is in the basement. Nobody is in the basement.

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EntailmentIf 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".

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EntailmentA 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.

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HedgingIt is often believed that academic writing, particularly scientific writing, is factual, simply to convey facts and information. However it is now recognised that an important feature of academic writing is the concept of cautious language, often called "hedging" or "vague language". In other words, it is necessary to make decisions about your stance on a particular subject, or the strength of the claims you are making. Different subjects prefer to do this in different ways.

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Language used in hedging:

Introductory words:e.g. seem, tend, look like, appear to be, think,

believe, doubt, be sure, indicate, suggest Certain lexical verbs:

e.g. believe, assume, suggestCertain modal verbs:

e.g. will, must, would, may, might, could

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Language used in hedging:

Adverbs of frequency:e.g. often, sometimes, usually

Modal adverbs:e.g. certainly, definitely, clearly, probably,

possibly, perhaps, conceivablyModal adjectives:

e.g. certain, definite, clear, probable, possible

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Language used in hedging:

Modal nouns:e.g. assumption, possibility, probability

That clauses:e.g. It could be the case that ...e.g. It might be suggested that …e.g. There is every hope that…

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Language used in hedging:

To clause + adjectivee.g. It may be possible to obtain…e.g. It is important to develop…e.g. It is useful to study…

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Compare the following: It may be said that the

commitment to some of the social and economic concepts was less strong than it is now.

The commitment to some of the social and economic concepts was less strong than it is now.

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Compare the following: Weismann suggested that animals

become old because, if they did not, there could be no successive replacement of individuals and hence no evolution.

Weismann proved that animals become old because, if they did not, there could be no successive replacement of individuals and hence no evolution.

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Compare the following: Nowadays the urinary symptoms

seem to be of a lesser order.

Nowadays the urinary symptoms are of a lesser order.

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Identify the hedging expressions in the following sentences.

1. There is no difficulty in explaining how a structure such as an eye or a feather contributes to survival and reproduction; the difficulty is in thinking of a series of steps by which it could have arisen.

2. For example, it is possible to see that in January this person weighed 60.8 kg for eight days.

3. For example, it may be necessary for the spider to leave the branch on which it is standing, climb up the stem, and walk out along another branch.

4. Escherichia coli , when found in conjunction with urethritis, often indicate infection higher in the uro-genital tract.

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Identify the hedging expressions in the following sentences.

5. There is experimental work to show that a week or ten days may not be long enough and a fortnight to three weeks is probably the best theoretical period.

6. Conceivably, different forms, changing at different rates and showing contrasting combinations of characteristics, were present in different areas.

7. One possibility is that generalized latent inhibition is likely to be weaker than that produced by pre-exposure to the CS itself and thus is more likely to be susceptible to the effect of the long interval.

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Identify the hedging expressions in the following sentences.

8. For our present purpose, it is useful to distinguish two kinds of chemical reaction, according to whether the reaction releases energy or requires it.

9. It appears to establish three categories: the first contains wordings generally agreed to be acceptable, the second wordings which appear to have been at some time problematic but are now acceptable, and the third wordings which remain inadmissible.

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Logical equivalenceIf 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.

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Exercises1. Is the statement "good things are not cheap" logically

equivalent to the statement "cheap things are not good"?

2. 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.

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Exercises3. 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.

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Logic puzzle 1Stephen was looking at a photo. Someone asked him, "Whose picture are you looking at?" He replied: "I don't have any brother or sister, but this man's father is my father's son." So, whose picture was Stephen looking at?

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Logic puzzle 2There was a robbery in which a lot of goods were stolen. The robber(s) left in a truck. It is known that :

(1)Nobody else could have been involved other than A, B and C.

(2)C never commits a crime without A's participation.

(3)B does not know how to drive.

So, is A innocent or guilty?

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Logic puzzle 3Suppose there is such a little town : (1) No two inhabitants have exactly the same number of hairs. (2) No inhabitant has exactly 409 hairs. (3) There are more inhabitants than there are hairs on the head of any inhabitant. So, what is the largest possible number of inhabitants in that little town?

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Making inferencesInference is just a big word that means a conclusion or judgement. If you infer that something has happened, you do not see, hear, feel, smell, or taste the actual event. But from what you know, it makes sense to think that it has happened.

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An example of inferencesYou make inferences everyday. Most of the time you do so without thinking about it. Suppose you are sitting in your car stopped at a red signal light. You hear screeching tires, then a loud crash and breaking glass. You see nothing, but you infer that there has been a car accident. We all know the sounds of screeching tires and a crash. We know that these sounds almost always mean a car accident.

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Making inferences via general sense

The meaning of a word may be implied by the general sense of its context, as the meaning of the word incarcerated is implied in the following sentence:Murderers are usually incarcerated for longer periods of time than robbers.You may infer the meaning of incarcerated by answering the question: "What usually happens to those found guilty of murder or robbery?"

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Making inferences via example

When the meaning of the word is not implied by the general sense of its context, it may be implied by examples. For instance,Those who enjoy belonging to clubs, going to parties, and inviting friends often to their homes for dinner are gregarious.You may infer the meaning of gregarious by answering the question: "What word or words describe people who belong to clubs, go to parties a lot, and often invite friends over to their homes for dinner?"

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Making inferences via antonyms

The meaning of a word may be implied by an antonym or by a contrasting thought in a context. Antonyms are words that have opposite meanings, such as happy and sad. For instance,Ben is fearless, but his brother is timorous.You may infer the meaning of timorous by answering the question: "If Ben is fearless and Jim is very different from Ben with regard to fear, then what word describes Jim?"

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Making inferences via contrasts

A contrast in the following sentence implies the meaning of credence:Dad gave credence to my story, but Mom's reaction was one of total disbelief.You may infer the meaning of credence by answering the question: "If Mom's reaction was disbelief and Dad's reaction was very different from Mom's, what was Dad's reaction?"

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Exercises What political climate is the driver's request bringing to light? Why do you think the cartoonist drew this picture?

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What is this cartoon saying about President Obama?

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Make an inference about the following statements.

1. If she died, I wouldn’t go to her funeral.

2. As you give a speech in front of a large audience, you realize that people are laughing behind their hands and pointing to the region below your waist.

3. No, Honey, I don’t want you to spend a lot of money on my birthday present. Just having you for a husband is the only gift I need. In fact, I’ll just drive my old rusty bucket of bolts down to the mall and buy myself a little present. And if the poor old car doesn't break down, I’ll be back soon.