Introduction to Logic

Introduction to Logic Dr. Pedro Poitevin IDS 108-08

Consistency • A set of beliefs is consistent if the beliefs are compatible with each other. • A set of beliefs may itself be consistent while not being consistent with the known facts. • A set of beliefs is consistent if these beliefs could all be true in some possible situation.

Example 1 • “It would be wrong to censor violent programs on television, because people’s behavior isn’t really affected by what they see on the screen. All the same it would be a good idea to have more programs showing the good side of our national way of life, because it would change the ways of some of the people who are always knocking down our country.”

Example 2 • “During the last five years I have been involved in three major accidents and several minor ones, while driving my car. After two of the major accidents, courts held me responsible. But basically I’m a thoroughly safe driver; I’ve simply had a run of bad luck.”

Example 3 • “The surface of the earth is flat (apart from mountains, oceans, and other relatively small bumps and dips). When people think they have sailed round the earth, all they have really done is to set out from one place and finish up in another place exactly like the one they started from, but several thousand miles away.”

Example 4 • “I have invented an amazing new sedative that makes people faster and more excited.”

Exercise 1A • You know that human beings normally have two legs. Try to convince yourself that they normally have five.

Exercise 1B Are the following sets of beliefs consistent? • “I’ve never drawn anything in my life. But if I sat down to it now, it would take me two minutes to produce a drawing worth as much as anything by Picasso.” • “I knew I would never get pregnant. But somehow it just happened.” • “There is no housing shortage in Lincoln today – just a rumour that is put about by people who have nowhere to live.” • “Walter joined the friendly club two years ago, and has been one of its most loyal members ever since. Last year he paid for the holidays of precisely those club members who didn’t pay for their own holidays.”

Beliefs and Words • Many sentences do not naturally state beliefs. (eg: “What time is it?”) • One sentence may have two different meanings. (eg: “Tariq is a Persian carpet importer.”) • One sentence can be used on different occasions to talk about different things. (eg: “Tomorrow will be rainy.”) • The sentences that do express beliefs are called declarative sentences.

Declarative Sentences • A declarative sentence of English is a grammatical English sentence which can be put in place of ‘x’ in Is it true that x? • Example: “The price of beef has fallen.” • We call a string of words grammatical if most speakers of the language would accept it as correctly formed. • A sentence may be slightly off from being grammatical. In that case we say it is a perturbation of a grammatical sentence. • A sentence is a selection violation if it puts concepts together in impossible ways but is otherwise grammatically okay.

Examples • “These pages must not be removed or defaced.” • “Your breast will not lie by the breast Of your beloved in sleep.” • “The very so not was, wasn’t it?” • “echo foxtrot golf hotel” • “Did you be angry with Sally?” • “Her train was decorated with silver gorgeous lace.” • “Government hastening collapse of economy.” • “By swaggering could I never thrive.”

Selection Violation: Examples • “The pianist then played a red hat topped with geraniums and wisdom.” • “He ate a slice of boredom.” • “Nothing spoke her more than a hot bath.” • “The civilization of the ancient Persians fervently knew at least two inches.”

Selection Violation and Poetry • Selection violations are easy to recognize by their bizarre and poetic feel. • In fact, they play an important role in poetic or metaphorical writing. • By committing a selection violation deliberately, a writer can force her readers to forget the literal sense of what she says; since they can make nothing of her words if they take them literally, they have to notice the colors and the overtones.

Exercise 3A Classify the following sentences: • “Colorless green ideas sleep furiously.” • “Furiously sleep ideas green colorless.” • “Please pass me a butter.” • “I bet there’s dozens of people you’ve forgotten to invite.” • “singing each morning out of each night my father moved through depths of height” • “My father moved through theys of we” • “You can’t do nothing with nobody that doesn’t want to win.” • “His face was calm and relaxed, like the face of an asleep child.” • “Not, Father, further do prolong Our necessary defeat.” 10. “Time and the bell have buried the day.”

Exercise 3B Which of the following are declarative sentences? • “Twice two is four.” • “Please write a specimen of your signature in the space provided.” • “Would you believe you’re standing where Cromwell once stood?” • “That’s true.” • “I promise not to peep.” • “Blackmail is wicked.”

Ambiguity • Lexical ambiguity occurs when a single word can be understood in more than one way. • Example: “I thought it was rum.” • Structural ambiguity occurs when the words in the string can be grouped together in different ways. • Example: “I heard about him at school.” • Ambiguity of cross-reference: it occurs when a word or phrase refers back to something mentioned elsewhere, but it isn’t clear which thing is it that it refers to. • Example: “Tasmin’s stuck at home with a migraine, but Diane said she’d be dealing with the Kaplan file today.”

Exercise 4A What kinds of ambiguity occur in the following? • “Launching the ship with impressive ceremony, the Admiral’s lovely daughter smashed a bottle of champaign over her stern as she slid gracefully down the slipways.” • “Miss Crichton pluckily extinguished the blaze while Herr Eckold pulled the orchestra through a difficult passage.” • “The font so generously presented by Mrs Smith will be set in position at the East end of the Church. Babies may now be baptized at both ends.” • “The Government were strongly urged to take steps to put a stop to the growing evil of methylated spirit drinking by the Liverpool justices at their quarterly meetings.”

Exercise 4B Rewrite each string in two non-ambiguous ways. • “I shall wear no clothes to distinguish me from my fellow citizens.” • “He only relaxes on Sundays.” • “He gave each guest a glass of rum or gin and tonic.” • “Most of the nation’s assets are in the hands of just one person.” • “Dogs must be carried.”

Referential Failure “Suppose that the greatest thing that can be conceived doesn’t exist outside our minds. Then it is not as great as it would have been if it had existed. Therefore we can conceive something greater than it; which is impossible. Therefore our original supposition is incorrect.” - Anselm

Scaling Adjectives • Adjectives like fat, funny, happy, expensive, heavy, unpleasant, etc. are scaling adjectives. One can be fatter, or happier than someone else. One’s jokes may be funnier than someone else’s. Some things are heavier than others, and some events are more unpleasant than others.

Non-Scaling Adjectives • Some adjectives are not scaling. • Examples: straight, silent, perfect, daily. • What about the adjective “true”?

Bizarre Situations • Example: the two cerebral hemispheres of the human brain are joined by a structure called the corpus callosum. Surgeons sometimes cut through the corpus callosum in order to control epilepsy. • People whose corpus callosum has been severed are quite normal in most ways, but they show one or two strange symptoms. • Suppose we show an object to such a person, and then ask the person to indicate what we showed him or her by writing a mark on a piece of paper. • If the object fell in the left half of his or her field of vision, then the person can answer our question with his or her left hand but not with his or her right. • If the object fell in the right half then the person can answer with his or her right hand but not with his or her left. • Does a person with a severed corpus callosum, who can see a mouse to the left of him or her, know that he or she is seeing a mouse? • It seems her or she knows with one half of his or her brain but not with the other. • In this case, is it appropriate to say simply that he or she knows? • In a bizarre situation it may be impossible to say whether a sentence is true-not because we are stupid or lack the facts, but because language fails us.

Misleading Statements • A witness in the case of Thumptmann v. Thumper states that “Mr. Tumper hit Mr. Thumptmann three times with the camera tripod, and Mr. Thumptmann fell to the floor.” • What the witness actually saw was that Mr. Thumptmann fell to the floor just before Mr. Thumper came into the room, and Mr. Thumper hit him three times with the camera tripod before he could get up. • After the office party, a man admits to his wife “I did kiss some of the girls.” • In fact, he kissed all nineteen of them. • After another office party, another man boasts to his wife “All the girls kissed me.” • In fact there were no girls at the party.

Possible Situations • In logic, a situation is described as possible if it could have been the actual situation, forgetting what we know about the world. • There are some limits to what could have been: two times two could never have been equal to four. • One of the smartest philosophy books of all time starts “If kangaroos had no tails, they would topple over” and proceeds to work out the details of how to make sense of the meaning of that sentence.

Applications • Counterfactual reasoning is the art of reasoning with possible situations. • Counterfactual reasoning helps us distinguish between concepts that appear to be the same concept. • Distinguishing between concepts is very useful, for a variety of reasons.

Mistake and Accident • How to distinguish between mistake and accident? • Imagine a situation in which an accident has occurred, but no mistake has occurred. • Example: “You have a donkey, so have I, and they graze in the same field. The day comes when I conceive a dislike for mine. I go to shoot it, draw a bead on it, fire… but as I do so, the beasts move, and to my horror yours falls.” • This is clearly an accident, but I didn’t make a mistake. • Nevermind the fact that the situation hasn’t happened, we have made it clear that the two concepts are not the same.

Exercise 8 Distinguish between the following pairs of sentences: • The senator was lying. What the senator said was untrue. • It’s in your best interests to go to Corsica. It would do you good to go to Corsica. • He knows I’m at home. He thinks I’m at home, and I am. 4. Brutus killed Caesar. Brutus caused Caesar to die. 5. A crow is a kind of bird. The word “crow” is used to denote a kind of bird.

Introduction to Logic