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Introduction to Logic

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Title: Introduction to Logic


1
Introduction to Logic
  • Dr. Pedro Poitevin
  • IDS 108-08

2
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.

3
Example 1
  • It would be wrong to censor violent programs on
    television, because peoples behavior isnt
    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.

4
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 Im a thoroughly safe driver Ive
    simply had a run of bad luck.

5
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.

6
Example 4
  • I have invented an amazing new sedative that
    makes people faster and more excited.

7
Exercise 1A
  • You know that human beings normally have two
    legs. Try to convince yourself that they
    normally have five.

8
Exercise 1B
  • Are the following sets of beliefs consistent?
  • Ive 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 didnt pay for
    their own holidays.

9
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.

10
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.

11
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, wasnt 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.

12
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.

13
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.

14
Exercise 3A
  • Classify the following sentences
  • Colorless green ideas sleep furiously.
  • Furiously sleep ideas green colorless.
  • Please pass me a butter.
  • I bet theres dozens of people youve 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 cant do nothing with nobody that doesnt
    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.

15
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 youre standing where Cromwell
    once stood?
  • Thats true.
  • I promise not to peep.
  • Blackmail is wicked.

16
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 isnt clear which thing is it
    that it refers to.
  • Example Tasmins stuck at home with a migraine,
    but Diane said shed be dealing with the Kaplan
    file today.

17
Exercise 4A
  • What kinds of ambiguity occur in the following?
  • Launching the ship with impressive ceremony, the
    Admirals 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.

18
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 nations assets are in the hands of
    just one person.
  • Dogs must be carried.

19
Referential Failure
  • Suppose that the greatest thing that can be
    conceived doesnt 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

20
Scaling Adjectives
  • Adjectives like fat, funny, happy, expensive,
    heavy, unpleasant, etc. are scaling adjectives.
    One can be fatter, or happier than someone else.
    Ones jokes may be funnier than someone elses.
    Some things are heavier than others, and some
    events are more unpleasant than others.

21
Non-Scaling Adjectives
  • Some adjectives are not scaling.
  • Examples straight, silent, perfect, daily.
  • What about the adjective true?

22
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.

23
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.

24
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.

25
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.

26
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 didnt make a
    mistake.
  • Nevermind the fact that the situation hasnt
    happened, we have made it clear that the two
    concepts are not the same.

27
Exercise 8
  • Distinguish between the following pairs of
    sentences
  • The senator was lying.
  • What the senator said was untrue.
  • Its in your best interests to go to Corsica.
  • It would do you good to go to Corsica.
  • He knows Im at home.
  • He thinks Im 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.
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