Docstoc

Logic

Document Sample
Logic Powered By Docstoc
					Logic
              What is Logic?

   LOGIC = is the study of the principles, the
    nature, and the character of GOOD
    REASONING OR GOOD ARGUMENTS.
   In Logic, we also study fallacies or ways in
    which people typically make logical errors.
                 Reasoning

   Reasoning is a process of thought where
    one moves from a belief in the truth of
    one claim to a belief in the truth of some
    other claim.
   Reasoning appears in arguments given in
    natural language.
     Example argument

We should never allow the killing of a
   human being, no matter what the
 circumstances. Therefore, we should
      eliminate the death penalty.
         Argument diagram
(Reason:) We should never allow the killing
   of a human being, no matter what the
              circumstances.

                      Step of reasoning


(Conclusion:) We should eliminate the death
                  penalty.
          Another example

We should not have attacked Iraq because it
 didn’t have any major weapons of mass
 destruction.
            Diagram

Iraq didn’t have any weapons of mass
               destruction.




 We should not have attacked Iraq.
          Another Example

Occupying Iraq is the best way to fight the
 global threat of terrorism. So Bush was
 correct to attack and occupy Iraq.
               Diagram
Occupying Iraq is the best way to fight the
        global threat of terrorism.



Bush was correct to attack and occupy Iraq.
Deductive and Inductive Arguments
Traditionally, there are two kinds of
  arguments:
 Deductive

 Inductive
         Deductive Arguments

   Deductive argument: The step of
    reasoning is intended to be deductively
    valid. (It might not be.)
   Deductively Valid: The conclusion is
    guaranteed to be true 100% if the reasons
    are true. There is no possible way for the
    conclusion to be false.
             Example

We should never allow the killing of a
  human being, no matter what the
           circumstances.
                    Deductively Valid?: is the
                    conclusion guaranteed to be
                    true if the reason were true?

We should eliminate the death penalty.
                Example

Abortion is the intentional killing of an
 innocent human being. No one should
 ever intentionally kill an innocent human
 being. So no one should ever have an
 abortion.
         Deductive argument
(1) Abortion is the intentional killing of an
  innocent human being.
(2) No one should ever intentionally kill an
  innocent human being.

                       Deductively Valid ?


No one should ever have an abortion.
                 Example

The soul survives the death of the body.
 Thus the soul is not part of the body. In
 addition, the soul is where the self resides.
 Consequently, the self is not part of the
 body.
               Diagram
The soul survives the death of the body.

                      Deductively Valid?



  (1) The soul is not part of the body.
+ (2) The soul is where the self resides.
                      Deductively Valid?



    The self is not part of the body.
          Inductive Arguments
   An inductive argument is not intended to
    be deductively valid.
   It is intended to be Logically Strong if it
    is supposed to be a good inductive
    argument.
   The conclusion of an inductive argument
    contains information that goes beyond
    what is contained in the reasons.
            Inductive Arguments

   Degrees of Logical Strength
       Deductively Valid
       Strong
       Moderate
       Weak
       Nil
           Strong Arguments

   Strong: The conclusion might be false if
    the reasons are true, but it is highly
    unlikely.
   The conclusion will be true beyond a
    Reasonable Doubt.
            Murder Trial Example
   You are a juror and you have this
    evidence:
       There is a video of the defendant hitting the
        innocent victim and then shooting him to
        death.
       There are three witnesses to the whole affair.
       The DNA evidence matches the defendant.
       He confesses.
   Would you find the defendant guilty?
                    Strong
   You will probably find the defendant guilty
    of murder, but the evidence does not
    guarantee that the defendant is guilty. It
    is possible that he is innocent, but he
    guilty beyond a reasonable doubt. The
    logical connection between the evidence
    and the conclusion that he is guilty is
    strong, not deductively valid.
       Different kinds of inductive
               arguments
   Enumerative
   Analogical
   Argument to the best explanation
Enumerative Inductive Argument

  Every scientific theory that we have
developed in the past has eventually been
            shown to be false.


 All scientific theories we develop in the
future will eventually be shown to be false
                    as well.
Analogical Inductive Argument

(1) The human heart is like a motor pump.
(2) If a motor pump breaks, it can be fixed.



If the human heart breaks down, it can be
                   fixed.
 Argument to the Best Explanation

(1) Occasionally, I hear creaking sounds in
  my home at night.
(2) The best explanation for the noise is
  that my walls and/or roof are moving
  slightly.


  My walls and/or roof are moving slightly.
            Rating Arguments
   Ask the Magic Question: If I pretend
    that the reasons are all true, is it still
    possible for the conclusion to be false?
   Try to think of a possibility. If you can,
    then the argument is not deductively
    valid. The more likely the possibility, the
    weaker the logical strength of the
    argument: Strong, Moderate, Weak, Nil. If
    the possibility is highly unlikely, then the
    argument is strong.
         Rate the argument

There are 150 people in Dr. Goldberg’s
 History class.




There is at least one woman in his class.
      Rate the argument

      John owns an automobile.




John owns a car, van, SUV, truck, dune
   buggy, go-cart, golf cart, dragster,
     motorcycle, or motorized bike.
        Rate the argument

 Nothing can go faster than the speed of
                   light.




Electrons can’t go faster than the speed of
                    light.
        Rate the argument

Einstein’s theory says that nothing can go
       faster than the speed of light.




Electrons can’t go faster than the speed of
                    light.
    Rate the argument

(1) The Bible says that God exists.
  (2) What the Bible says is true.




           God exists.
 Rate the argument

I am thinking at this moment.



I exist at this same moment.
         Rate the argument

(1) Yesterday, I met Billy for the first
          time after his brain surgery.
(2) He did not recognize me or any
          member of his family.



Billy must have become a different person.
          Rate the Argument
(1) We all have the moral obligation to perform
  those acts that make as many people happy as
  possible.
(2) Giving $2,000 to a homeless shelter will make
  more people happy than spending it on a TV set.



Spending your $2,000 on a wide-screen TV set is a
  morally wrong action.
       Rate the argument

(1) Only those beings (or things) are free
     who can act in unpredictable ways.
(2) Computers are always programmed to
       act only in predictable ways.



      Computers can never be free.
         Rate the Argument
           (1) Abortion kills a fetus
         (2) A fetus is a human being
(3) It is morally wrong to kill a human being



         Abortion is morally wrong.
  Make the argument deductively
              valid
(1) Anybody who voluntarily decides to
  destroy one’s own body is irrational.
(2) [What is the missing reason?]



       Smokers are irrational people.
Make the argument deductively valid
 (1) If my brain stops functioning, then it will
        not be possible for me to have any
                    thoughts.
       (2) [What is the missing reason?]



 When I die, it will not be possible for me to
              have any thoughts.
All Good Arguments Must Pass Two
              Tests
   (1) Deductively valid or strong
   (2) All the reasons are true
              Categorical Logic:
              Aristotelian Logic
   Every statement is analyzable in terms of classes or
    categories and their relations. Two classes of things
    could be related in 4 ways using 4 kinds of standard-
    form statements (standard-form categorical
    propositions).

 STANDARD FORMS                   EXAMPLE
 All S is P               All cows are animals.
 No S is P                No cats are dogs.
 Some S is P              Some Christians are Catholics.
 Some S is not P          Some fish are not dolphins.
      The Parts of the Basic
        Statement Forms
Note: the copula is always some form of the verb
  “to be” (with “not” in the last statement form).
  For instance, it is appropriate to use “were,” “are,”
  and “will be.”
                    Subject                Predicate
Quantifier          Term        Copula     Term

ALL               S            IS            P
NO                S            IS            P
SOME              S            IS            P
SOME              S            IS NOT        P
    Names of the Four Standard
        Statement Forms
A   All S is P        Universal Affirmative
E   No S is P         Universal Negative
I   Some S is P       Particular Affirmative
O   Some S is not P   Particular Negative
               Distribution
Distribution: a subject or predicate term is
  distributed when the statement refers to
  all the members of the class in question.

All S is P     S term is distributed.
No S is P      S and P terms are distributed
Some S is P         none distributed
Some S is not P     P term is distributed.
            Square of Opposition

   All S is P        contraries
                                     No S is P



  subalternation   contradictories     subalternation




Some S is P        subcontraries     Some S is not P
        Categorical Syllogisms
A syllogism is an argument that has 2 reasons and one
  conclusion and uses standard categorical statements.


All Catholics are Christians. (Major premise)
All Popes are Catholics.       (Minor premise)
All Popes are Christians.     (Conclusion)

   Major term = the predicate of the conclusion – Christians.
   Minor Term = the subject of the conclusion – Popes.
   Middle Term = the term left out of the conclusion, but it
    appears once in both reasons – Catholics.
    6 Rules for Determining the Deductive
            Validity of a Syllogism

   1. The middle term must distributed at least once.
   2. If a term is distributed in the conclusion, then it
    must be distributed in a premise.
   3. There must be at least one affirmative
    (nonnegative) premise.
   4. If the conclusion is negative, then one premise
    must be negative. If one premise is negative, then
    the conclusion must be negative.
   5. There must be exactly three terms, each one
    repeated twice.
   6. If the conclusion is particular (I or O), then at
    least one particular premise (I or O).
     Modern Symbolic Logic and
     Rules of Inference (Barker)
   Capital Letters
   Connectives
   Parentheses
   Rules of Inference
            Modern Symbolic Logic
   Capital Letters: P, Q, R, S, T, U, V, etc.
       They are used to stand for statements
   Symbolize these statements:
       The house is red.
       The second quarter shows that the economy is
        losing steam.
       Democracy is the best form of government.
                   Symbolic Logic
   Connectives:
      NOT = ~          ~P
      AND = &          P&Q
      OR = v           PvQ
      IF, THEN =      PQ
   Symbolize the following:
       The house is not red.
       I’m big, and you’re smart.
       The light’s on or it is off.
       If the mortgage rate lowers, then we can refinance.
       Either you are with us or the terrorists.
               Symbolic Logic
   Answer with “true,” “false,” or “can’t say”:


    P is false. ~P is __________?
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


P is true. P v Q is _________?
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


P is false. P & Q is _________?
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:

P is true. Q is false. P  Q is _______?
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


Q is true. P v Q is ________?
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


Q is false. P v Q is ________?
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


P is true. P & Q is _________?
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


P is false. P  Q is __________?
                Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


            P is false. R is true.

            ~P & R is ______?
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


       P v ~R is true. P is false.

            R is ______?
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


            P & ~Q is true.

            Q v ~P is _______?
             Symbolic Logic

   Parentheses are used to make more
    complex statements.
     Not (P & Q)
     P & (P v Q)

     P  ~(R & Q)

     S & (~P v ~(R v S))
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


       P is true. Q is true.

       ~(P & Q) is ______?
              Symbolic Logic

   Answer with “true,” “false,” or “can’t say”:


P is false. Q is true.

~P & ~(P v ~Q) is ______?
         Barker: Rules of Inference

   1.   Double Negation
   2.   Disjunctive Argument
   3.   Valid Conjunctive Argument
   4.   Modus Ponens
   5.   Modus Tollens
   6.   Hypothetical Syllogism
           Double Negation

It is not the case that the house is not
  warm. (Not-not P) (or ~~P)



          The house is warm. (P)
  Disjunctive Syllogism

(1) P or Q    (1) P or Q
(2) Not-P     (2) Not-Q



     Q             P
      Conjunctive Argument

(1) Not (P and Q)   (1) Not (P and Q)
(2) P               (2) Q



    Not-Q               Not-P
Modus Ponens

 (1) If P, then Q
 (2) P



       Q
Modus Tollens

 (1) If P, then Q
 (2) Not-Q



     Not-P
Hypothetical Syllogism

    (1) If P, then Q
    (2) If Q, then R



      If P, then R

				
DOCUMENT INFO