# Logic by suchenfz

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```									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

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

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