Transcript INTRO LOGIC

INTRO LOGIC
DAY 02
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Schedule for Unit 1
Day 1 Intro
Day 2 Chapter 1
warm-up
Day 3 Chapter 2
Day 4 Chapter 3
40% of
Exam 1
Day 5 Chapter 4
Day 6 Chapter 4
60% of
Exam 1
Day 7 Chapter 4
Day 8 EXAM #1
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Chapter 1
Basic Concepts
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What is logic?
Logic is the science of reasoning,
which is to say:
the academic discipline that
investigates reasoning.
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What is reasoning?
reasoning is inferring (deducing)
to infer is
to draw conclusions (output)
from premises (input).
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Aside
words/ideas related to ‘draw’
an often used cognate of ‘draw’
is ‘draft’ (‘draught’)
both words mean to pull
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Examples of Reasoning
You see smoke,
and you infer (deduce) that
there is fire.
You count 19 in a group,
which originally had 20,
and you infer (deduce) that
someone is missing.
(input)
(output)
(input)
(input)
(output)
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Basic Idea
Logic evaluates reasoning
in terms of arguments.
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What is an argument?
ar·gu·ment (är“gy…-m…nt) n.
1.a. A discussion in which disagreement is expressed; a debate. b. A quarrel; a dispute. c.
Archaic. A reason or matter for dispute or contention: “sheath'd their swords for lack of
argument” (Shakespeare).
2.a. A course of reasoning aimed at demonstrating truth or falsehood: presented a
careful argument for extraterrestrial life. b. A fact or statement put forth as proof or
evidence; a reason: The current low mortgage rates are an argument for buying a house
now.
3.a. A summary or short statement of the plot or subject of a literary work. b. A topic; a
subject: “You and love are still my argument” (Shakespeare).
4. Logic. The minor premise in a syllogism.
5. Mathematics. a. The independent variable of a function. b. The amplitude of a
complex number.
6. Computer Science. A value used to evaluate a procedure or subroutine.
[Middle English, from Old French, from Latin arg¿mentum, from arguere, to make clear.
See ARGUE.]
[American Heritage Dictionary]
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For purposes of logic…
an argument is
a collection of statements,
one of which is designated as
the conclusion,
and the remainder of which
are designated as
the premises.
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What is a statement?
A statement is
a declarative sentence,
i.e., a sentence that is capable of
being true or false.
Kinds of sentence
 declarative
 interrogative
 imperative
 exclamatory
 performative
Example
the window is shut
is the window shut?
shut the window
$%&@!!!!
I hereby …
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Examples of Arguments
there is smoke
therefore,
there is fire
(premise)
(conclusion)
there are 19 persons currently
(premise 1)
there were 20 persons originally (premise 2)
therefore,
someone is missing
(conclusion)
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2 questions about an argument
1. are all the premises true?
2. does the conclusion follow from
the premises?
Alternatively,
1. do the premises rest on
the facts?
2. does the conclusion rest on
the premises?
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conclusion
premises
facts
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Definitions
an argument is
if and only if
factually correct
all its premises are true
valid
its conclusion follows
from its premises
sound
it is both factually correct
and valid.
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Example 1
Parish is taller than McHale
McHale is taller than Bird
therefore
Parish is taller than Bird
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Example 1
Parish is taller than McHale
T
McHale is taller than Bird
T
/ Parish is taller than Bird
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factually correct?
YES
valid?
YES
sound?
YES
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Example 2
Bird is taller than McHale
F
McHale is taller than Parish
F
/ Bird is taller than Parish
F
factually correct?
NO
valid?
YES
sound?
NO
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Example 3
Parish is taller than McHale
T
Parish is taller than Bird
T
/ McHale is taller than Bird
T
factually correct?
YES
valid?
NO
sound?
NO
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Example 4
McHale is taller than Parish
F
McHale is taller than Bird
F
/ Bird is taller than Parish
F
factually correct?
NO
valid?
NO
sound?
NO
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Fundamental Principle Of Logic
Whether an argument is
valid or invalid
is determined entirely
by its form.
validity is a function of form
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In other words…
If an argument is valid,
then every argument
with the same form
is also valid.
If an argument is invalid,
then every argument
with the same form
is also invalid.
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Method Of Counterexamples
In order to show that
an argument is invalid,
it is sufficient to
find a counterexample to it.
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Definition of ‘counterexample’
Consider an argument; call it .
Then a counterexample to
is (by definition) any argument
with the following properties:
1.
2.
3.

*
* has the same form as ;
* has all true premises;
* has a false conclusion.
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Example 1
Argument
Parish is taller than McHale
Parish is taller than Bird
/ McHale is taller than Bird
T
T
T
Form
X is taller than Y
X is taller than Z
/ Y is taller than Z
Counterexample
The Library is taller than PeeWee Herman
The Library is taller than Arnold Swarzenegger
/ PeeWee H. is taller than Arnold S.
T
T
F
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Example 2
Argument
all UMass students are high school graduates
some high school graduates are athletes
/ some UMass students are athletes
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T
T
Form
all X are Y
some Y are Z
/ some X are Z
Counterexample
all UMass students are high school graduates
some high school graduates are U.S. senators
/ some UMass students are U.S. senators
T
T
F
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THE END
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