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INTRO LOGICDAY 02

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Schedule for Unit 1

IntroDay 1

Chapter 4Day 7

EXAM #1Day 8

Chapter 4Day 6

Chapter 4Day 5

Chapter 3Day 4

Chapter 2Day 3

Chapter 1Day 2

40% of Exam 1

60% of Exam 1

warm-up

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Chapter 1Basic 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 isto draw conclusions (output)

from premises (input).

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Aside

words/ideas related to ‘draw’

both words mean to pull

an often used cognate of ‘draw’is ‘draft’ (‘draught’)

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Examples of Reasoning

You see smoke, (input)and you infer (deduce) that

there is fire. (output)

You count 19 in a group, (input)which originally had 20, (input)

and you infer (deduce) thatsomeone is missing. (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 whichare 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

Examplethe window is shutis the window shut?shut the window$%&@!!!!I hereby …

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Examples of Arguments

there is smoke (premise)therefore,there is fire (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?

1. do the premises rest onthe facts?

2. does the conclusion rest onthe premises?

Alternatively,

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facts

premises

conclusion

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Definitions

it is both factually correctand valid.sound

its conclusion follows from its premisesvalid

all its premises are truefactually correct

if and only ifan argument is

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Example 1

therefore

McHale is taller than Bird

Parish is taller than McHale

Parish is taller than Bird

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Example 1

YESsound?

YESvalid?

YESfactually correct?

T/ Parish is taller than Bird

TMcHale is taller than Bird

TParish is taller than McHale

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Example 2

NOsound?

YESvalid?

NOfactually correct?

F/ Bird is taller than Parish

FMcHale is taller than Parish

FBird is taller than McHale

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Example 3

NOsound?

NOvalid?

YESfactually correct?

T/ McHale is taller than Bird

TParish is taller than Bird

TParish is taller than McHale

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Example 4

NOsound?

NOvalid?

NOfactually correct?

F/ Bird is taller than Parish

FMcHale is taller than Bird

FMcHale is taller than Parish

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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 formis also valid.

If an argument is invalid,then every argument

with the same formis 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. ´* has the same form as ´;2. ´* has all true premises;3. ´* has a false conclusion.

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Example 1

ArgumentParish is taller than McHaleParish is taller than Bird/ McHale is taller than Bird

TTT

TTF

FormX is taller than YX is taller than Z/ Y is taller than Z

CounterexampleThe Library is taller than PeeWee HermanThe Library is taller than Arnold Swarzenegger/ PeeWee H. is taller than Arnold S.

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Example 2

Argumentall UMass students are high school graduatessome high school graduates are athletes/ some UMass students are athletes

TTT

TTF

Formall X are Ysome Y are Z/ some X are Z

Counterexampleall UMass students are high school graduatessome high school graduates are U.S. senators/ some UMass students are U.S. senators