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Introductory Logic Introductory Logic

Introductory Logic - PowerPoint Presentation

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Introductory Logic - PPT Presentation

PHI 120 Presentation Truth Tables Sentences Homework Review WFFs Can you read sentences correctly Print Truth Tables handout Building TTs Sentences and Sequents Connectives when are they false ID: 419850

false truth statement true truth false true statement negation amp main simple theorem step sentences wff tables logic disjunction

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Slide1

Introductory LogicPHI 120

Presentation: "Truth Tables – Sentences"Slide2

HomeworkReviewWFFsCan you read sentences correctly?

Print:

Truth Tables

handout

"Building TTs: Sentences and Sequents"

"Connectives – when are they false"

Allen/Hand

Section 2.1, esp. pages 40-41

p. 47-8: “tautology,” “inconsistency &

contingent sentence”Slide3

In ClassHave in hand

Truth Tables Handout

See especially “Building Truth Tables” sectionSlide4

Review – Logical FormSentences (WFFs)Slide5

Well-formed FormulasSimple WFFsP, Q,

R

,

S

, ….

Complex

WFFs

Negation

~

Φ Conjunction Φ & Ψ Disjunction Φ v Ψ Conditional Φ -> Ψ Biconditional Φ <-> Ψ and nothing else

Binary Structure

Unary StructureSlide6

The Concept of Truth ValueTruth TablesSlide7

Theorem of the LogicAny statement (WFF) is either True or FalseT v ~TThis is a theorem of logic

Theorems are

tautologies

Tautologies

are necessarily true

“A statement is true.” =

TSlide8

Theorem of the LogicAny statement (WFF) is either True or FalseΦ v ~Φ

This is a theorem of logic

Theorems are

tautologies

Tautologies

are necessarily trueSlide9

Theorem of the LogicAny statement (WFF) is either True or FalseP v ~PThis is a theorem of logic

Theorems are

tautologies

Tautologies

are necessarily trueSlide10

Theorem of the LogicAny statement (WFF) is either True or False(P&~Q) v ~(P&~Q)This is a theorem of logic

Theorems are

tautologies

Tautologies

are necessarily trueSlide11

The Key to Recognizing SentencesWhich connective is the weakest link in a sequence of symbols? (or as I like to ask

)

Where can you

most easily

bend

the sentence?

See

page

9

Strongest

~& and/or v-><->WeakestSlide12

What kind of sentence?~P~P & ~Q

P v Q -> R

P v Q <-> R -> P

negation:

~

Φ

conjunction:

Φ

& Ψconditional: Φ -> Ψbiconditional: Φ <-> Ψ

“the main connective”

Metaphor of the Binding of a BookSlide13

Building Truth TablesSentences (WFFs)Slide14

The SimpleThe truth-value of an atomic sentence

PSlide15

The SimpleThe truth-value of an atomic sentence

P

1

T

2

F

1Slide16

Simple NegationThe truth-value of a simple negation

P

~

P

1

T

2

F

1

2

3

A negation (~) takes the opposite value of the statement being negated.Slide17

Simple NegationThe truth-value of a simple negation

P

~

P

1

T

F

2

F

T

1

2

3

A negation (~) takes the opposite value of the statement being negated.Slide18

Building a Truth TableRead the sentenceP v ~PSlide19

Building a Truth TableRead the sentenceP v ~PThe wedge is the main connective.Hence this is a disjunction.

Φ v ~Φ

P v ~P is an instance of our theoremSlide20

Step 1P v ~ PA Truth Table has two main columns

Left

main column: ATOMIC SENTENCES

Right

column: the WFF.

This row represents a header row.

P

P

v

~PSlide21

Step 2P v ~ PDetermine the number of rows for the WFF:Rows = 2 (power of simple statements)

P

P

v

~

P

1

2Slide22

Step 3P v ~ PFill in left main column first.

P

P

v

~

P

1

T

2

F

1

2

3

4

5Slide23

Step 4P v ~ PRight main columnassign truth-values for

negation of simple statements

.

P

P

v

~

P

1

T

2

F

1

2

3

4

5Slide24

Step 4P v ~ PRight main columnassign truth-values for

negation of simple statements

.

P

P

v

~

P

1

T

F

2

F

T

1

2

3

4

5

Notice that only one connective remains.Slide25

Skip to Last StepP v ~ PAssign truth-values for the remaining wedge.

CONNECTIVES – when they are false

A negation is false

if

the statement being negated (Φ) is true

Φ & Ψ

A conjunction is false

if

one or both

of the conjuncts is false

Φ v Ψ

A disjunction is false

only if

both disjuncts

are false

Φ -> Ψ

An conditional is false

only if

antecedent

(Φ)

true

and

consequent

(Ψ) false

Φ <-> Ψ

A biconditional is false

only if

the

two conditions

have a

different

truth value

See bottom of

Truth Tables HandoutSlide26

Step 6bP v ~ PRight main columnMain (or governing) connective

A disjunction (a “v” statement) is FALSE only when both

disjuncts

are F.

P

P

v

~

P

1

T

F

2

F

T

1

2

3

4

5Slide27

Step 5 & 6P v ~ PRight main columnMain (or governing) connective

A disjunction (a “v” statement) is FALSE only when both

disjuncts

are F.

P

P

v

~

P

1

T

T

F

2

F

T

1

2

3

4

5Slide28

Step 5 & 6P v ~ PRight main columnMain (or governing) connective

A disjunction (a “v” statement) is FALSE only when both

disjuncts

are F.

P

P

v

~

P

1

T

T

F

2

F

T

T

1

2

3

4

5Slide29

Theorems are Necessarily TrueThis WFF is a Tautology.

regardless of whether P is true.

regardless of whether P is false.

P

P

v

~

P

1

T

T

F

2

F

T

T

1

2

3

4

5Slide30

HomeworkReviewWFFsCan you read sentences correctly?

Print:

Truth Tables

handout

"Building TTs: Sentences and Sequents"

"Connectives – when are they false"

Allen/Hand

Section 2.1, esp. pages 40-41

p. 47-8: “tautology,” “inconsistency & contingent sentence”