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Introductory Logic PHI 120PowerPoint Presentation

Introductory Logic PHI 120

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Presentation: "Truth Tables – Sentences". Introductory Logic PHI 120. Homework. Review WFFs Can 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

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### Introductory LogicPHI 120

Presentation: "Truth Tables – Sentences"

Homework Allen/Hand

- Review
- WFFs
- Can you read sentences correctly?

- WFFs
- Print: Truth Tables handout
- "Building TTs: Sentences and Sequents"
- "Connectives – when are they false"

- Section 2.1, esp. pages 40-41
- p. 47-8: “tautology,” “inconsistency & contingent sentence”

Review – Logical Form

Well-formed Formulas

- Simple WFFs
- P, Q, R, S, ….

- Complex WFFs
- Negation~Φ
- ConjunctionΦ&Ψ
- DisjunctionΦvΨ
- ConditionalΦ->Ψ
- BiconditionalΦ<->Ψ
- and nothing else

Unary Structure

Binary Structure

The Concept of Truth Value

Theorem of the Logic

Any statement (WFF) is either True or False

T v ~T

- This is a theorem of logic
- Theorems are tautologies
- Tautologies are necessarily true

“A statement is true.” = T

Theorem of the Logic

Any statement (WFF) is either True or False

Φ v ~Φ

- This is a theorem of logic
- Theorems are tautologies
- Tautologies are necessarily true

Theorem of the Logic

Any statement (WFF) is either True or False

P v ~P

- This is a theorem of logic
- Theorems are tautologies
- Tautologies are necessarily true

Theorem of the Logic

Any statement (WFF) is either True or False

(P&~Q) v ~(P&~Q)

- This is a theorem of logic
- Theorems are tautologies
- Tautologies are necessarily true

The Key to Recognizing Sentences

- Which connective is the weakest link in a sequence of symbols?
(or as I like to ask)

- Where can you most easily bend the sentence?

Strongest

~

&and/orv

->

<->

Weakest

See

page

9

What kind of sentence?

~P

~P & ~Q

P v Q -> R

P v Q <-> R -> P

negation: ~Φ

conjunction: Φ&Ψ

conditional: Φ->Ψ

biconditional: Φ<->Ψ

“the main connective”

Metaphor of the Binding of a Book

Building Truth Tables

The Simple

- The truth-value of an atomic sentence

The Simple

- The truth-value of an atomic sentence

Simple Negation

- The truth-value of a simple negation

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

Simple Negation

- The truth-value of a simple negation

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

Building a Truth Table

- Read the sentence
P v ~P

Building a Truth Table

- Read the sentence
P v ~P

The wedge is the main connective.

Hence this is a disjunction.

Φ v ~Φ

P v ~P is an instance of our theorem

Step 1P v ~ P

- A Truth Table has two main columns
- Left main column: ATOMIC SENTENCES
- Right column: the WFF.
- This row represents a header row.

Step 2P v ~ P

- Determine the number of rows for the WFF:
- Rows = 2 (power of simple statements)

Step 3P v ~ P

- Fill in left main column first.

Step 4P v ~ P

- Right main column
- assign truth-values for negation of simple statements.

Step 4P v ~ P

- Right main column
- assign truth-values for negation of simple statements.

Notice that only one connective remains.

Skip to Last StepP v ~ P

- Assign truth-values for the remaining wedge.

See bottom of Truth Tables Handout

Step 6bP v ~ P

- Right main column
- Main (or governing) connective

A disjunction (a “v” statement) is FALSE only when both disjuncts are F.

Step 5 & 6P v ~ P

- Right main column
- Main (or governing) connective

A disjunction (a “v” statement) is FALSE only when both disjuncts are F.

Step 5 & 6P v ~ P

- Right main column
- Main (or governing) connective

A disjunction (a “v” statement) is FALSE only when both disjuncts are F.

Theorems are Necessarily True

- This WFF is a Tautology.
- regardless of whether P is true.
- regardless of whether P is false.

Homework Allen/Hand

- Review
- WFFs
- Can you read sentences correctly?

- WFFs
- Print: Truth Tables handout
- "Building TTs: Sentences and Sequents"
- "Connectives – when are they false"

- Section 2.1, esp. pages 40-41
- p. 47-8: “tautology,” “inconsistency & contingent sentence”

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