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# Introductory Logic PHI 120 - PowerPoint PPT Presentation

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

• 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

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

• Have in hand

Truth Tables HandoutSee especially “Building Truth Tables” section

Review – Logical Form

• 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

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

Any statement (WFF) is either True or False

Φ v ~Φ

• This is a theorem of logic

• Theorems are tautologies

• Tautologies are necessarily true

Any statement (WFF) is either True or False

P v ~P

• This is a theorem of logic

• Theorems are tautologies

• Tautologies are necessarily true

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

• 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

~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 truth-value of an atomic sentence

• The truth-value of an atomic sentence

• The truth-value of a simple negation

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

• The truth-value of a simple negation

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

P v ~P

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.

• 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.

• 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

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