Introductory Logic PHI 120 Presentation: "Truth Tables – Sentences"

Homework 1.Review – WFFs Can you read sentences correctly? 2.Print: Truth Tables handoutTruth Tables a."Building TTs: Sentences and Sequents" b."Connectives – when are they false" 3.Allen/Hand 1.Section 2.1, esp. pages p. 47-8: “tautology,” “inconsistency & contingent sentence”

In Class Have in hand Truth Tables Handout Truth Tables Handout See especially “Building Truth Tables” section Truth Tables Handout Truth Tables Handout See especially “Building Truth Tables” section

REVIEW – LOGICAL FORM Sentences (WFFs)

Well-formed Formulas Simple WFFs – P, 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 Structure

THE CONCEPT OF TRUTH VALUE Truth Tables

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? See page 9 Strongest ~ & and/or v -> Weakest

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 Sentences (WFFs)

The Simple The truth-value of an atomic sentence P

The Simple The truth-value of an atomic sentence P 1 T 2 F 1

Simple Negation The truth-value of a simple negation P~P 1 T 2 F 123 A negation (~) takes the opposite value of the statement being negated.

Simple Negation The truth-value of a simple negation P~P 1 TF 2 FT 123 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 1 P v ~ P A Truth Table has two main columns – Left main column: ATOMIC SENTENCES – Right column: the WFF. This row represents a header row. PPv~P

Step 2 P v ~ P Determine the number of rows for the WFF: – Rows = 2 (power of simple statements) PPv~P 1 2

Step 3 P v ~ P Fill in left main column first. PPv~P 1 T 2 F 12345

Step 4 P v ~ P Right main column – assign truth-values for negation of simple statements. PPv~P 1 T 2 F 12345

Step 4 P v ~ P Right main column – assign truth-values for negation of simple statements. PPv~P 1 TF 2 FT Notice that only one connective remains.

Skip to Last Step P v ~ P Assign truth-values for the remaining wedge. CONNECTIVES – when they are false ~ΦA negation is false ifthe statement being negated (Φ) is true Φ & ΨA conjunction is false ifone or both of the conjuncts is false Φ v ΨA disjunction is false only ifboth disjuncts are false Φ -> ΨAn conditional is false only ifantecedent (Φ) true and consequent (Ψ) false Φ ΨA biconditional is false only ifthe two conditions have a different truth value See bottom of Truth Tables HandoutTruth Tables Handout

Step 6b P v ~ P Right main column – Main (or governing) connective A disjunction (a “v” statement) is FALSE only when both disjuncts are F. PP~P 1 TF 2 FT 12345

Step 5 & 6 P v ~ P Right main column – Main (or governing) connective A disjunction (a “v” statement) is FALSE only when both disjuncts are F. PP~P 1 TF 2 FT 12345

Step 5 & 6 P v ~ P Right main column – Main (or governing) connective A disjunction (a “v” statement) is FALSE only when both disjuncts are F. PP~P 1 TF 2 FT 12345

Theorems are Necessarily True This WFF is a Tautology. – regardless of whether P is true. – regardless of whether P is false. PP~P 1 TF 2 FT 12345

Homework 1.Review – WFFs Can you read sentences correctly? 2.Print: Truth Tables handoutTruth Tables a."Building TTs: Sentences and Sequents" b."Connectives – when are they false" 3.Allen/Hand 1.Section 2.1, esp. pages p. 47-8: “tautology,” “inconsistency & contingent sentence”