Introductory Logic PHI 120

1. Presentation: "Truth Tables – Sentences" Introductory LogicPHI 120

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

3. In Class Have in hand Truth Tables HandoutSee especially “Building Truth Tables” section

4. Sentences (WFFs) Review – Logical Form

5. 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

6. Truth Tables The Concept of Truth Value

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. Sentences (WFFs) Building Truth Tables

14. The Simple • The truth-value of an atomic sentence

15. The Simple • The truth-value of an atomic sentence

16. Simple Negation • The truth-value of a simple negation A negation (~) takes the opposite value of the statement being negated.

17. Simple Negation • The truth-value of a simple negation A negation (~) takes the opposite value of the statement being negated.

18. Building a Truth Table • Read the sentence P v ~P

19. 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

20. 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.

21. Step 2P v ~ P • Determine the number of rows for the WFF: • Rows = 2 (power of simple statements)

22. Step 3P v ~ P • Fill in left main column first.

23. Step 4P v ~ P • Right main column • assign truth-values for negation of simple statements.

24. Step 4P v ~ P • Right main column • assign truth-values for negation of simple statements. Notice that only one connective remains.

25. Skip to Last StepP v ~ P • Assign truth-values for the remaining wedge. See bottom of Truth Tables Handout

26. Step 6bP v ~ P • Right main column • Main (or governing) connective A disjunction (a “v” statement) is FALSE only when both disjuncts are F.

27. 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.

28. 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.

29. Theorems are Necessarily True • This WFF is a Tautology. • regardless of whether P is true. • regardless of whether P is false.

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