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Understanding Well-Formed Formulas in Sentential Logic

This presentation offers a comprehensive overview of well-formed formulas (WFFs) in sentential logic. It covers the fundamental compositions of WFFs, including simple WFFs and complex forms such as negation, conjunction, disjunction, conditional, and biconditional. Through various exercises, participants will learn to identify and read sentences formulated in logical syntax. The importance of binding strength and the correct interpretation of complex sentences are emphasized, aiding in the development of solid logical reasoning skills.

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Understanding Well-Formed Formulas in Sentential Logic

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  1. Presentation: “Basic Concepts Review " Introductory LogicPHI 120

  2. Review of WFFs Identifying and Reading Sentences

  3. WFFs Identifying Form

  4. Sentential Logic • Simple WFFs • P, Q, R, S, …. • Complex WFFs • Negation (~Φ) • Conjunction (Φ&Ψ) • Disjunction (ΦvΨ) • Conditional (Φ->Ψ) • Biconditional (Φ<->Ψ) • and nothing else Learn these five forms especially!

  5. Exercise: Seeing Form • ~Φ (negation) • ~P • ~(P & Q)

  6. Exercise: Seeing Form • ~Φ (negation) • ~P • ~(P & Q) • Φ&Ψ (conjunction) • P & Q • ~P & ~Q

  7. Exercise: Seeing Form • ~Φ (negation) • ~P • ~(P & Q) • Φ & Ψ (conjunction) • P & Q • ~P & ~Q • ΦvΨ (disjunction) • P v Q • (P & Q) v R

  8. Exercise: Seeing Form • ~Φ (negation) • ~P • ~(P & Q) • Φ & Ψ (conjunction) • P & Q • ~P & ~Q • Φ v Ψ (disjunction) • P v Q • (P & Q) v R • Φ->Ψ (conditional) • P -> Q • P -> (Q <-> R)

  9. Exercise: Seeing Form • ~Φ (negation) • ~P • ~(P & Q) • Φ & Ψ (conjunction) • P & Q • ~P & ~Q • Φ v Ψ (disjunction) • P v Q • P v (Q & R) • Φ -> Ψ (conditional) • P -> Q • P -> (Q <-> R) • Φ<->Ψ (biconditional) • P <-> Q • (P -> Q) <-> (R <->S)

  10. WFFs Reading Sentences

  11. The Key is Binding Strength Strongest ~ &and/orv -> <-> Weakest

  12. Exercise: Reading Complex Sentences • P & (Q & R) What kind of sentence is this?

  13. Exercise: Reading Complex Sentences • P & (Q & R) • Obviously an & (“ampersand”) kind of WFF • Φ&Ψ This is the form of a conjunction (or ampersand) kind of statement • Φ&Ψ is a binary. • It has a left side (Φ) and a right side (Ψ).

  14. Exercise: Reading Complex Sentences • P & (Q & R) • Obviously an & (“ampersand”) kind of WFF • Φ & Ψ • Question • Look at the sentence as written: • What is the first conjunct (Φ)? • What is the second conjunct (Ψ)?

  15. Exercise: Reading Complex Sentences • P & (Q & R) • Obviously an & (“ampersand”) kind of WFF • Φ & Ψ • Answer • Φ = P • Ψ = Q & R • This second conjunct is, itself, a conjunction (Q & R) • Q is the first conjunct • R is the second conjunct

  16. Exercise: Reading Complex Sentences • P & (Q & R) • Obviously an & (“ampersand”) kind of WFF • Φ & Ψ • Answer • Φ = P • Ψ = Q & R • This second conjunct is, itself, a conjunction • Q is the first conjunct • R is the second conjunct • Why are there parentheses around the 2nd conjunct?

  17. Exercise: Reading Complex Sentences • P & Q -> R What kind of sentence is this?

  18. Exercise: Reading Complex Sentences • P & Q -> R • Could be an & (“ampersand”) or -> (“arrow”) kind of WFF • Φ&Ψ • Φ->Ψ • Question • Look at the sentence as written: • What is the weaker connective: the & or the ->?

  19. Exercise: Reading Complex Sentences • P & Q -> R • Not obviously an & (“ampersand”) or -> (“arrow”) kind of WFF • Φ & Ψ • Φ -> Ψ • Answer • The -> binds more weakly than the & • You can break the sentence most easily here • Φ - “the antecedent”: P & Q • Ψ - “the consequent”: R

  20. Exercise: Reading Complex Sentences () • P & Q -> R • Not obviously an & (“ampersand”) or -> (“arrow”) kind of WFF • Φ & Ψ • Φ -> Ψ • Answer • The -> binds more weakly than the & • You can break the sentence most easily here • Antecedent: P & Q • Consequent: R • Why are there no parentheses around the antecedent?

  21. Exercise: Reading Complex Sentences • R <-> P v (R & Q) What kind of sentence is this?

  22. Exercise: Reading Complex Sentences • R <-> P v (R & Q) • Either • Φ<->Ψ • ΦvΨ • Φ&Ψ • Question • Which is the main connective? Conjunction is embedded within parentheses.

  23. Exercise: Reading Complex Sentences • R <-> P v (R & Q) • Either • Φ <-> Ψ • ΦvΨ • Φ & Ψ • Answer • Φ<->Ψ

  24. Exercise: Reading Complex Sentences • R<->P v (R & Q) • What is first condition? • R • What is the second condition? • P v (R & Q) • Is this WFF a disjunction (v) or a conjunction (&)? • It is a v (a disjunction) • First disjunct: P • Second disjunct: R & Q • Question: can you see why are there parentheses around the second disjunct (R & Q)?

  25. Grammar and Syntax - Non-Sense- Ambiguity- Well-formed formulas

  26. Non-Sense Formula Exercise 1.2.1: v (page 8) A –> (

  27. Ambiguous Formula Exercise 1.2.3: v (page 10) P -> R & S -> T

  28. Well-Formed Formula Exercise 1.2.3: iii (page 10) P v Q -> R <-> S

  29. Well-Formed Formula P v Q -> (R <-> S)

  30. Sentential Logic • Simple WFFs • P, Q, R, S, …. • Complex WFFs • Negation (~Φ) • Conjunction (Φ&Ψ) • Disjunction (ΦvΨ) • Conditional (Φ->Ψ) • Biconditional (Φ<->Ψ) • and nothing else

  31. The end.

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