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Chapter Four

Chapter Four. Proofs. 1. Argument Forms. An argument form is a group of sentence forms such that all of its substitution instances are arguments. Argument Forms, continued. If an argument form has no substitution instances that are invalid, it is said to be a valid argument form .

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Chapter Four

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  1. Chapter Four Proofs

  2. 1. Argument Forms An argument form is a group of sentence forms such that all of its substitution instances are arguments.

  3. Argument Forms, continued • If an argument form has no substitution instances that are invalid, it is said to be a valid argument form. • An argument form that has even one invalid argument as a substitution instance is called an invalid argument form.

  4. 2. The Method of Proof: Modus Ponens and Modus Tollens • Truth tables give us a decision procedure for any sentential argument. • There is another method available to demonstrate validity of sentential arguments: the method of proof, or natural deduction.

  5. The Method of Proof, continued A proof of an argument is a series of steps that starts with premises; each step beyond the premises is derived from a valid argument form by being a substitution instance of it; the last step is the conclusion.

  6. Method of Proof, continued Modus Ponens (MP): p ⊃ q p Therefore, q.

  7. Method of Proof, continued Modus Tollens (MT): p⊃q ˜q Therefore, ˜p

  8. Method of Proof, continued Do not confuse either MP or MT with the invalid arguments that resemble them: Affirming the Consequent: p ⊃ q q Therefore, p

  9. Method of Proof, continued Denying the Antecedent: p ⊃ q ˜p Therefore, ˜q.

  10. 3. Disjunctive Syllogism (DS) and Hypothetical Syllogism (HS) Another valid argument form is the Disjunctive Syllogism (DS). This has two forms: p∨q ˜p Therefore, q And p∨q ˜q Therefore, q

  11. DS and HS, continued Another valid argument form is the Hypothetical Syllogism (HS): p⊃q q⊃r Therefore, p⊃r

  12. 4. Simplification and Conjunction Another valid argument form is Simplification (Simp.), which has two forms: p.q Therefore, p And p.q Therefore, q

  13. Simplification and conjunction, continued Conjunction (Conj.) is another valid argument form: p q Therefore, p.q

  14. 5. Addition and Constructive Dilemma Another valid argument form is Addition (Add.): p Therefore, p∨q

  15. Addition and constructive dilemma, continued Another valid argument form is Constructive Dilemma (CD): p ∨ q p ⊃ r q ⊃ r Therefore, r ∨ s

  16. 6. Principles of Strategy • Look for forms that correspond to valid rules of inference • Remember: small sentences are your friends! • Once you have mastered the rules of inference, you will find completing many proofs much easier by working backwards from the conclusion. • Trace the connections between the letters in the argument, starting with those in the conclusion. • Begin the proof with the letter most distant from those in the conclusion.

  17. 7. Double Negation (DN) and DeMorgan’s Theorem (DeM) Double Negation (DN) is an equivalence argument form: p Therefore, ˜˜p And ˜˜p Therefore, p

  18. DN and DeM, continued DeMorgan’s Theorem (DeM): ˜(p . q) is equivalent to ˜p ∨ ˜q And ˜(p ∨ q) is equivalent to ˜p . ˜q

  19. 8. Commutation (Comm.), Association (Assoc.), and Distribution (Dist.) There are three more valid equivalence argument forms: Commutation (Comm.): p ∨ q is equivalent to q ∨ p p . q is equivalent to q . p

  20. Comm., Assoc., and Dist., continued Association (Assoc.): p ∨ (q ∨ r) is equivalent to (p ∨ q) ∨ r p . (q . r) is equivalent to (p . q) . R

  21. Comm., Assoc., and Dist., continued Distribution (Dist): p . (q ∨ r) is equivalent to (p . q) ∨ (p . r) p ∨ (q . r) is equivalent to (p∨q) . (p∨r)

  22. 9. Contraposition, Implication, and Exportation Contraposition (Contra.): p ⊃ q is equivalent to ˜q ⊃ ˜p Implication (Impl.): p ⊃ q is equivalent to ˜p ∨ q

  23. Contraposition, Implication, and Exportation, continued Exportation (Exp.): (p . q) ⊃ r is equivalent to p ⊃ (q ⊃ r)

  24. 10. Tautology and Equivalence Another valid equivalence form is Tautology (Taut.): p is equivalent to p . p p is equivalent to p ∨ p

  25. Tautology and Equivalence, continued There are two valid argument forms called Equivalence (Equiv.): p ≡ q is equivalent to (p ⊃ q) . (q ⊃ p ) p ≡ q is equivalent to (p . q) ∨ (˜p . ˜q)

  26. 11. More Principles of Strategy • Break down complex sentences with DeM, Simp, and Equiv. • Use DeM and Dist to isolate “excess baggage” • Use Impl when you have a mix of conditionals and disjunctions • Work backward from the conclusion

  27. 12. Common Errors in Problem Solving • Using implicational forms on parts of lines • Reluctance to use Addition • Reluctance to use Distribution • Trying to Prove What Cannot be Proved • Failure to Notice the Scope of a Negation Sign

  28. Key Terms • Argument form • Completeness • Decision procedure • Equivalence argument form • Expressive completeness • Implicational argument form • Invalid argument form • Valid argument form

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