Argument Forms and Formalizing Arguments: Validity and Necessary Conditions, Study notes of Reasoning

Download Argument Forms and Formalizing Arguments: Validity and Necessary Conditions and more Study notes Reasoning in PDF only on Docsity! Review Argument forms Formalizing Arguments Math 10130 Lecture 11 21 September 2007 10130 Lecture 11 Review Argument forms Formalizing Arguments Last time Checking validity of argument forms by truth tables. Modus ponens: for any sentential formulas A and B, the argument form A, A⇒ B ∴ B is valid. Ex falso: if A is an inconsistent formula and B is any sentential formula, then the argument form A ∴ B is valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Last time Checking validity of argument forms by truth tables. Modus ponens: for any sentential formulas A and B, the argument form A, A⇒ B ∴ B is valid. Ex falso: if A is an inconsistent formula and B is any sentential formula, then the argument form A ∴ B is valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Valid or Invalid? P ∨ Q ¬Q ∨ ¬R P ⇒ Q ∴ ¬R P Q R P ∨ Q ¬Q ∨ ¬R P ⇒ Q ¬R T T T T F T F T T F T X T F T T T F F T F F T T F T F T T T F T F F T F T X F F T F T T F F F F F T T T Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Valid or Invalid? P ∨ Q ¬Q ∨ ¬R P ⇒ Q ∴ ¬R P Q R P ∨ Q ¬Q ∨ ¬R P ⇒ Q ¬R T T T T F T F T T F T X T F T T T F F T F F T T F T F T T T F T F F T F T X F F T F T T F F F F F T T T Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Valid or Invalid? P ∨ Q ¬Q ∨ ¬R P ⇒ Q ∴ ¬R P Q R P ∨ Q ¬Q ∨ ¬R P ⇒ Q ¬R T T T T F T F T T F T X T F T T T F F T F F T T F T F T T T F T F F T F T X F F T F T T F F F F F T T T Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Valid or Invalid? P ∨ Q ¬Q ∨ ¬R P ⇒ Q ∴ ¬R P Q R P ∨ Q ¬Q ∨ ¬R P ⇒ Q ¬R T T T T F T F T T F T X T F T T T F F T F F T T F T F T T T F T F F T F T X F F T F T T F F F F F T T T Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Valid or Invalid? P ∨ Q ¬Q ∨ ¬R P ⇒ Q ∴ ¬R P Q R P ∨ Q ¬Q ∨ ¬R P ⇒ Q ¬R T T T T F T F T T F T T X T F T T T F F T F F T T F T F T T T F T F F T F T T X F F T F T T F F F F F T T T Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Valid or Invalid? P ∨ Q ¬Q ∨ ¬R P ⇒ Q ∴ ¬R P Q R P ∨ Q ¬Q ∨ ¬R P ⇒ Q ¬R T T T T F T F T T F T T T T X T F T T T F F T F F T T F T F T T T F T F F T F T T T T X F F T F T T F F F F F T T T Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Valid or Invalid? P ∨ Q ¬Q ∨ ¬R P ⇒ Q ∴ ¬R P Q R P ∨ Q ¬Q ∨ ¬R P ⇒ Q ¬R T T T T F T F T T F T T T T X T F T T T F F T F F T T F T F T T T F T F F T F T T T T X F F T F T T F F F F F T T T Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Valid or Invalid? P ∨ Q ¬Q ∨ ¬R P ⇒ Q ∴ ¬R P Q R P ∨ Q ¬Q ∨ ¬R P ⇒ Q ¬R T T T T F T F T T F T T T T X T F T T T F F T F F T T F T F T T T F T F F T F T T T T X F F T F T T F F F F F T T T Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalization Natural language argument: If you finish your dinner, then you can have some dessert. You did finish your dinner. Therefore, you can have some dessert. Assign sentence letters: P “You finish your dinner” Q “You can have some dessert” Argument form: P ⇒ Q P ∴ Q Valid argument form. (Modus ponens) 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalization Natural language argument: If you finish your dinner, then you can have some dessert. You did finish your dinner. Therefore, you can have some dessert. Assign sentence letters: P “You finish your dinner” Q “You can have some dessert” Argument form: P ⇒ Q P ∴ Q Valid argument form. (Modus ponens) 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalization Natural language argument: If you finish your dinner, then you can have some dessert. You did finish your dinner. Therefore, you can have some dessert. Assign sentence letters: P “You finish your dinner” Q “You can have some dessert” Argument form: P ⇒ Q P ∴ Q Valid argument form. (Modus ponens) 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalization Natural language argument: If you finish your dinner, then you can have some dessert. You did finish your dinner. Therefore, you can have some dessert. Assign sentence letters: P “You finish your dinner” Q “You can have some dessert” Argument form: P ⇒ Q P ∴ Q Valid argument form. (Modus ponens) 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalization Natural language argument: If you finish your dinner, then you can have some dessert. You did finish your dinner. Therefore, you can have some dessert. Assign sentence letters: P “You finish your dinner” Q “You can have some dessert” Argument form: P ⇒ Q P ∴ Q Valid argument form. (Modus ponens) 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalization Natural language argument: If you finish your dinner, then you can have some dessert. You did finish your dinner. Therefore, you can have some dessert. Assign sentence letters: P “You finish your dinner” Q “You can have some dessert” Argument form: P ⇒ Q P ∴ Q Valid argument form. (Modus ponens) 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalizing conditionals “You can have dessert only if you finish your dinner.” How to formalize? Hint: when would the statement be a lie? P “You can have dessert.” Q “You finish your dinner.” Original statement is false exactly when P is true and Q is false. Formalization: P ⇒ Q “only if” is translated in place as ⇒ 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalizing conditionals “You can have dessert only if you finish your dinner.” How to formalize? Hint: when would the statement be a lie? P “You can have dessert.” Q “You finish your dinner.” Original statement is false exactly when P is true and Q is false. Formalization: P ⇒ Q “only if” is translated in place as ⇒ 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalizing conditionals “You can have dessert only if you finish your dinner.” How to formalize? Hint: when would the statement be a lie? P “You can have dessert.” Q “You finish your dinner.” Original statement is false exactly when P is true and Q is false. Formalization: P ⇒ Q “only if” is translated in place as ⇒ 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalizing conditionals “You can have dessert only if you finish your dinner.” How to formalize? Hint: when would the statement be a lie? P “You can have dessert.” Q “You finish your dinner.” Original statement is false exactly when P is true and Q is false. Formalization: P ⇒ Q “only if” is translated in place as ⇒ 10130 Lecture 11 Review Argument forms Formalizing Arguments Formalizing conditionals “You can have dessert only if you finish your dinner.” How to formalize? Hint: when would the statement be a lie? P “You can have dessert.” Q “You finish your dinner.” Original statement is false exactly when P is true and Q is false. Formalization: P ⇒ Q “only if” is translated in place as ⇒ 10130 Lecture 11 Review Argument forms Formalizing Arguments Necessary and sufficient conditions Assume the formula P ⇒ Q is true. Two points of view: 1 When P is true, Q must be true... so P is a sufficient condition for Q. 2 When Q is false, P cannot be true... so Q is a necessary condition for P. Remember: [Sufficient Cond.]⇒[Necessary Cond.] 10130 Lecture 11 Review Argument forms Formalizing Arguments Necessary and sufficient conditions Assume the formula P ⇒ Q is true. Two points of view: 1 When P is true, Q must be true... so P is a sufficient condition for Q. 2 When Q is false, P cannot be true... so Q is a necessary condition for P. Remember: [Sufficient Cond.]⇒[Necessary Cond.] 10130 Lecture 11 Review Argument forms Formalizing Arguments Necessary and sufficient conditions Assume the formula P ⇒ Q is true. Two points of view: 1 When P is true, Q must be true... so P is a sufficient condition for Q. 2 When Q is false, P cannot be true... so Q is a necessary condition for P. Remember: [Sufficient Cond.]⇒[Necessary Cond.] 10130 Lecture 11 Review Argument forms Formalizing Arguments Necessary and sufficient conditions Assume the formula P ⇒ Q is true. Two points of view: 1 When P is true, Q must be true... so P is a sufficient condition for Q. 2 When Q is false, P cannot be true... so Q is a necessary condition for P. Remember: [Sufficient Cond.]⇒[Necessary Cond.] 10130 Lecture 11 Review Argument forms Formalizing Arguments Examples of necessary and sufficient conditions P: Ed goes online. Q: Ed checks his e-mail. Going online is necessary for Ed to check his e-mail. Q ⇒ P For Ed to check his e-mail, it suffices that he goes online. P ⇒ Q A necessary condition for Ed to go online is that he checks his e-mail. P ⇒ Q Going online is both necessary and sufficient for Ed to check his e-mail. (P ⇒ Q) &(Q ⇒ P) Equivalently: P ⇔ Q Ed checks his e-mail only if he goes online. Q ⇒ P 10130 Lecture 11 Review Argument forms Formalizing Arguments Examples of necessary and sufficient conditions P: Ed goes online. Q: Ed checks his e-mail. Going online is necessary for Ed to check his e-mail. Q ⇒ P For Ed to check his e-mail, it suffices that he goes online. P ⇒ Q A necessary condition for Ed to go online is that he checks his e-mail. P ⇒ Q Going online is both necessary and sufficient for Ed to check his e-mail. (P ⇒ Q) &(Q ⇒ P) Equivalently: P ⇔ Q Ed checks his e-mail only if he goes online. Q ⇒ P 10130 Lecture 11 Review Argument forms Formalizing Arguments Examples of necessary and sufficient conditions P: Ed goes online. Q: Ed checks his e-mail. Going online is necessary for Ed to check his e-mail. Q ⇒ P For Ed to check his e-mail, it suffices that he goes online. P ⇒ Q A necessary condition for Ed to go online is that he checks his e-mail. P ⇒ Q Going online is both necessary and sufficient for Ed to check his e-mail. (P ⇒ Q) &(Q ⇒ P) Equivalently: P ⇔ Q Ed checks his e-mail only if he goes online. Q ⇒ P 10130 Lecture 11 Review Argument forms Formalizing Arguments Examples of necessary and sufficient conditions P: Ed goes online. Q: Ed checks his e-mail. Going online is necessary for Ed to check his e-mail. Q ⇒ P For Ed to check his e-mail, it suffices that he goes online. P ⇒ Q A necessary condition for Ed to go online is that he checks his e-mail. P ⇒ Q Going online is both necessary and sufficient for Ed to check his e-mail. (P ⇒ Q) &(Q ⇒ P) Equivalently: P ⇔ Q Ed checks his e-mail only if he goes online. Q ⇒ P 10130 Lecture 11 Review Argument forms Formalizing Arguments Examples of necessary and sufficient conditions P: Ed goes online. Q: Ed checks his e-mail. Going online is necessary for Ed to check his e-mail. Q ⇒ P For Ed to check his e-mail, it suffices that he goes online. P ⇒ Q A necessary condition for Ed to go online is that he checks his e-mail. P ⇒ Q Going online is both necessary and sufficient for Ed to check his e-mail. (P ⇒ Q) &(Q ⇒ P) Equivalently: P ⇔ Q Ed checks his e-mail only if he goes online. Q ⇒ P 10130 Lecture 11 Review Argument forms Formalizing Arguments Examples of necessary and sufficient conditions P: Ed goes online. Q: Ed checks his e-mail. Going online is necessary for Ed to check his e-mail. Q ⇒ P For Ed to check his e-mail, it suffices that he goes online. P ⇒ Q A necessary condition for Ed to go online is that he checks his e-mail. P ⇒ Q Going online is both necessary and sufficient for Ed to check his e-mail. (P ⇒ Q) &(Q ⇒ P) Equivalently: P ⇔ Q Ed checks his e-mail only if he goes online. Q ⇒ P 10130 Lecture 11 Review Argument forms Formalizing Arguments An argument to formalize Eve can enter the library only if she has her ID. But, she doesn’t have her ID. So, Eve can’t enter the library. Assign sentence letters: P: “Eve can enter the library.” Q: “Eve has her ID.” Argument form: P ⇒ Q (Q is necessary for P) ¬Q ∴ ¬P 10130 Lecture 11 Review Argument forms Formalizing Arguments An argument to formalize Eve can enter the library only if she has her ID. But, she doesn’t have her ID. So, Eve can’t enter the library. Assign sentence letters: P: “Eve can enter the library.” Q: “Eve has her ID.” Argument form: P ⇒ Q (Q is necessary for P) ¬Q ∴ ¬P 10130 Lecture 11 Review Argument forms Formalizing Arguments An argument to formalize Eve can enter the library only if she has her ID. But, she doesn’t have her ID. So, Eve can’t enter the library. Assign sentence letters: P: “Eve can enter the library.” Q: “Eve has her ID.” Argument form: P ⇒ Q (Q is necessary for P) ¬Q ∴ ¬P 10130 Lecture 11 Review Argument forms Formalizing Arguments An argument to formalize Eve can enter the library only if she has her ID. But, she doesn’t have her ID. So, Eve can’t enter the library. Assign sentence letters: P: “Eve can enter the library.” Q: “Eve has her ID.” Argument form: P ⇒ Q (Q is necessary for P) ¬Q ∴ ¬P 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P ⇒ Q ¬Q ∴ ¬P P Q P ⇒ Q ¬Q ¬P T T T F F T F F T F F T T F T F F T X Valid. Called modus tollens (=“method of denying”). (Also called “denying the consequent”.) 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P ⇒ Q ¬Q ∴ ¬P P Q P ⇒ Q ¬Q ¬P T T T F F T F F T F F T T F T F F T X Valid. Called modus tollens (=“method of denying”). (Also called “denying the consequent”.) 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P ⇒ Q ¬Q ∴ ¬P P Q P ⇒ Q ¬Q ¬P T T T F F T F F T F F T T F T F F T T T X Valid. Called modus tollens (=“method of denying”). (Also called “denying the consequent”.) 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P ⇒ Q ¬Q ∴ ¬P P Q P ⇒ Q ¬Q ¬P T T T F F T F F T F F T T F T F F T T T X Valid. Called modus tollens (=“method of denying”). (Also called “denying the consequent”.) 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P ⇒ Q ¬Q ∴ ¬P P Q P ⇒ Q ¬Q ¬P T T T F F T F F T F F T T F T F F T T T X Valid. Called modus tollens (=“method of denying”). (Also called “denying the consequent”.) 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P ⇒ Q ¬Q ∴ ¬P P Q P ⇒ Q ¬Q ¬P T T T F F T F F T F F T T F T F F T T T X Valid. Called modus tollens (=“method of denying”). (Also called “denying the consequent”.) 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P ⇒ Q ¬Q ∴ ¬P P Q P ⇒ Q ¬Q ¬P T T T F F T F F T F F T T F T F F T T T X Valid. Called modus tollens (=“method of denying”). (Also called “denying the consequent”.) 10130 Lecture 11 Review Argument forms Formalizing Arguments Another argument to formalize You don’t graduate unless you learn logic. Moreover, you learn logic only if you read this book. If you don’t graduate, you don’t get into law school. Therefore, if you get into law school, you learn logic. (p. 141 #3) Assign sentence letters: P: “You graduate.” Q: “You learn logic.” R: “You read this book.” S : “You get into law school.” Argument form: P ⇒ Q (Q is necessary for P) Q ⇒ R (R is necessary for Q) ¬P ⇒ ¬S ∴ S ⇒ Q 10130 Lecture 11 Review Argument forms Formalizing Arguments Another argument to formalize You don’t graduate unless you learn logic. Moreover, you learn logic only if you read this book. If you don’t graduate, you don’t get into law school. Therefore, if you get into law school, you learn logic. (p. 141 #3) Assign sentence letters: P: “You graduate.” Q: “You learn logic.” R: “You read this book.” S : “You get into law school.” Argument form: P ⇒ Q (Q is necessary for P) Q ⇒ R (R is necessary for Q) ¬P ⇒ ¬S ∴ S ⇒ Q 10130 Lecture 11 Review Argument forms Formalizing Arguments Another argument to formalize You don’t graduate unless you learn logic. Moreover, you learn logic only if you read this book. If you don’t graduate, you don’t get into law school. Therefore, if you get into law school, you learn logic. (p. 141 #3) Assign sentence letters: P: “You graduate.” Q: “You learn logic.” R: “You read this book.” S : “You get into law school.” Argument form: P ⇒ Q (Q is necessary for P) Q ⇒ R (R is necessary for Q) ¬P ⇒ ¬S ∴ S ⇒ Q 10130 Lecture 11 Review Argument forms Formalizing Arguments Another argument to formalize You don’t graduate unless you learn logic. Moreover, you learn logic only if you read this book. If you don’t graduate, you don’t get into law school. Therefore, if you get into law school, you learn logic. (p. 141 #3) Assign sentence letters: P: “You graduate.” Q: “You learn logic.” R: “You read this book.” S : “You get into law school.” Argument form: P ⇒ Q (Q is necessary for P) Q ⇒ R (R is necessary for Q) ¬P ⇒ ¬S ∴ S ⇒ Q 10130 Lecture 11 Review Argument forms Formalizing Arguments Another argument to formalize You don’t graduate unless you learn logic. Moreover, you learn logic only if you read this book. If you don’t graduate, you don’t get into law school. Therefore, if you get into law school, you learn logic. (p. 141 #3) Assign sentence letters: P: “You graduate.” Q: “You learn logic.” R: “You read this book.” S : “You get into law school.” Argument form: P ⇒ Q (Q is necessary for P) Q ⇒ R (R is necessary for Q) ¬P ⇒ ¬S ∴ S ⇒ Q 10130 Lecture 11 Review Argument forms Formalizing Arguments Another argument to formalize You don’t graduate unless you learn logic. Moreover, you learn logic only if you read this book. If you don’t graduate, you don’t get into law school. Therefore, if you get into law school, you learn logic. (p. 141 #3) Assign sentence letters: P: “You graduate.” Q: “You learn logic.” R: “You read this book.” S : “You get into law school.” Argument form: P ⇒ Q (Q is necessary for P) Q ⇒ R (R is necessary for Q) ¬P ⇒ ¬S ∴ S ⇒ Q 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P Q R S P ⇒ Q Q ⇒ R ¬P ⇒ ¬S S ⇒ Q T T T T T X T T T F T X T T F T T F T T T T F F T F T T T F T T F T T F T F T F F T T T T F F T F T T F T F F F F T T T F T T T T T F T F T T F T X F T F T T F F T F T F F T F T T F F T T T T F F F F T F T X F F F T T T F F F F F F T X Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P Q R S P ⇒ Q Q ⇒ R ¬P ⇒ ¬S S ⇒ Q T T T T T X T T T F T X T T F T T F T T T T F F T F T T T F T T F T T F T F T F F T T T T F F T F T T F T F F F F T T T F T T T T T F T F T T F T X F T F T T F F T F T F F T F T T F F T T T T F F F F T F T X F F F T T T F F F F F F T X Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P Q R S P ⇒ Q Q ⇒ R ¬P ⇒ ¬S S ⇒ Q T T T T T X T T T F T X T T F T T F T T T T F F T F T T T F T T F T T F T F T F F T T T T F F T F T T F T F F F F T T T F T T T T T F T F T T F T X F T F T T F F T F T F F T F T T F F T T T T F F F F T F T X F F F T T T F F F F F F T X Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P Q R S P ⇒ Q Q ⇒ R ¬P ⇒ ¬S S ⇒ Q T T T T T X T T T F T X T T F T T F T T T T F F T F T T T F T T F T T F T F T F F T T T T F F T F T T F T F F F F T T T F T T T T T F T F T T F T X F T F T T F F T F T F F T F T T F F T T T T F F F F T F T X F F F T T T F F F F F F T X Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P Q R S P ⇒ Q Q ⇒ R ¬P ⇒ ¬S S ⇒ Q T T T T T T T T X T T T F T T T T X T T F T T F T T T T F F T F T T T F T T F T T F T F T F F T T T T F F T F T T F T F F F F T T T F T T T T T F T F T T F T T T T X F T F T T F F T F T F F T F T T F F T T T T F F F F T F T T T T X F F F T T T F F F F F F T T T T X Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P Q R S P ⇒ Q Q ⇒ R ¬P ⇒ ¬S S ⇒ Q T T T T T T T T X T T T F T T T T X T T F T T F T T T T F F T F T T T F T T F T T F T F T F F T T T T F F T F T T F T F F F F T T T F T T T T T F T F T T F T T T T X F T F T T F F T F T F F T F T T F F T T T T F F F F T F T T T T X F F F T T T F F F F F F T T T T X Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Checking validity P Q R S P ⇒ Q Q ⇒ R ¬P ⇒ ¬S S ⇒ Q T T T T T T T T X T T T F T T T T X T T F T T F T T T T F F T F T T T F T T F T T F T F T F F T T T T F F T F T T F T F F F F T T T F T T T T T F T F T T F T T T T X F T F T T F F T F T F F T F T T F F T T T T F F F F T F T T T T X F F F T T T F F F F F F T T T T X Valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Preview Notice: 16 lines in the truth table, but we only really care about 5! We need a short cut! Idea: Don’t draw the whole truth table. Instead, think about lines in the truth table where all premises are true and conclusion is false. If we find such a line, argument form is invalid. If we show that there is no such line, the argument form is valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Preview Notice: 16 lines in the truth table, but we only really care about 5! We need a short cut! Idea: Don’t draw the whole truth table. Instead, think about lines in the truth table where all premises are true and conclusion is false. If we find such a line, argument form is invalid. If we show that there is no such line, the argument form is valid. 10130 Lecture 11 Review Argument forms Formalizing Arguments Preview Notice: 16 lines in the truth table, but we only really care about 5! We need a short cut! Idea: Don’t draw the whole truth table. Instead, think about lines in the truth table where all premises are true and conclusion is false. If we find such a line, argument form is invalid. If we show that there is no such line, the argument form is valid. 10130 Lecture 11