Q is necessary P is sufficient for Q, if P It is cloudy, if it is cloudy rainy If P, then Q for P Q Q provided that It is cloudy provided Q is necessary P is sufficient for cloudy If P, then Q P that it is rainy. for P Q It is rainy only if it is Q is necessary P is sufficient for P only if Q rainy cloudy If P, then Q cloudy. for P Q When it is it is Q is necessary P is sufficient for When P then Q rainy cloudy If P, then Q cloudy. for P Q All rainy days are rainy cloudy Q is necessary P is sufficient for All Ps are Qs If P, then Q cloudy days. days days for P Q 4. Contrapositive a bachelor is sufficient for being a male, since being a bachelor is enough to be a male. Being a male is necessary for being a bachelor, since it is required that to be a bachelor one is male. Contrapositive The contrapositive of P then Q = not Q then not P’. The two sentences are logically equivalent. This us that ‘If ‘If is sufficient for not-P, and not-P is necessary for not-Q. Example 3.4: Contrapositive The contrapositive of it is rainy, then it is cloudy” is it is not cloudy, then it is not “If “If rainy”. Notice that in constructing a contrapositive you must perform two operations: (1) switch the consequent and the antecedent, and (2) put negations in of the antecedent and the consequent. Students often forget to do one or the other. Chapter 4: Validity Material covered in this chapter 4.1 Valid and invalid arguments
Q, if P It is cloudy, if it is cloudy rainy Q is necessary for P P is sufficient for Q If P, then Q Q provided that P It is cloudy provided that it is rainy. cloudy Q is necessary for P P is sufficient for Q If P, then Q P only if Q It is rainy only if it is cloudy. rainy cloudy Q is necessary for P P is sufficient for Q If P, then Q When P then Q When it is it is cloudy. rainy cloudy Q is necessary for P P is sufficient for Q If P, then Q All Ps are Qs All rainy days are cloudy days. rainy days cloudy days Q is necessary for P P is sufficient for Q If P, then Q 4. Contrapositive a bachelor is sufficient for being a male, since being a bachelor is enough to be a male. Being a male is necessary for being a bachelor, since it is required that to be a bachelor one is male. Contrapositive The contrapositive of ‘If P then Q = ‘If not Q then not P’. The two sentences are logically equivalent. This us that is sufficient for not-P, and not-P is necessary for not-Q. Example 3.4: Contrapositive The contrapositive of “If it is rainy, then it is cloudy” is “If it is not cloudy, then it is not rainy”. Notice that in constructing a contrapositive you must perform two operations: (1) switch the consequent and the antecedent, and (2) put negations in of the antecedent and the consequent. Students often forget to do one or the other. Chapter 4: Validity Material covered in this chapter 4.1 Valid and invalid arguments