Math 102 Section 08, Fall 2010 Solutions Practice for Formal Proofs of Arguments Basic valid arguments: 1) Law of Detachment, p, q 2) Law of Contraposition, q, p 3) Law of Syllogism, q r, p r 4) Disjunctive Syllogism p q, p, q p q, q, p 5) Simplification p q, p p q, q 6) Addition p, p q q, p q Logically equivalent statements 1) Double negation p = ( p) 2) Contraposition = q p 3) De Morgan s Law (p q) = ( p) ( q) (p q) = ( p) ( q) 4) Conditional to Disjunction = ( p) q 5) Commutative p q = q p p q = q p 1
How to write a formal proof for an argument. Write a 2-column table: In the left column we write a sequence of true statements. Sequence means that we label the statements with numbers for future reference (see Exercise 1). On the right column we write the. Possibilities for s are: premise, one of the above basic valid arguments, or one of the above logically equivalent statements. Premises are given to be true, so we write them first with : premise. If some of the previous statements in the left column are the premises of a Basic valid argument, then we write its conclusion as the next true statement. The corresponding is the name of the basic valid argument with the numbers of the corresponding premises we have used. We can write a statement that is logically equivalent to one of the previous statements in the left column. The reason is the name of the quality with the number of the statement we have used. When we have written the conclusion of the given argument we have proved that the argument is valid and we are done. 2
1. Exercise Following is a possible list of statements in a formal proof of the argument form: (p q) p q Provide reasons for each statement: 1) (p q) premise 2) p premise 3) ( p) ( q) De Morgan on statement 1 4) q Disjunctive Syllogism on statements 2 and 3 2. Exercise Construct a formal proof for the following: r ( q) r p Sol. 1) premise 2) r ( q) premise 3) r premise 4) q Law of Detachment on 3 and 2 5) q p Contrapositive of 1 6) p Detachment on 4 and 5 3
3. Exercise Construct a formal proof for the following: d z z w r ( w) d r Sol. 1) d z premise 2) z w premise 3) r ( w) premise 4) d premise 5) d w Syllogism on 1 and 2 6) w Detachment on 4 and 5 7) w r Contrapositive of 3 8) r Detachment on 6 and 7 4. Exercise Construct a formal proof for the following: q p s s 4
Sol. 1) premise 2) q premise 3) p s premise 4) q p Contrapositive of 1 5) p Detachment on 2 and 4 6) s Disjunctive Syllogism on 3 and 5 5. Exercise Construct a formal proof for the following: r ( q) p ( r) Sol. 1) premise 2) r ( q) premise 3) q r Contrapositive of 2 4) p r Syllogism on 1 and 3 6. Exercise Construct a formal proof for the following: s p r () s r 5
Sol. 1) s premise 2) p r premise 3) () s premise 4) s () Contrapositive of 3 5) () Detachment on 1 and 4 6) ( p q) Conditional to Disjunction 7) ( p) q De Morgan on 6 8) p q Double Negation on 7 9) p Simplification on 8 10) r Detachment on 2 and 9 7. Exercise Construct a formal proof for the following: p (r s) ( r) ( p) p s 6
Sol. 1) p (r s) premise 2) ( r) ( p) premise 3) p premise 4) (r s) Detachment on 1 and 3 5) p r Contrapositive of 2 6) r Detachment on 3 and 5 7) s Detachment on 4 and 6 8. Exercise In the following, first write the argument in symbolic form and then construct a formal proof. If the cup is gold, then it is heavier than water. If John can carry the cup, then it is not heavier than water. Therefore, if the cup is gold, then John cannot carry it. Sol. We start by labeling the statements: p = the cup is gold q = cup is heavier than water r = John can carry the cup In symbols, the argument becomes r q p r This is exactly exercise 5, that we have already proved. 7
9. Exercise In the following, first write the argument in symbolic form and then construct a formal proof. If prices are high, then wages are high. Prices are high or there are price controls. It there are price controls, then there is no inflation. There is inflation. Therefore wages are high. Sol. Label the statements: p = prices are high q = wages are high r = there are price controls s = there is inflation In symbols p r r s s q Now we prove the argument 8
1) premise 2) p r premise 3) r s premise 4) s premise 5) s r Contrapositive of 3 6) r Detachment on 4 and 5 7) p Disjunctive Syllogism on 2 and 6 8) q Detachment on 1 and 7 10. Exercise In the following, first write the argument in symbolic form and then construct a formal proof. If the hour is late, then I become sleepy. I am not sleepy or I take a nap. I am not taking a nap. Therefore, the hour is not late. Sol. Label the statements: p = hour is late q = I become sleepy r = I take a nap In symbols q r r p 9
1) premise 2) q r premise 3) r premise 4) q Disjunctive Syllogism on 2 and 3 5) q p Contrapositive of 1 6) p Detachment on 4 and 5 11. Exercise In the following, first write the argument in symbolic form and then construct a formal proof. If health care is not improved, then the quality of life will not be high. If health care is improved and the quality of life is high, then the incumbents will be reelected. The quality of life is high. Therefore, the incumbents will be reelected. Sol. Label statements: p = health care is improved q = quality of life is high r = incumbents will be reelected In symbols: p q (p q) r q r 10
1) p q premise 2) (p q) r premise 3) q premise 4) q p Contrapositive of 1 5) p Detachment on 3 and 4 6) p q we just put 3 and 5 together 7) r Detachment on 2 and 6 11