COMP Intro to Logic for Computer Scientists. Lecture 13

Transcription

1 COMP 1002 Intro to Logic for Computer Scientists Lecture 13 B 5 2 J

2 Admin stuff Assignments schedule? Split a2 and a3 in two (A2,3,4,5), 5% each. A2 due Feb 17 th. Midterm date? March 2 nd. No office hour on Feb 9 th

3 Puzzle 11 Let SS = xx N xx iiii eeeeeeee} xx N xx iiii oooooo} Prove or disprove: xx SS, xx dddddddd nnnnnn dddddddddddd xx 2

4 Puzzle 11 Let SS = xx N xx iiii eeeeeeee} xx N xx iiii oooooo} S = Prove or disprove: xx SS, xx dddddddd nnnnnn dddddddddddd xx 2 Let P(x)= xx dddddddd nnnnnn dddddddddddd xx 2 To disprove, can give a counterexample Find an element in S such that P(x) is true But there is no such element in S, because there are no elements in S at all! To prove, enough to check that it holds for all elements of S. There is none, so it does hold for every element in S. Another way: Since S is defined as a subset of natural numbers, can read xx SS PP xx as xx N xx SS PP xx. Since xx SSS is always false, xx SS PP xx is true for every xx N Call a statement xx PP xx vacuously true.

5 Universal Modus Ponens All men are mortal Socrates is a man Therefore, Socrates is mortal All cats like fish Molly likes fish Therefore, Molly is a cat

6 Universal Modus Ponens xx, PP xx QQ xx PP aa QQ aa Q P All men are mortal ( xx, MMMMMM xx MMMMMMMMMMMM xx ) Socrates is a man (MMMMMM SSSSSSSSSSSSSSSS ) Therefore, Socrates is mortal (MMMMMMMMMMMM(SSSSSSSSSSSSSSSS) All numbers are either odd or even 2 is a number Therefore, 2 is either odd or even. Mortals Men All trees drop leaves Pine does not drop leaves Therefore, pine is not a tree

7 Universal Modus Ponens All men are mortal Socrates is a man Therefore, Socrates is mortal Mortal Men All cats like fish Molly likes fish Therefore, Molly is a cat Like fish cats

8 Instantiation/generalization In general, if xx SS FF xx is true for some formula FF xx, if you take any specific element aa SS, then FF aa must be true. This is called the universal instantiation rule. xx N xx > 1 5 > 1 If you prove FF aa without any assumptions about aa other than aa SS, then xx SS, FF xx This is called universal generalization.

9 Instantiation/generalization If you can find an element aa SS such that FF aa, then xx SS, FF xx This is called existential generalization. Alternatively, if xx SS FF xx is true, then you can give that element of SS for which FF xx is true a name, as long as that name has not been used elsewhere. This is called the existential instantiation rule. xx N xx 5 = 0 kk = 0 + 5

10 Existential instantiation If xx SS FF xx is true, then you can give that element of SS for which FF xx is true a name, as long as that name has not been used elsewhere. Let n be an even number. Then n=2k for some k. xx N EEEEEEEE xx yy N xx = 2 yy Important to have a new name! Let n and m be two even numbers. Then n=2k and m=2k is wrong! xx 1, xx 2 N EEEEEEEE xx 1 EEEEEEEE xx 2 yy 1, yy 2 N xx 1 = 2 yy 1 xx 2 = 2 yy 2 Let n and m be two even numbers. Then n=2k and m=2l

11 Other inference rules Combining universal instantiation with tautologies, get other types of arguments: pp qq q rr pp rr xx PP xx QQ xx xx QQ xx RR xx xx PP xx RR xx For any x, if xx > 3, then xx > 2 For any x, if xx > 2, then xx 1 For any x, if xx > 3, then xx 1 (This particular rule is called transitivity )

12 Types of proofs Direct proof of xx FF xx Show that FF xx holds for arbitrary x, then use universal generalization. Often, FF xx is of the form GG xx HH(xx) Example: A sum of two even numbers is even. Proof by cases If can write xx FF xx as xx(gg 1 xx GG 2 xx GG kk xx ) HH(xx), prove GG 1 xx HH xx (GG 2 xx HH xx ) (GG kk xx HH(xx)) Example: triangle inequality ( xx + yy xx + yy ) Proof by contraposition To prove xx GG xx HH(xx), prove xx HH xx GG(xx) Example: If square of an integer is even, then this integer is even. Proof by contradiction To prove xx FF xx, prove xx FF xx FFFFFFFFFF Example: 2 is not a rational number. Example: There are infinitely many primes.

13 Puzzle: better than nothing Nothing is better than eternal bliss A burger is better than nothing Therefore, a burger is better than eternal bliss. Is there anything wrong with this argument?