CS322

1. Week 1 - Wednesday CS322

2. Last time • Course overview • Propositional logic • Truth tables • AND, OR, NOT • Logical equivalence

3. Questions?

4. Logical warmup • You come to a fork in the road • Two men stand beneath a sign that reads: • Ask for the way, but waste not your breath • One road is freedom, the other is death • Just one of the pair will lead you aright • For one is a Knave, the other a Knight • What single yes or no question can you ask to determine which fork to take?

5. De Morgan’s Laws • What’s an expression that logically equivalent to ~(p  q) ? • What about logically equivalent to ~(p  q) ? • De Morgan’s Laws state: • ~(p  q)  ~p ~q • ~(p  q)  ~p ~q • Essentially, the negation flips an AND to an OR and vice versa

6. Tautologies and Contradictions

7. Tautology • A tautology is something that is true no matter what • Examples: • T • p ~p • pp • The final column in a truth table for a tautology is all true values • The book sometimes writes a statement which is a tautology as a t

8. Contradiction • A contradiction is something that is false no matter what • Examples: • F • p ~p • ~(pp) • The final column in a truth table for a contradiction is all false values • The book sometimes writes a statement which is a contradiction as a c

9. Laws of Boolean algebra

10. Back to Implications

11. What are you implying? • You can construct all possible outputs using combinations of AND, OR, and NOT • But, sometimes it’s useful to introduce notation for common operations • This truth table is for pq

12. If… • We use  to represent an if-then statement • Let p be “The moon is made of green cheese” • Let q be “The earth is made of rye bread” • Thus, pq is how a logician would write: • If the moon is made of green cheese, then the earth is made of rye bread • Here, p is called the hypothesis and q is called the conclusion • What other combination of p and q is logically equivalent to pq ?

13. Why is implication used that way? • pq is true when: • p is true and q is true • p is false • Why? • For the whole implication to be true, the conclusion must always be true when the hypothesis is true • If the hypothesis is false, it doesn’t matter what the conclusion is • “If I punch the tooth fairy in the face, I will be Emperor of the World” • What’s the negation of an implication?

14. Contrapositive • Given a conditional statement pq, its contrapositive is ~q ~p • Conditional: “If a murderer cuts off my head, then I will be dead.” • Contrapositive: “If I am not dead, then a murderer did not cut off my head.” • What’s the relationship between a conditional and its contrapositive?

15. Converse and inverse • Given a conditional statement pq: • Its converse is qp • Its inverse is ~p ~q • Consider the statement: • “If angry ham sandwiches explode, George Clooney will become immortal.” • What is its converse? • What is its inverse? • How are they related?

16. Biconditional • Sometimes people say “if and only if”, as in: • “A number is prime if and only if it is divisible only by itself and 1.” • This can be written piffq or pq • This is called the biconditional and has this truth table: • What is the biconditional logically equivalent to?

17. Arguments

18. Arguments • An argument is a list of statements (called premises) followed by a single statement (called a conclusion) • Whenever all of the premises are true, the conclusion must also be true, in order to make the argument valid

19. Examples • Are the following arguments valid? • pq ~r (premise) • qp r (premise) •  pq (conclusion) • p (q  r)(premise) • ~r (premise) •  pq (conclusion)

20. Common argument tools • Modus ponens is a valid argument of the following form: • pq • p •  q • Modus tollens is a contrapositive reworking of the argument, which is also valid: • pq • ~q •  ~p • Give verbal examples of each • We call these short valid arguments rules of inference

21. Generalization • The following are also valid rules of inference: • p •  p q • q •  p q • English example: “If pigs can fly, then pigs can fly or swans can breakdance.”

22. Specialization • The following are also valid rules of inference: • p q •  p • p q •  q • English example: “If the beat is out of control and the bassline just won’t stop, then the beat is out of control.”

23. Conjunction • The following is also a valid rule of inference: • p • q •  p q • English example: “If the beat is out of control and the bassline just won’t stop, then the beat is out of control and the bassline just won’t stop.”

24. Elimination • The following are also valid rules of inference: • p q • ~q •  p • p q • ~p •  q • English example: “If you’re playing it cool or I’m maxing and relaxing, and you’re not playing it cool, then I’m maxing and relaxing.”

25. Transitivity • The following is also a valid rule of inference: • p  q • q  r •  p r • English example: “If you call my mom ugly I will call my brother, and if I call my brother he will beat you up, then if you call my mom ugly my brother will beat you up.”

26. Division into cases • The following is also a valid rule of inference: • p  q • p  r • q  r •  r • English example: “If am fat or sassy, and being fat implies that I will give you trouble, and being sassy implies that I will give you trouble, then I will give you trouble.”

27. Contradiction Rule • The following is also a valid rule of inference: • ~p  c •  p • English example: “If my water is at absolute zero then the universe does not exist, thus my water must not be at absolute zero.”

28. Fallacies • A fallacy is an argument that is not valid • It could mean that the conclusion is not true in only a single case in the truth table • But, if the conclusion is ever false whenever all the premises are true, the argument is a fallacy • Most arguments presented by politicians are fallacies for one reason or another

29. Common fallacies • Converse error • If Joe sings a sad song, then Joe will make it better. • Joes makes it better. • Conclusion: Joe sings a sad song. FALLACY • Inverse error • If you eat too much, you will get sick. • You are not eating too much. • Conclusion: You will not get sick. FALLACY

30. Digital Logic Circuits

31. Digital logic circuits • Digital logic circuits are the foundation of all computer hardware • Circuits are built out of components called gates • A gate has one or more inputs and an output • Gates model Boolean operations • Usually, in digital logic, we use a 1 for true and a 0 for false

32. Common gates • The following gates have the same function as the logical operators with the same names: • NOT gate: • AND gate: • OR gate:

33. Example • Draw the digital logic circuit corresponding to: (p ~q)  ~(p  r) • What’s the corresponding truth table?

34. Upcoming

35. Next time… • Predicate logic • Universal quantifier • Existential quantifier

36. Reminders • Read Chapter 3