Mathematical Logic Part Two

Transcription

1 Mathematical Logic Part Two

2 A Note on the Career Fair

3 Recap from Last Time

4 Recap So Far A propositional variable is a variable that is either true or false. The propositional connectives are as follows: Negation: p Conjunction: p q Disjunction: p q Implication: p q Biconditional: p q True: False:

5 Take out a sheet of paper!

6 What's the truth table for the connective?

7 What's the negation of p q?

8 New Stuff!

9 First-Order Logic

10 What is First-Order Logic? First-order logic is a logical system for reasoning about properties of objects. Augments the logical connectives from propositional logic with predicates that describe properties of objects, functions that map objects to one another, and quantifiers that allow us to reason about multiple objects.

11 Some Examples

12 Likes(You, Eggs) Likes(You, Tomato) Likes(You, Shakshuka) AreNetflixingAndChilling(You, Him) WatchesNetflix(You) WatchesNetflix(Him) IsStranded(MattDamon) WillBringHome(MattDamon)

13 Likes(You, Eggs) Likes(You, Tomato) Likes(You, Shakshuka) AreNetflixingAndChilling(You, Him) WatchesNetflix(You) WatchesNetflix(Him) IsStranded(MattDamon) WillBringHome(MattDamon) These These blue blue terms terms are are called called constant constant symbols. symbols. Unlike Unlike propositional propositional variables, variables, they they refer refer to to objects, objects, not not propositions. propositions.

14 Likes(You, Eggs) Likes(You, Tomato) Likes(You, Shakshuka) AreNetflixingAndChilling(You, Him) WatchesNetflix(You) WatchesNetflix(Him) IsStranded(MattDamon) WillBringHome(MattDamon) The The red red things things that that look look like like function function calls calls are are called called predicates. predicates. Predicates Predicates take take objects objects as as arguments arguments and and evaluate evaluate to to true true or or false. false.

15 Likes(You, Eggs) Likes(You, Tomato) Likes(You, Shakshuka) AreNetflixingAndChilling(You, Him) WatchesNetflix(You) WatchesNetflix(Him) IsStranded(MattDamon) WillBringHome(MattDamon) What What remains remains are are traditional traditional propositional propositional connectives. connectives. Because Because each each predicate predicate evaluates evaluates to to true true or or false, false, we we can can connect connect the the truth truth values values of of predicates predicates using using normal normal propositional propositional connectives. connectives.

16 Reasoning about Objects To reason about objects, first-order logic uses predicates. Examples: ExtremelyCute(Quokka) DeadlockEachOther(Democrats, Republicans) Predicates can take any number of arguments, but each predicate has a fixed number of arguments (called its arity). The arity and meaning of each predicate are typically specified in advance. Applying a predicate to arguments produces a proposition, which is either true or false.

17 First-Order Sentences Sentences in first-order logic can be constructed from predicates applied to objects: LikesToEat(V, M) Near(V, M) WillEat(V, M) Cute(t) Dikdik(t) Kitty(t) Puppy(t) x < 8 x < 137 The The notation notation xx<<88 isis just just aa shorthand shorthand for for something something like like LessThan(x, LessThan(x,8). 8). Binary Binary predicates predicates inin math math are are often often written written like like this, this, but but symbols symbols like like << are are not not aa part part of of first-order first-order logic. logic.

18 Equality First-order logic is equipped with a special predicate = that says whether two objects are equal to one another. Equality is a part of first-order logic, just as and are. Examples: TomMarvoloRiddle = LordVoldemort MorningStar = EveningStar Equality can only be applied to objects; to state that two propositions are equal, use.

19 For notational simplicity, define as x y (x = y)

20 Let's see some more examples.

21 FavoriteMovieOf(You) FavoriteMovieOf(Her) StarOf(FavoriteMovieOf(You)) = StarOf(FavoriteMovieOf(Her)) These These purple purple terms terms are are functions. functions. Functions Functions take take objects objects as as input input and and produce produce objects objects as as output. output.

22 Functions First-order logic allows functions that return objects associated with other objects. Examples: LengthOf(path) MedianOf(x, y, z) x+y As with predicates, functions can take in any number of arguments, but each function has a fixed arity. As with predicates, the arity and interpretation of functions are specified in advance. Functions evaluate to objects, not propositions. There is no syntactic way to distinguish functions and predicates; you'll have to look at how they're used.

23 Objects and Predicates When working in first-order logic, be careful to keep objects (actual things) and predicates (true or false) separate. You cannot apply connectives to objects: Venus TheSun You cannot apply functions to propositions: StarOf(IsRed(Sun) IsGreen(Mars)) Ever get confused? Just ask!

24 One last (and major) change

25 x. (IsPuppy(x) IsAdorable(x)) ( All puppies are adorable ) a. (IsAdorable(a) IsPuppy(a)) ( Something is adorable but not a puppy ) These These green green terms terms are are called called quantifiers. quantifiers. The The symbol symbol isis read read for for all, all, and and the the symbol symbol isis read read there there exists exists

26 x. (IsPuppy(x) IsAdorable(x)) ( All puppies are adorable ) a. (IsAdorable(a) IsPuppy(a)) ( Something is adorable but not a puppy ) The The teal teal terms terms are are variables. variables. Each Each quantifier quantifier introduces introduces aa variable variable that that itit quantifies quantifies over. over. In In the the first first case, case, we're we're saying saying that that for for any any choice choice of of x, x, ifif xx isis aa puppy, puppy, then then xx isis adorable. adorable. In In the the second, second, we're we're saying saying that that there there exists exists some some choice choice of of aa where where aa isis adorable adorable and and aa isis not not aa puppy. puppy.

27 x. (IsPuppy(x) IsAdorable(x)) ( All puppies are adorable ) a. (IsAdorable(a) IsPuppy(a)) ( Something is adorable but not a puppy )

28 Quantifiers The biggest change from propositional logic to first-order logic is the use of quantifiers. A quantifier is an operator that expresses that some statement is true for some or all choices that could be made.

29 For any natural number n, n is even iff n2 is even n. (n ℕ (Even(n) Even(n2))) isis the the universal universalquantifier quantifier and and says says for for any any choice choice of of n, n, the the following following isis true. true.

30 The Universal Quantifier A statement of the form x. ψ asserts that for every choice of x, the statement ψ is true when we plug in that choice of x. Examples: v. (Puppy(v) Cute(v)) x. (IsMillennial(x) IsSpecial(x)) Tallest(SK) x. (SK x ShorterThan(x, SK))

31 Some muggles are intelligent. m. (Muggle(m) Intelligent(m)) isis the the existential existentialquantifier quantifier and and says says for for some some choice choice of of m, m, the the following following isis true. true.

32 The Existential Quantifier A statement of the form x. ψ asserts that there is some choice of x where ψ is true when we plug in that x. Examples: x. (Even(x) Prime(x)) x. (TallerThan(x, me) LighterThan(x, me)) ( x. Appreciates(x, me)) Happy(me)

33 Variables and Quantifiers Each quantifier has two parts: the variable that is introduced, and the statement that's being quantified. The variable introduced is scoped just to the statement being quantified. ( x. Loves(You, x)) ( y. Loves(y, You)) The The variable variable xx just just lives lives here. here. The The variable variable yy just just lives lives here. here.

34 Variables and Quantifiers Each quantifier has two parts: the variable that is introduced, and the statement that's being quantified. The variable introduced is scoped just to the statement being quantified. ( x. Loves(You, x)) ( x. Loves(x, You)) The The variable variable xx just just lives lives here. here. AA different different variable, variable, also also named named x, x, just just lives lives here. here.

35 Operator Precedence (Again) When writing out a formula in first-order logic, the quantifiers and have precedence just below. The statement x. P(x) R(x) Q(x) is parsed like this: ( ( x. P (x) ) R (x) ) Q ( x ) This is syntactically invalid because the variable x is out of scope in the back half of the formula. To ensure that x is properly quantified, explicitly put parentheses around the region you want to quantify: x. (P(x) R(x) Q(x))

36 Time-Out for Announcements!

37 Checkpoints Graded The PS1 checkpoint assignment has been graded. Review your feedback on GradeScope. Contact the staff (via Piazza or by stopping by office hours) if you have any questions. Some notes: Make sure to list your collaborators through GradeScope. There's a space to list your collaborators when you submit the assignment. If you forget to do this, they won't get credit for the assignment! Make sure to check your grade ASAP. For the reason listed above, make sure you have a grade recorded. If not, contact the course staff. Plus, that way, you can take our feedback into account when writing up your answers to the rest of the problem set questions. Best of luck on the rest of the problem set!

38 WiCS Casual CS Dinner WiCS (Women in Computer Science) will be holding a Casual CS Dinner next Wednesday, January 20 in the Women's Community Center at 6PM. These events are fantastic and are a great way to meet cool people, find study groups, and get Good Life Advice. Highly recommended!

39

40 Your Questions

41 How did you first learn discrete math? Any advice for first-timers? II first first learned learned discrete discrete math math when when II took took CS103 CS103 many many years years back! back! The The class class was was aa lot lot of of work, work, but but II learned learned aa ton! ton! My My advice: advice: don't don't settle settle for for aa partial partial understanding understanding of of aa topic. topic. If If you you kinda kinda get get something, something, ask ask questions questions about about it! it! Don't Don't be be afraid afraid of of the the concepts concepts you you don't don't understand. understand. Take Take them them head-on. head-on. You'll You'll learn learn so so much much ifif you you do. do.

42 If you can prove that the contrapositive of a theorem is false, does that count as a proof that the theorem is false? (I understand that a true contrapositive is sufficient to prove a theorem true, but I want to know if it is necessary) Yep! Yep! AA statement statement isis logically logically equivalent equivalent to to its its contrapositive, contrapositive, so so ifif you you can can disprove disprove the the contrapositive, contrapositive, you've you've disproven disproven the the original original statement. statement.

43 What are your favorite books? Any reading recommendations? II highly highly recommend recommend Whistling Whistling Vivaldi Vivaldi by by Claude Claude Steele. Steele. This This book book talks talks about about stereotype stereotype threat, threat, its its causes, causes, and and how how to to address address it. it. It's It's fundamentally fundamentally changed changed the the way way II teach teach my my classes. classes. It's It's also also aa quick quick and and fascinating fascinating read. read. If If you you like like long-form long-form essays, essays, read read Scott Scott and and Scurvy, Scurvy, the the story story of of how how we we lost lost the the cure cure to to scurvy scurvy due due to to scientific scientific progress progress inin other other areas. areas. It's It's aa fascinating fascinating read read and and will will change change the the way way you you think think about about scientific scientific progress. progress. I'd I'd also also recommend recommend Command Command and and Control: Control: Nuclear Nuclear Weapons, Weapons, The The Damascus Damascus Incident Incident and and the the Illusion Illusion of of Nuclear Nuclear Safety Safety by by Eric Eric Schlosser. Schlosser. It's It's aa fascinating fascinating look look at at the the history history of of nuclear nuclear weapons weapons and and institutional institutional dysfunction. dysfunction. Fun Fun fact fact some some of of the the ridiculous ridiculous scenes scenes from from Dr. Dr. Strangelove Strangelove and and War War Games Games actually actually happened. happened.

44 Back to CS103!

45 Translating into First-Order Logic

46 Translating Into Logic First-order logic is an excellent tool for manipulating definitions and theorems to learn more about them. Applications: Determining the negation of a complex statement. Figuring out the contrapositive of a tricky implication.

47 Translating Into Logic Translating statements into first-order logic is a lot more difficult than it looks. There are a lot of nuances that come up when translating into first-order logic. We'll cover examples of both good and bad translations into logic so that you can learn what to watch for. We'll also show lots of examples of translations so that you can see the process that goes into it.

48 Using the predicates - Puppy(p), which states that p is a puppy, and - Cute(x), which states that x is cute, write a sentence in first-order logic that means all puppies are cute.

49 An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x)) This This should should work work for for any any choice choice of of x, x, including including things things that that aren't aren't puppies. puppies.

50 An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x)) This This should should work work for for any any choice choice of of x, x, including including things things that that aren't aren't puppies. puppies.

51 An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x)) AA statement statement of of the the form form x. x. ψ ψ isis true true only only when when ψψ isis true true for for every every choice choice of of x. x.

52 An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x)) This This first-order first-order statement statement isis false false even even though though the the English English statement statement isis true. true. Therefore, Therefore, itit can't can't be be aa correct correct translation. translation.

53 An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x)) The The issue issue here here isis that that this this statement statement asserts asserts that that everything everything isis aa puppy. puppy. That's That's too too strong strong of of aa claim claim to to make. make.

54 A Better Translation All puppies are cute! x. (Puppy(x) Cute(x)) This This should should work work for for any any choice choice of of x, x, including including things things that that aren't aren't puppies. puppies.

55 A Better Translation All puppies are cute! x. (Puppy(x) Cute(x)) This This should should work work for for any any choice choice of of x, x, including including things things that that aren't aren't puppies. puppies.

56 A Better Translation All puppies are cute! x. (Puppy(x) Cute(x)) AA statement statement of of the the form form x. x. ψ ψ isis true true only only when when ψψ isis true true for for every every choice choice of of x. x.

57 All P's are Q's translates as x. (P(x) Q(x))

58 Useful Intuition: Universally-quantified statements are true unless there's a counterexample. x. (P(x) Q(x)) If If xx isis aa counterexample, counterexample, itit must must have have property property PP but but not not have have property property Q. Q.

59 Using the predicates - Blobfish(b), which states that b is a blobfish, and - Cute(x), which states that x is cute, write a sentence in first-order logic that means some blobfish is cute.

60 An Incorrect Translation Some blobfish is cute. x. (Blobfish(x) Cute(x))

61 An Incorrect Translation Some blobfish is cute. x. (Blobfish(x) Cute(x))

62 An Incorrect Translation Some blobfish is cute. x. (Blobfish(x) Cute(x))

63 An Incorrect Translation Some blobfish is cute. x. (Blobfish(x) Cute(x)) AA statement statement of of the the form form x. x. ψ ψ isis true true only only when when ψψ isis true true for for some some choice choice of of x. x.

64 An Incorrect Translation Some blobfish is cute. x. (Blobfish(x) Cute(x)) This This first-order first-order statement statement isis true true even even though though the the English English statement statement isis false. false. Therefore, Therefore, itit can't can't be be aa correct correct translation. translation.

65 An Incorrect Translation Some blobfish is cute. x. (Blobfish(x) Cute(x)) The The issue issue here here isis that that implications implications are are true true whenever whenever the the antecedent antecedent isis false. false. This This statement statement accidentally accidentally isis true true because because of of what what happens happens when when xx isn't isn't aa blobfish. blobfish.

66 A Correct Translation Some blobfish is cute. x. (Blobfish(x) Cute(x))

67 A Correct Translation Some blobfish is cute. x. (Blobfish(x) Cute(x)) AA statement statement of of the the form form x. x. ψ ψ isis true true only only when when ψψ isis true true for for some some choice choice of of x. x.

68 Some P is a Q translates as x. (P(x) Q(x))

69 Useful Intuition: Existentially-quantified statements are false unless there's a positive example. x. (P(x) Q(x)) If If xx isis an an example, example, itit must must have have property property PP on on top top of of property property Q. Q.

70 Good Pairings The quantifier usually is paired with. The quantifier usually is paired with. In the case of, the connective prevents the statement from being false when speaking about some object you don't care about. In the case of, the connective prevents the statement from being true when speaking about some object you don't care about.