Mathematical Logic Part Two

Recap from Last Time

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

Take out a sheet of paper!

What's the truth table for the connective?

What's the negation of p q?

New Stuff!

First-Order Logic

What First-Order Logic? First-order logic 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.

Some Examples

Likes(You, ComicBooks) Likes(You, GoodMovies) Likes(You, AwesomeWomenInTech) Likes(You, BlackPanther) Learns(You, Htory) ForeverRepeats(You, Htory) DrinksTooMuch(Me) IsAnIssue(That) IsOkay(Me)

Likes(You, Eggs) Likes(You, Tomato) Likes(You, Shakshuka) Learns(You, Htory) ForeverRepeats(You, Htory) In(MyHeart, Havana) TookBackTo(Him, Me, EastAtlanta)

Likes(You, Eggs) Likes(You, Tomato) Likes(You, Shakshuka) Learns(You, Htory) ForeverRepeats(You, Htory) In(MyHeart, Havana) TookBackTo(Him, Me, EastAtlanta) 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.

Likes(You, Eggs) Likes(You, Tomato) Likes(You, Shakshuka) Learns(You, Htory) ForeverRepeats(You, Htory) In(MyHeart, Havana) TookBackTo(Him, Me, EastAtlanta)

Likes(You, Eggs) Likes(You, Tomato) Likes(You, Shakshuka) Learns(You, Htory) ForeverRepeats(You, Htory) In(MyHeart, Havana) TookBackTo(Him, Me, EastAtlanta) 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.

Likes(You, Eggs) Likes(You, Tomato) Likes(You, Shakshuka) Learns(You, Htory) ForeverRepeats(You, Htory) In(MyHeart, Havana) TookBackTo(Him, Me, EastAtlanta)

Likes(You, Eggs) Likes(You, Tomato) Likes(You, Shakshuka) Learns(You, Htory) ForeverRepeats(You, Htory) In(MyHeart, Havana) TookBackTo(Him, Me, EastAtlanta) 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.

Reasoning about Objects To reason about objects, first-order logic uses predicates. Examples: Cute(Quokka) Likes(DrLee, CS103) Likes(DrLee, Quokka) Cute(Mosquito) Likes(DrLee, Mosquito) Applying a predicate to arguments produces a proposition, which either true or false. Typically, when you re working in FOL, you ll have a lt of predicates, what they stand for, and how many arguments they take. It ll be given separately than the formulas you write.

First-Order Sentences Sentences in first-order logic can be constructed from predicates applied to objects: Cute(a) Dikdik(a) Kitty(a) Puppy(a) Succeeds(You) Practices(You) x < 8 x < 137 The The less-than less-than sign sign just just another another predicate. predicate. Binary Binary predicates predicates are are sometimes sometimes written written in in infix infix notation notation th th way. way. Numbers Numbers are are not not built built in in to to first-order first-order logic. logic. They re They re constant constant symbols symbols just just like like You You and and a a above. above.

Equality First-order logic equipped with a special predicate = that says whether two objects are equal to one another. Equality 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.

Let's see some more examples.

FavoriteMovieOf(You) FavoriteMovieOf(Date) StarOf(FavoriteMovieOf(You)) = StarOf(FavoriteMovieOf(Date))

FavoriteMovieOf(You) FavoriteMovieOf(Date) StarOf(FavoriteMovieOf(You)) = StarOf(FavoriteMovieOf(Date)) 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.

FavoriteMovieOf(You) FavoriteMovieOf(Date) StarOf(FavoriteMovieOf(You)) = StarOf(FavoriteMovieOf(Date))

Functions First-order logic allows functions that return objects associated with other objects. Examples: ColorOf(Sky) MedianOf(x, y, z) x + y As with predicates, functions can take in any number of arguments, but always return a single value. Functions evaluate to objects, not propositions.

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!

The Type-Checking Table operate on... and produce Connectives (,, etc.) propositions a proposition Predicates (=, etc.) objects a proposition Functions objects an object

Type Inference Consider Consider the the following following formula formula in in first-order first-order logic: logic: R(y) R(y) (S(x, (S(x, y) y) = T(x)) T(x)) Assuming Assuming that that th th formula formula syntactically syntactically correct, correct, which which of of R, R, S, S, and and T are are predicates predicates and and which which are are functions? functions? A. A. R R a a predicate, predicate, S S a a predicate, predicate, and and T T a a predicate. predicate. B. B. R R a a predicate, predicate, S S a a predicate, predicate, and and T T a a function. function. C. C. R R a a predicate, predicate, S S a a function, function, and and T T a a predicate. predicate. D. D. R R a a predicate, predicate, S S a a function, function, and and T T a a function. function. E. E. R R a a function, function, S S a a predicate, predicate, and and T T a a predicate. predicate. F. F. R R a a function, function, S S a a predicate, predicate, and and T T a a function. function. G. G. R R a a function, function, S S a a function, function, and and T T a a predicate. predicate. H. H. R R a a function, function, S S a a function, function, and and T T a a function. function. Answer Answer at at PollEv.com/cs103 PollEv.com/cs103 or or text text CS103 CS103 to to 22333 22333 once once to to join, join, then then A, A, B, B, C, C,,, or or H. H.

One last (and major) change

Some muggle intelligent.

Some muggle intelligent. m. (Muggle(m) Intelligent(m))

Some muggle intelligent. m. (Muggle(m) Intelligent(m)) the the extential extential quantifier quantifier and and says says for for some some choice choice of of m, m, the the following following true. true.

The Extential Quantifier A statement of the form x. some-formula true if, for some choice of x, the statement some-formula true when that x plugged into it. Examples: x. (Even(x) Prime(x)) x. (TallerThan(x, me) LighterThan(x, me)) ( w. Will(w)) ( x. Way(x))

The Extential Quantifier x. Smiling(x)

The Extential Quantifier x. Smiling(x) Since Since Smiling(x) Smiling(x) true true for for some some choice choice of of x, x, th th statement statement evaluates evaluates to to true. true.

The Extential Quantifier x. Smiling(x)

The Extential Quantifier x. Smiling(x) Since Since Smiling(x) Smiling(x) not not true true for for any any choice choice of of x, x, th th statement statement evaluates evaluates to to false. false.

In In th th world, world, th th first-order first-order logic logic statement statement A. A. true. true. B. B. false. false. C. C. neither neither true true nor nor false. false. The Extential Quantifier Answer Answer at at PollEv.com/cs103 PollEv.com/cs103 or or text text CS103 CS103 to to 22333 22333 once once to to join, join, then then A, A, B, B, or or C. C. ( x. Smiling(x)) ( y. WearingHat(y))

The Extential Quantifier ( x. Smiling(x)) ( y. WearingHat(y))

The Extential Quantifier Is Is th th part part of of the the statement statement true true or or false? false? ( x. Smiling(x)) ( y. WearingHat(y))

The Extential Quantifier ( x. Smiling(x)) ( y. WearingHat(y)) Is Is th th part part of of the the statement statement true true or or false? false?

The Extential Quantifier Is Is th th overall overall statement statement true true or or false? false? ( x. Smiling(x)) ( y. WearingHat(y))

Fun with Edge Cases x. Smiling(x)

Fun with Edge Cases x. Smiling(x) Extentially-quantified Extentially-quantified statements statements are are false false in in an an empty empty world, world, since since it s it s not not possible possible to to choose choose an an object! object!

Some Technical Details

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

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

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

For any natural number n, n even iff n 2 even

For any natural number n, n even iff n 2 even n. (n N (Even(n) Even(n 2 )))

For any natural number n, n even iff n 2 even n. (n N (Even(n) Even(n 2 ))) the the universal universal quantifier quantifier and and says says for for any any choice choice of of n, n, the the following following true. true.

The Universal Quantifier A statement of the form x. some-formula true if, for every choice of x, the statement some-formula true when x plugged into it. Examples: p. (Puppy(p) Cute(p)) m. (IsMillennial(m) IsSpecial(m)) Tallest(SultanKösen) x. (SultanKösen x ShorterThan(x, SultanKösen))

The Universal Quantifier x. Smiling(x)

The Universal Quantifier x. Smiling(x) Since Since Smiling(x) Smiling(x) true true for for every every choice choice of of x, x, th th statement statement evaluates evaluates to to true. true.

The Universal Quantifier x. Smiling(x)

The Universal Quantifier x. Smiling(x) Since Since Smiling(x) Smiling(x) false false for for th th choice choice x, x, th th statement statement evaluates evaluates to to false. false.

The Universal Quantifier ( x. Smiling(x)) ( y. WearingHat(y))

The Universal Quantifier ( x. Smiling(x)) ( y. WearingHat(y)) Is Is th th part part of of the the statement statement true true or or false? false?

The Universal Quantifier Is Is th th part part of of the the statement statement true true or or false? false? ( x. Smiling(x)) ( y. WearingHat(y))

The Universal Quantifier ( x. Smiling(x)) ( y. WearingHat(y)) Is Is th th overall overall statement statement true true or or false false in in th th scenario? scenario?

Fun with Edge Cases x. Smiling(x)

Fun with Edge Cases Universally-quantified Universally-quantified statements statements are are vacuously vacuously true true in in empty empty worlds. worlds. x. Smiling(x)

Time-Out for Announcements!

Problem Set One Problem Set One due Friday at 2:30PM. Want to use your late days? You can submit up to 2:30PM on Sunday. Remember that late days are 24-hour extensions, so submitting on Sunday would use two late days. Problem Set Two goes out Friday. Checkpoint assignment due next Monday at 2:30PM. Remaining problems are due next Friday at 2:30PM. Play around with propositional and first-order logic and sharpen your proof-writing skills!

Back to CS103!

Translating into First-Order Logic

Translating Into Logic First-order logic an excellent tool for manipulating definitions and theorems to learn more about them. Need to take a negation? Translate your statement into FOL, negate it, then translate it back. Want to prove something by contrapositive? Translate your implication into FOL, take the contrapositive, then translate it back.

Translating Into Logic Translating statements into first-order logic 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.

Using the predicates - Puppy(p), which states that p a puppy, and - Cute(x), which states that x cute, write a sentence in first-order logic that means all puppies are cute. Which Which of of these these first-order first-order logic logic statements statements a proper proper translation? translation? A. A. p. p. (Puppy(p) (Puppy(p) Cute(p)) Cute(p)) B. B. p. p. (Puppy(p) (Puppy(p) Cute(p)) Cute(p)) C. C. p. p. (Puppy(p) (Puppy(p) Cute(p)) Cute(p)) D. D. p. p. (Puppy(p) (Puppy(p) Cute(p)) Cute(p)) E. E. More More than than one one of of these. these. F. F. None None of of these. these. Answer Answer at at PollEv.com/cs103 PollEv.com/cs103 or or text text CS103 CS103 to to 22333 22333 once once to to join, join, then then A, A, B, B, C, C, D, D, E, E, or or F. F.

An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x))

An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x)) Th Th 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.

An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x)) A statement statement of of the the form form x. x. something true true only only when when something something true true for for every every choice choice of of x. x.

An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x))

An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x)) Th Th first-order first-order statement statement false false even even though though the the Englh Englh statement statement true. true. Therefore, Therefore, it it can't can't be be a a correct correct translation. translation.

An Incorrect Translation All puppies are cute! x. (Puppy(x) Cute(x)) The The sue sue here here that that th th statement statement asserts asserts that that everything everything a a puppy. puppy. That's That's too too strong strong of of a a claim claim to to make. make.

A Better Translation All puppies are cute! x. (Puppy(x) Cute(x))

A Better Translation All puppies are cute! x. (Puppy(x) Cute(x)) Th Th 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.

A Better Translation All puppies are cute! x. (Puppy(x) Cute(x))

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

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

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

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

Using the predicates - Blobfh(b), which states that b a blobfh, and - Cute(x), which states that x cute, write a sentence in first-order logic that means some blobfh cute. Which Which of of these these first-order first-order logic logic statements statements a proper proper translation? translation? A. A. b. b. (Blobfh(b) (Blobfh(b) Cute(b)) Cute(b)) B. B. b. b. (Blobfh(b) (Blobfh(b) Cute(b)) Cute(b)) C. C. b. b. (Blobfh(b) (Blobfh(b) Cute(b)) Cute(b)) D. D. b. b. (Blobfh(b) (Blobfh(b) Cute(b)) Cute(b)) E. E. More More than than one one of of these. these. F. F. None None of of these. these. Answer Answer at at PollEv.com/cs103 PollEv.com/cs103 or or text text CS103 CS103 to to 22333 22333 once once to to join, join, then then A, A, B, B, C, C, D, D, E, E, or or F. F.

An Incorrect Translation Some blobfh cute. x. (Blobfh(x) Cute(x))

An Incorrect Translation Some blobfh cute. x. (Blobfh(x) Cute(x)) A statement statement of of the the form form x. x. something true true only only when when something something true true for for at at least least one one choice choice of of x. x.

An Incorrect Translation Some blobfh cute. x. (Blobfh(x) Cute(x))

An Incorrect Translation Some blobfh cute. x. (Blobfh(x) Cute(x)) Th Th first-order first-order statement statement true true even even though though the the Englh Englh statement statement false. false. Therefore, Therefore, it it can't can't be be a a correct correct translation. translation.

An Incorrect Translation Some blobfh cute. x. (Blobfh(x) Cute(x)) The The sue sue here here that that implications implications are are true true whenever whenever the the antecedent antecedent false. false. Th Th statement statement accidentally accidentally true true because because of of what what happens happens when when x x n't n't a a blobfh. blobfh.

A Correct Translation Some blobfh cute. x. (Blobfh(x) Cute(x))

A Correct Translation Some blobfh cute. x. (Blobfh(x) Cute(x)) A statement statement of of the the form form x. x. something true true only only when when something something true true for for at at least least one one choice choice of of x. x.

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

Useful Intuition: Extentially-quantified statements are false unless there's a positive example. x. (P(x) Q(x)) If If x x an an example, example, it it must must have have property property P on on top top of of property property Q. Q.

Good Pairings The quantifier usually paired with. x. (P(x) Q(x)) The quantifier usually paired with. x. (P(x) Q(x)) 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.

Next Time First-Order Translations How do we translate from Englh into first-order logic? Quantifier Orderings How do you select the order of quantifiers in first-order logic formulas? Negating Formulas How do you mechanically determine the negation of a first-order formula? Expressing Uniqueness How do we say there s just one object of a certain type?