Multiple Quanti ers - eliza.newhaven.edueliza.newhaven.edu/discrete/attach/L10-Multiple.pdf · Outline Multiple Quanti ers Universal Modus Ponens and Modus Tollens Instantiation Rules

OutlineMultiple Quantifiers

Universal Modus Ponens and Modus TollensInstantiation

Rules of InferencePractice

Multiple Quantifiers

Alice E. Fischer

CSCI 1166 Discrete Mathematics for ComputingFebruary, 2018

Alice E. Fischer Multiple Quantifiers. . . 1/28




1 Multiple QuantifiersExamples and VarietiesOrder of QuantifiersMultiple Quantifiers and Negations

2 Universal Modus Ponens and Modus Tollens

3 InstantiationUniversalExistential

4 Rules of InferenceUniversal Modus PonensUniversal Modus TollensUniversal Transitivity

5 Practice





Examples and VarietiesOrder of QuantifiersMultiple Quantifiers and Negations

Multiple Quantifiers

Examples and VarietiesOrder of Quantifiers

Negations of Multiply Quantified Statements






For All, There Exists

A statement can have multiple quantifiers:

All married people have a spouse.∀m ∈ People,Married(m)→ ∃s ∈ People,Spouse(m, s).

All squares on a completed Sudoko board hold a digit 1 . . . 9∀s ∈ squares, ∃d ∈ 1 . . . 9, d is written in s.

All columns on a completed Sudoko board contain a squarewith the digit 1∀c ∈ Sudoku columns, ∃s ∈ squares, s ∈ c ∧ 1 ∈ s.






Turning quantified statements into English.

Let the predicate T (p, x) mean p teaches x ,

Example ∀p ∈ Pros ∃x ∈ Skaters,T (p, x).

Wrong English: For all pros, there exists a skater such that allpros teach the skater.

Right: you need to use the x and y in the English statement.

For all pros p, there exists a skater x, such that p teaches x.

If you focus on any particular pro, p, there exists a skater x,such that p teaches x.






Same Quantifier Twice

A statement can have two quantifiers that are alike.

There are pairs of men and women who are not married.∃ m ∈ men, ∃w ∈ women,m is not married to w .

For all digits d ∈ 1 . . . 9, and for all rows r ∈ a completedSudoko board, d occurs once in r.∀ d ∈ 1 . . . 9, and ∀r ∈ completed Sudoku rows, Once(d , r).






Two or More Quantifiers

Given two consecutive quantifiers in a series of two or more:

If they are the same quantifier, the order does not matter.

(∃y ∈ Student)(∃x ∈ Teacher)(∃z ∈ Course), Teaches(x , y , z)(∃x ∈ Teacher)(∃z ∈ Course)(∃y ∈ Student), Teaches(x , y , z)

If they are different quantifiers, the order matters greatly.

(∀x ∈ Z+)(∃y ∈ Z−), x + y = 0. (true)(∃y ∈ Z−)(∀x ∈ Z+), x + y = 0. (false)






Order of quantifiers: All-Exists vs. Exists-All

∀x∃y is not at all the same as ∃y∀xa) Every man is descended from a mother.

(∀m ∈ Men) (∃f ∈ femalePeople), f is the mother of m.

b) A woman is the ancestor of every man.(∃f ∈ femalePeople)(∀m ∈ Men), f is the ancestor of m.

To disprove a, find a man without a mother.

To disprove b, find two men who do not have a common ancestor.(Remember Eve!)






Practice: Interpreting Quantified Statements

CS Students Ann,Bob,Cal ,Don,Eva,Flo belong to three clubs.

Ann Don EvaProgramming team

Ann Cal DonFloHacking team

AnnBobCalEvaRobotics club

Given the diagram, say whether each statement is true or false.

1 ∀z ∈ Students ∃w ∈ Students,SameClub(z ,w)

2 ∃s ∈ Students ∀c ∈ clubs,Member(s, c).

3 ∃t ∈ Students ∃d ∈ clubs,∼ Member(t, d).

4 ∀b ∈ clubs ∃v ∈ Students,Member(v , b)

5 ∃s ∈ Students ∀z ∈ Students,SameClub(s, z).

6 ∃x ∈ Students ∃y ∈ Students,∼ SameClub(x , y)






Negating a Statement with Two or More Quantifiers

To negate a multiply quantified statement, negate each quantifierin order, from left to right. Start with this true predicate that says“there is no largest or smallest integer”:

(∀y ∈ Z )(∃x ∈ Z )(∃z ∈ Z ), x < y < z .

To prove this, choose y=any integer, x=y-1 and z=y+1.

Now let us negate it; the result will be a predicate that is false:∼ (∀y ∈ Z )(∃x ∈ Z )(∃z ∈ Z ), x < y < z .(∃y ∈ Z ) ∼ (∃x ∈ Z )(∃z ∈ Z ), x < y < z .(∃y ∈ Z )(∀x ∈ Z ) ∼ (∃z ∈ Z ), x < y < z .(∃y ∈ Z )(∀x ∈ Z )(∀z ∈ Z ), ∼ (x < y < z .)(∃y ∈ Z )(∀x ∈ Z )(∀z ∈ Z ), x ≥ y ≥ z .)

To disprove this, choose x=1 and z=10.





Universal Modus Ponens and Modus Tollens

Universal Modus Ponens

∀x ,P(x)→ Q(x)P(k) for a particular k

∴ Q(k) for that k.

Example:If an integer is even,then its square is even.The integer 10 is even.Therefore, 100 is even.

Universal Modus Tollens

∀x ,P(x)→ .Q(x)∼ Q(k) for a particular k

∴∼ P(k) for that k

Example:All cows have hoofs.I do not have hoofs.Therefore, I am not a cow.





Inverse Error and Converse Error

Inverse Error

∀x ,P(x)→ Q(x)Q(k) for a particular k

∴ P(k) for that k. Invalid!

Example:All dogs have four legs,My pet fluffy has four legs.Therefore Fluffy is a dog.Not true, Fluffy is a cat.

Converse Error

∀x ,P(x)→ .Q(x)∼ P(k) for a particular k

∴∼ Q(k) for that k Invalid!

Example:All lawyers are educated.I am not a lawyer.Therefore I am not educated.Not true!





UniversalExistential

Instantiation

Making an argument using quantified statements uses the samerules of inference as for propositional calculus. But we need oneextra concept.

Instantiation is the process of applying a general quantifiedstatement to a particular element(s) from the domain for which itapplies.

Sometimes we want to answer general questions: do all birds fly?Other times we might be more particular: which bird doesn’t fly?Instantiation allows us to focus parts of an argument on particularindividuals in the domain to answer a specific question.






Universal Instantiation

Universal instantiation states that a quantified statement that istrue for every element of a domain is equally true for one specificelement of the domain.

Example - All men are nerds∀x ∈ Men,Nerd(x)

If this statement is true, and Fred is a man, then Fred is a nerd,Nerd(Fred).

This version of instantiation is used commonly in everyday speech.






Existential Instantiation

Existential instantiation states that if a quantified statement istrue for at least one element of a domain, we can give such anelement an arbitrary name.

Example - Some men are nerds∃x ∈ Men,Nerd(x)

Using existential instantiation, we can give a convenient name toone of the qualifying men, say, SmartGuy.

This version of instantiation is used commonly in mathematics.





Universal Modus PonensUniversal Modus TollensUniversal Transitivity

Rules of Inference

When we made arguments in propositional calculus, we presentedmany different rules of inference that could be used to generatenew statements that eventually led to a conclusion. The samerules apply to predicate calculus, with the help of instantiation.

We will look at three of the most useful rules: modus ponens,modus tollens, and transitivity.






Universal Modus Ponens

Universal Modus Ponens states:

∀x ,P(x)→ Q(x)

P(k) for a particular k

∴ Q(k) for that k.

In other words, if the general premise of P(x)→ Q(x) is true forall x in the domain, and we know that the fact P(k) is true for aparticular element, k, then we can conclude that the fact Q(k) isalso true.






An Example

Here is a classic argument to which universal modus ponensapplies:

All men are mortal.Socrates is a man.∴ Socrates is mortal.

∀x ∈ People,Man(x)→ Mortal(x)Man(Socrates)∴ Mortal(Socrates)

This can be demonstrated using Venn diagrams:






Beware the Converse Error

Consider this argument:

All men are mortal.Socrates is mortal.∴ Socrates is a man.

∀x ∈ People,Man(x)→ Mortal(x)Mortal(Socrates)∴ Man(Socrates)

This invalid argument, suffers from the converse error:






Beware the Inverse Error

Consider this argument:

All men are mortal.Zeus is not a man.∴ Zeus is not mortal.

∀x ∈ People,Man(x)→ Mortal(x)∼ Man(Zeus)∴ ∼ Mortal(Zeus)

This invalid argument, suffers from the inverse error:






Universal Modus Tollens

Universal Modus Tollens states

∀x ,P(x)→ Q(x)

∼ Q(k) for a particular k

∴∼ P(k) for that k

In other words, if the general premise of P(x)→ Q(x) is true forall x in the domain, and we know that the fact Q(k) is false for aparticular element, k, then we can conclude that the fact P(k) isalso false.

This is based on the fact that the contrapositive version of astatement is equivalent to the statement.






An Example

Here is a valid argument to which universal modus tollens applies:

All men are mortal.Zeus is not mortal.∴ Zeus is not a man.

∀x ∈ People,Man(x)→ Mortal(x)∼ Mortal(Zeus)∴ ∼ Man(Zeus)







Transitivity

The Law of Universal Transitivity states:

∀x ,P(x)→ Q(x)

∀x ,Q(x)→ R(x)

∴ ∀x ,P(x)→ R(x)

In other words, if the general premise of P(x)→ Q(x) is true forall x in the domain, and it is also true that Q(x)→ R(x), then wecan conclude that P(x)→ R(x).

This can be validated by specifying P(k) is true and usinguniversal modus ponens to show that R(k) is true, for all k.






An Example

Here is a valid argument to which universal transitivity applies:

All men are mortal.All mortals die.∴ All men die.Fred will be a dead man.

∀x ∈ People,Man(x)→ Mortal(x)∀x ∈ People,Mortal(x)→ Die(x)∴ ∀x ∈ People,Man(x)→ Die(x)Man(Fred)→ Die(Fred)






Some example arguments

Consider these facts:

1 Marcus was a man

2 Marcus was a Pompeian

3 All Pompeians were Romans

4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated him

6 Everyone is loyal to someone

7 Men only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar





Converting From English to Predicates

1 Marcus was a man

2 Marcus was a Pompeian

3 All Pompeians were Romans

4 Caesar was a ruler

5 All Romans were either loyalto Caesar or hated him

6 Everyone is loyal to someone

7 Men only try to assassinaterulers they are not loyal to

8 Marcus tried to assassinateCaesar

1 Man(Marcus)

2 Pompeian(Marcus)

3 ∀x ∈ People,Pompeian(x)→ Roman(x)

4 Ruler(Caesar)

5 ∀x ∈ People,Roman(x)→(Loyalto(x ,Caesar) ∨ Hate(x ,Caesar)) ∧∼ (Loyalto(x ,Caesar) ∧ Hate(x ,Caesar))

6 ∀x ∈ People, ∃y ∈ People, Loyalto(x , y)

7 ∀x ∈ People, ∀y ∈ People,Man(x) ∧ Ruler(y) ∧Tryassassinate(x , y)→∼ Loyalto(x , y)

8 Tryassassinate(Marcus,Caesar)





Did Marcus hate Caesar?

9. Loyalto(Marcus, Caesar)Wrong: Tried instantiation with 6, picking x as Marcus and yas Caesar. It was ok to pick x as Marcus because of universalinstantiation. It was not ok to pick Caesar as the model for yusing existential instantiation, because Caesar has specialproperties that are not shared by all Romans.

9. ∼ Loyalto(Marcus,Caesar)Use universal modus ponens with 1, 4, 8 and 7.





Did Marcus hate Caesar?

10. Roman(Marcus)

Use universal modus ponens with 2 and 3

11. (Loyalto(Marcus,Caesar) ∨ Hate(Marcus,Caesar)) ∧∼ (Loyalto(Marcus,Caesar) ∧ Hate(Marcus,Caesar))Use universal modus ponens with 10 and 5

12. Hate(Marcus,Caesar)Use 9 and 11 and the definitions of And and Or


Documents

Multiple Quanti ers - eliza.newhaven.edueliza.newhaven.edu/discrete/attach/L10-Multiple.pdf · Outline Multiple Quanti ers Universal Modus Ponens and Modus Tollens Instantiation Rules