Philosophy assignment 10 questions

The Exam Has Been Scheduled

• Most of you will take the exam on WED 15-Apr, 1200-15:00, C9001

• Some of you will take the exam at the CAL.

• Some of you may qualify for “hardship”: • You have three exams within 24 hours.

• You have an examination at one location (e.g. the Burnaby campus) followed immediately by an exam at another location (e.g., the Surrey campus).

PHIL 110; Spring 2020; Lecture 16 1

1 6 : T h e U n i v e r s a l Q u a n t i f i e r P H I L 1 1 0 ; S p r i n g 2 0 2 0 ; To m D o n a l d s o n

PHIL 110; Spring 2019; Lecture 14 3

Contains Chocolate Contains Garlic

Contains Cream

Domain: Traditional Dishes

PHIL 110; Spring 2020; Lecture 16 4

Contains Chocolate Contains Garlic

Contains Cream

Domain: Traditional Dishes

PHIL 110; Spring 2020; Lecture 16 5

Contains Chocolate Contains Garlic

Contains Cream

Domain: Traditional Dishes

PHIL 110; Spring 2020; Lecture 16 6

Contains Chocolate Contains Garlic

Contains Cream

Domain: Traditional Dishes

PHIL 110; Spring 2020; Lecture 16 7

Contains Chocolate Contains Garlic

Contains Cream

Domain: Traditional Dishes

x

PHIL 110; Spring 2020; Lecture 16 8

Contains Chocolate Contains Garlic

Contains Cream

Domain: Traditional Dishes

x

PHIL 110; Spring 2020; Lecture 16 9

Contains Chocolate Contains Garlic

Contains Cream

Domain: Traditional Dishes

x

x

Valid or not?

(1) No A are B. (2) All A are B.

Some A is C. No C are A.

Therefore: Therefore:

Some C is not B. No C are B.

Valid! Not valid!

PHIL 110; Spring 2020; Lecture 16 10

Valid or not?

(1) No A are B.

Some A is C.

Therefore:

Some C is not B.

PHIL 110; Spring 2020; Lecture 16 11

Valid or not?

(2) All A are B.

No C are A.

Therefore:

No C are B.

PHIL 110; Spring 2020; Lecture 16 12

1 : T h e N e e d f o r S y m b o l s

The Complexities of English

Sometimes when one uses the phrase “a bear”, you mean to talk about some particular bear; sometimes you mean to generalise about all bears.

A universal generalization: A bear is a mammal.

An existential generalization: A bear is in my garden.

PHIL 110; Spring 2020; Lecture 16 14

Ambiguities in English Quantification

• “Everyone at the party isn’t dancing.” • It is not true that everyone at the party is dancing.

• Nobody at the party is dancing.

• “Everything is caused by something.” • There is some particular thing (God perhaps?) which causes everything. • Everything has a cause (though perhaps different things have different

causes).

PHIL 110; Spring 2020; Lecture 16 15

Goals for this lecture …

• Introduce the symbol which we use to express universal generalizations – what we call the “universal quantifier”.

• Introduce one of the inference rules associated with this symbol.

PHIL 110; Spring 2020; Lecture 16 16

2 : N a m e s a n d P r e d i c a t e s

Proper Names A proper name is a word that represents an individual member of the domain of quantification. For example, if our domain is singers, we might use the following names:

• “Celine Dion”

• “Beyoncé”

• “Pavarotti”

Similarly, if our domain is countries, we might use the following names:

• “Canada”

• “Germany”

• “India”

PHIL 110; Spring 2020; Lecture 16 18

Predicates

If you take a declarative sentence and remove one or more proper names from it, the result is a predicate. (If you like, a predicate is a name with one or more proper-name-shaped holes in it.)

• A one-place predicate has one hole.

• A two-place predicate has two holes.

• A three-place predicate has three holes.

• (And so on!)

PHIL 110; Spring 2020; Lecture 16 19

Predicates

Some one-place predicates:

• “____ likes dancing.”

• “____ likes muffins”

• “____ likes swimming.”

Some two-place predicates:

• “____ and ____ are friends.”

• “____ is taller than ____.”

A three-place predicate:

• “____ and ____ together ate more food than ____.”

PHIL 110; Spring 2020; Lecture 16 20

Predicates

One can make a sentence by taking a predicate, and then “filling in” the hole with a name (or filling in the holes with names):

“Ashni” + “____ likes muffins” = “Ashni likes muffins”.

For the moment, we’ll restrict our attention to one-place predicates.

PHIL 110; Spring 2020; Lecture 16 21

Names and Predicates in Our Symbolism • We will use lower-case letters (usually from the beginning of the alphabet) as

names. We will use capital letters as predicates:

Proper Names Predicates

a: Ashni D: “____ likes dancing.”

b: Ben M: “____ likes muffins.”

c: Chiara S: “____ likes swimming.”

• One can form a sentence in our symbolism by writing a predicate, and then the appropriate number of names.

• For example, “Ma” means Ashni likes muffins. • We can form longer statements using the now-familiar statement operators.

• For example, “(Ma & Mb)” means Ashni and Ben both like muffins.

PHIL 110; Spring 2020; Lecture 16 22

Examples Proper Names Predicates

a: Ashni D: “____ likes dancing.”

b: Ben M: “____ likes muffins.”

c: Chiara S: “____ likes swimming.”

PHIL 110; Spring 2020; Lecture 16 23

English sentence Sentence in our symbolism

Ashni likes dancing. Da

Ben likes swimming. Sb

Ashni likes dancing and Ben likes swimming.

(Da & Sb)

PHIL 110; Spring 2020; Lecture 16 24

English sentence Sentence in our symbolism

Either Ashni or Ben likes dancing.

Chiara and Ben like muffins, but Ashni doesn’t.

If Chiara likes swimming, so do Ben and Ashni.

Proper Names Predicates

a: Ashni D: “____ likes dancing.”

b: Ben M: “____ likes muffins.”

c: Chiara S: “____ likes swimming.”

PHIL 110; Spring 2020; Lecture 16 25

English sentence Sentence in our symbolism

(Da ↔ Db)

((Ma  Mb)  Mc)

(Sa & Sb)

Proper Names Predicates

a: Ashni D: “____ likes dancing.”

b: Ben M: “____ likes muffins.”

c: Chiara S: “____ likes swimming.”

3 : T h e U n i v e r s a l Q u a n t i f i e r

The Universal Quantifier

• Let’s suppose that my domain of quantification is people at my party. • There are five people in this domain: Ashni, Ben, Chiara, Deshaun, and

Emma. • I want to symbolise this statement: everyone likes dancing. • How do I do it? • I could write this: ((((Da & Db) & Dc) & Dd) & De) • But what if the domain is, say, people who attended the most recent

Canucks game? • What if the domain is, numbers? • What we need is a symbol that will allow us to ascribe some property to

ALL the things in the domain, even if the domain is very large. This is what the universal quantifier is for!

PHIL 110; Spring 2020; Lecture 16 27

The Universal Quantifier ∀x

• For any x …

• Whatever x may be …

• It’s true for every x that …

PHIL 110; Spring 2020; Lecture 16 28

English sentence Sentence in our symbolism

Everyone likes dancing. ∀x Dx

Everyone likes dancing and likes muffins.

∀x (Dx & Mx)

Everyone either likes swimming or likes muffins.

∀x (Sx  Mx)

The Universal Quantifier ∀x

• For any x …

• Whatever x may be …

• It’s true for every x that …

PHIL 110; Spring 2020; Lecture 16 29

Universal Quantifier

English sentence Sentence in our symbolism

Everyone likes dancing. ∀x Dx

Everyone likes dancing and likes muffins.

∀x (Dx & Mx)

Everyone either likes swimming or likes muffins.

∀x (Sx  Mx)

The Universal Quantifier ∀x

• For any x …

• Whatever x may be …

• It’s true for every x that …

PHIL 110; Spring 2020; Lecture 16 30

Variable

English sentence Sentence in our symbolism

Everyone likes dancing. ∀x Dx

Everyone likes dancing and likes muffins.

∀x (Dx & Mx)

Everyone either likes swimming or likes muffins.

∀x (Sx  Mx)

The Universal Quantifier ∀y

• For any y …

• Whatever y may be …

• It’s true for every y that …

PHIL 110; Spring 2020; Lecture 16 31

English sentence Sentence in our symbolism

Either everyone likes muffins or everyone likes swimming.

(∀x Mx  ∀y Sy)

If Ashni likes dancing, everyone likes dancing.

(Da → ∀y Dy)

The Universal Quantifier ∀y

• For any y …

• Whatever y may be …

• It’s true for every y that …

PHIL 110; Spring 2020; Lecture 16 32

English sentence Sentence in our symbolism

Either everyone likes muffins or everyone likes swimming.

(∀x Mx  ∀y Sy)

If Ashni likes dancing, everyone likes dancing.

(Da → ∀y Dy)

What do variables represent?

• Note that when you use a universal quantifier, the variable doesn’t represent any particular entity in the domain:

∀x Dx

• Rather, one might say, it represents “any object chosen freely from the domain”.

• This is very common in mathematics …

PHIL 110; Spring 2020; Lecture 16 33

Claim: For any whole number k, if k is an odd number so is k2.

Proof

Suppose that k is an odd number.

Then for some number j: k = 2j + 1

Then, k2 = (2j + 1)(2j + 1)

So: k2 = 4j2 + 4j + 1

So: k2 = 2(2j2 + 2j) + 1

So k2 is odd!

So, in conclusion, if k is odd number, so is k2. 

PHIL 110; Spring 2020; Lecture 16 34

What do variables represent?

• Note that when you use a universal quantifier, the variable doesn’t represent any particular entity in the domain:

∀x Dx

• Rather, one might say, it represents “any object chosen freely from the domain”.

• This is very common in mathematics …

• Something similar happens in English too: “When a dog is hot, he pants.

PHIL 110; Spring 2020; Lecture 16 35

4 : I n s t a n c e s

Instances

• Suppose you take a universal generalization. That is …

• … a statement that begins with “∀x”.

• You remove the initial “∀x” …

• … and replace every occurrence of “x” in the statement with a name for something in the domain (the same name each time!).

• The result is an instance of the universal generalization with which you started.

PHIL 110; Spring 2020; Lecture 16 37

Instances

Universal Generalization Instance

∀x Dx Da

∀y (Dy & My) (Db & Mb)

∀z (Sz  Mz) (Sc  Mc)

A universal generalization is true just in case all of its instances are true.1

1 I assume here that everything in the domain has a name …

PHIL 110; Spring 2020; Lecture 16 38

Instances

• To repeat, a universal generalization is true just in case all of its instances are true.

• Indeed, you might think of a universal generalization as a conjunction of all of its instances.

• Suppose that the domain of quantification is people at my party, and suppose that the people at my party are Ashni, Ben, Chiara, Deshaun, and Emma. Then the following two statements have the same truth value:

((((Da & Db) & Dc) & Dd) & De)

∀x Dx

PHIL 110; Spring 2020; Lecture 16 39

5 : S y m b o l i z i n g A a n d E s t a t e m e n t s

Symbolizing A statements

• Let’s write “M” for “____ is a mammal” and “W” for “____ is a whale”.

• Suppose that the domain of quantification is animals.

• How should we symbolize “Every whale is a mammal”?

• The standard symbolization is this:

∀x (Wx → Mx)

• This often puzzles students. (Where did the arrow come from?!)

• So let’s take a closer look ...

PHIL 110; Spring 2020; Lecture 16 41

PHIL 110; Spring 2020; Lecture 16 42

Whales Mammals

Domain: Animals

• To symbolize the claim that an animal x is inside the shaded region, we would write (Wx & Mx).

• To symbolize the claim that an animal x is outside the shaded region, we would write (Wx & Mx).

• We can symbolize “All whales are mammals” as ∀x (Wx & Mx).

• This is equivalent to ∀x (Wx → Mx).

Every whale is a mammal.

Symbolizing A Statements

• More generally, we symbolize A statements with a universal quantifier and an arrow:

Every whale is a mammal. ∀x (Wx → Mx)

Every dancer is happy. ∀x (Dx → Hx)

Every SFU student is clever. ∀x (Sx → Cx)

• I hope the last slide helped you to understand this!

• If not – that’s okay! You can just remember that this is how A statements are symbolized.

PHIL 110; Spring 2020; Lecture 16 43

Symbolizing E Statements

• Suppose that the domain of quantification is animals.

• Let’s write “R” for “____ is a reptile” and “W” for “____ is a whale”.

• How should we symbolize “No whale is a reptile”?

• Here’s one way of doing it. Note that “No whale is a reptile” is equivalent to “Every whale is a non-reptile”.

• This can be formalized: ∀x (Wx → Rx)

PHIL 110; Spring 2020; Lecture 16 44

6 : T h e U n i v e r s a l I n s t a n t i a t i o n R u l e

The UI Rule

• If you look at the inside of the back cover of your textbook, you’ll find a number of rules involving the universal quantifier …

• … some of them are rather complex! We’ll get to those later.

• For now, let’s focus on one rather simple rule – the UI rule:

From a universal quantification, you can infer any one of its instances.

PHIL 110; Spring 2020; Lecture 16 46

The UI Rule

For example, the following inferences are both valid …

Premise: ∀x Dx (Everyone likes dancing.)

Conclusion: Da (Ashni likes dancing.)

Premise: ∀x (Wx → Mx) (Every whale is a mammal.)

Conclusion: (Wd → Md) (If Moby Dick is a whale, he’s a mammal.)

PHIL 110; Spring 2020; Lecture 16 47

Example

Show that the following inference is valid, by giving a natural deduction proof:

Premise: ∀x (Wx → Mx) (Every whale is a mammal.)

Premise: Ma (Ashni is not a mammal.)

Conclusion: Wa (Ashni is not a whale.)

PHIL 110; Spring 2020; Lecture 16 48

Example

1. ∀x (Wx → Mx) Prem

2. Ma Prem

3. (Wa → Ma) 1, UI

4. Wa 2, 3 MT

PHIL 110; Spring 2020; Lecture 16 49

7 : S o m e P r a c t i c e

O: ____ likes opera. Universe of discourse: people.

C: ____ is a child.

S: ____ is a snob.

(1) Everyone likes opera.

(2) Every snob likes opera.

(3) Nobody likes opera.

(4) No child likes opera.

(5) Only snobs like opera.

(6) If a child likes opera, they’re a snob.

PHIL 110; Spring 2020; Lecture 16 51

A Quick Symbolization Exercise

Show that this inference is valid, using a natural deduction proof:

Premise: Every snob likes opera.

Premise: Sam is a snob.

Conclusion: Sam likes opera.

PHIL 110; Spring 2020; Lecture 16 52