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 17 1

1 7 : M o r e o n 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

1 : R e c a p

Symbolizing A Statements

• To symbolize an A statement, you use the universal quantifier “∀” and the arrow “→”.

• For example:

All whales are mammals. ∀x (Wx → Mx)

Every Canadian is polite. ∀x (Cx → Px)

• If you find this hard to understand, don’t worry! You can simply memorize the fact that this is how A statements are symbolized.

PHIL 110; Spring 2020; Lecture 17 4

Symbolizing E Statements

• To symbolize an E statement, you use the universal quantifier “∀”, the arrow “→”, and the negation operator “”.

• For example:

No children play bridge. ∀x (Cx → Px)

No mice understand calculus. ∀x (Mx → Cx)

• If you find this hard to understand, don’t worry! You can simply memorize the fact that this is how E statements are symbolized.

PHIL 110; Spring 2020; Lecture 17 5

Instances

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

• Suppose that the domain of quantification is animals.

• Suppose that “a” is a name for something in the domain.

• Here is a symbolization of Every whale is a mammal:

∀x (Wx → Mx)

PHIL 110; Spring 2020; Lecture 17 6

Instances

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

• Suppose that the domain of quantification is animals.

• Suppose that “a” is a name for something in the domain.

• Here is a symbolization of Every whale is a mammal:

∀x (Wx → Mx) A universal generalization, and ...

(Wa → Ma) ... one of its instances.

PHIL 110; Spring 2020; Lecture 17 7

Instances

• A universal generalization (i.e. a statement that starts with a “∀”) is true just in case all of its instances are true.

• A universal generalization is, in effect, a conjunction of all its instances.1

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

PHIL 110; Spring 2020; Lecture 17 8

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

C: ____ is a child.

S: ____ is a snob.

(1) Everyone likes opera.

(2) Every snob likes opera. (Hint: This is an A statement!)

(3) No child likes opera. (Hint: This is an E statement!)

(4) Nobody likes opera.

(5) Only snobs like opera.

PHIL 110; Spring 2020; Lecture 17 9

A Quick Symbolization Exercise

2 : 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 generalization, you can infer any one of its instances.

PHIL 110; Spring 2020; Lecture 17 11

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 17 12

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 17 13

Example

1. ∀x (Wx → Mx) Prem

2. Ma Prem

3. (Wa → Ma) 1, UI

4. Wa 2, 3 MT

PHIL 110; Spring 2020; Lecture 17 14

Exercise

Symbolize the following inference, and show that it is valid by giving a natural deduction proof:

Premise: Every SFU student is clever.

Premise: Dev is an SFU student.

Conclusion: Dev is clever.

PHIL 110; Spring 2020; Lecture 17 15

3 : S o m e Wo r d s o f C a u t i o n

1: The domain of quantification is sometimes called the universe of discourse.

PHIL 110; Spring 2020; Lecture 17 17

2: It’s bad practice to give one variable two jobs in one statement.

PHIL 110; Spring 2020; Lecture 17 18

• Suppose that you’re asked to symbolize “Everyone is dancing and everyone is smiling.”

• You could write:

(∀x Dx & ∀x Sx)

• This isn’t wrong, but it is potentially confusing, because you’ve used the variable “x” to do two different jobs in one statement.

• It would be much better to write:

(∀x Dx & ∀y Sy)

• In my lectures, I will assume that we adopt this convention!

PHIL 110; Spring 2020; Lecture 17 19

3: The Universal Generalizations in Our Symbolism are Strict …

PHIL 110; Spring 2020; Lecture 17 20

Our Universal Generalizations are Strict.

• This means that a universal generalizations in our symbolism can be refuted by a single counterexample.

• For example, the following inference is valid:

Premise: (Bp & Fp) (Pingu is a bird, but can’t fly.)

Conclusion: ∀x(Bx → Fx) (It is not true that every bird can fly.)

PHIL 110; Spring 2020; Lecture 17 21

Our Universal Generalizations are Strict.

1. (Bp & Fp) Prem

2. Bp 1, Simp

3. Fp 1, Simp

4. ∀x(Bx → Fx) Supp/RA

5. (Bp → Fp) 4, UI

6. Fp 2, 5 MP

7. ⊥ 3, 6 Conj

8. ∀x(Bx → Fx) 4-7, RA

PHIL 110; Spring 2020; Lecture 17 22

Our Universal Generalizations are Strict.

• It’s not possible to express loose universal generalizations in our symbolism …

• … but this is okay, since we’re trying to understand mathematical proof – and mathematicians don’t use loose generalizations in their proofs!

PHIL 110; Spring 2020; Lecture 17 23

4: Beware the following subtle error …

PHIL 110; Spring 2020; Lecture 17 24

On a Subtle Error in Proofs

Suppose there are twenty people at the party (one of whom is Ashni) and only sixty bottles of beer.

Domain of quantification: People at the party.

D “____ drinks three bottles of beer.”

R We will run out of beer.

a Ashni

Premise: (∀x Dx → R) (If everyone drinks three bottles of beer, we will run out.)

Premise: Da (Ashni drinks three bottle of beer.)

Conclusion: R (We will run out.)

PHIL 110; Spring 2020; Lecture 17 25

On a Subtle Error in Proofs

1. (∀x Dx → R) Prem

2. Da Prem

3. (Da → R) 1, UI

4. R 2, 3 MP

PHIL 110; Spring 2020; Lecture 17 26

5: Pay attention to the domain of quantification!

PHIL 110; Spring 2020; Lecture 17 27

Pay Attention to the Domain

• Suppose you’re asked to symbolize the statement “Everyone at the party who is dancing is happy.”

• If the domain of quantification for your symbolic sentences is people, you would write:

∀x((Px & Dx) → Hx)

• If the domain of quantification for your symbolic sentences is people at the party, you would write:

∀x(Dx → Hx)

PHIL 110; Spring 2020; Lecture 17 28

4 : F o r e s h a d o w i n g t h e U G R u l e

Square numbers: Rectangle numbers:

1  1 = 1 1  2 = 2

2  2 = 4 2  3 = 6

3  3 = 9 3  4 = 12

4  4 = 16 4  5 = 20

5  5 = 25 5  6 = 30

PHIL 110; Spring 2020; Lecture 17 30

Square numbers: Rectangle numbers:

1  1 = 1 1  2 = 2

2  2 = 4 2  3 = 6

3  3 = 9 3  4 = 12

4  4 = 16 4  5 = 20

5  5 = 25 5  6 = 30

PHIL 110; Spring 2020; Lecture 17 31

Square numbers: Rectangle numbers:

1  1 = 1 1  2 = 2

2  2 = 4 2  3 = 6

3  3 = 9 3  4 = 12

4  4 = 16 4  5 = 20

5  5 = 25 5  6 = 30

PHIL 110; Spring 2020; Lecture 17 32

Square numbers: Rectangle numbers:

1  1 = 1 1  2 = 2

2  2 = 4 2  3 = 6

3  3 = 9 3  4 = 12

4  4 = 16 4  5 = 20

5  5 = 25 5  6 = 30

PHIL 110; Spring 2020; Lecture 17 33

Square numbers: Rectangle numbers:

1  1 = 1 1  2 = 2

2  2 = 4 2  3 = 6

3  3 = 9 3  4 = 12

4  4 = 16 4  5 = 20

5  5 = 25 5  6 = 30

PHIL 110; Spring 2020; Lecture 17 34

Square numbers: Rectangle numbers:

1  1 = 1 1  2 = 2

2  2 = 4 2  3 = 6

3  3 = 9 3  4 = 12

4  4 = 16 4  5 = 20

5  5 = 25 5  6 = 30

Hypothesis:

• If you add together two consecutive rectangle numbers, the result is always twice a square.

• For any n, the sum of the nth rectangle number and the (n+1)th

rectangle number is always twice a square.

PHIL 110; Spring 2020; Lecture 17 35

Let n be any arbitrarily chosen natural number.

Then the nth rectangle number is: n(n+1)

Also, the (n+1)th rectangle number is: (n+1)(n+2)

So, the sum of the nth rectangle number and the (n+1)th rectangle number is:

n(n+1) + (n+1)(n+2)

= (n2 + n) + (n2 + n + 2n + 2)

= 2n2 + 4n + 2

= 2(n+1)2

This is indeed twice a square!

Therefore:

For any n, the sum of the nth rectangle number and the (n + 1)th rectangle number is twice a square.

PHIL 110; Spring 2020; Lecture 17 36

The UG Rule

• The statement we just proved is a universal generalization:

For any n, the sum of the nth rectangle number and the (n + 1)th rectangle number is twice a square.

• We proved it by proving that an “arbitrary instance” is true.

• This is an example of the UG rule at work.

• We’ll look at the rule in more detail next time …

PHIL 110; Spring 2020; Lecture 17 37

Exercise

Symbolize the following inference, and show that it is valid by giving a natural deduction proof:

Premise: Everyone who is drinking beer is dancing.

Premise: Everyone who is dancing is having fun.

Premise: Ashni is drinking beer.

Conclusion: Ashni is having fun.

PHIL 110; Spring 2020; Lecture 17 38