Philosophy assignment 10 questions

2 3 : I d e n t i t y 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

Exercise

Consider the following two inferences. You should assume that the domain of quantification is people. In each case:

• If the inference is valid, show that it is valid by giving a natural deduction proof.

• If the inference is not valid, that it is not valid by drawing an arrow diagram depicting a situation in which the premise is true and the conclusion false.

First Inference Second Inference

Premise: ∀x ∃y Lxy Premise: ∃y ∀x Lxy

Conclusion: ∃y ∀x Lxy Conclusion: ∀x ∃y Lxy

PHIL 110; Spring 2020; Lecture 23 2

Exercise

First Inference

Premise: ∀x ∃y Lxy

Conclusion: ∃y ∀x Lxy

PHIL 110; Spring 2020; Lecture 23 3

Exercise

First Inference

Premise: ∀x ∃y Lxy

Conclusion: ∃y ∀x Lxy

PHIL 110; Spring 2020; Lecture 23 4

Exercise

Second Inference

Premise: ∃y ∀x Lxy

Conclusion: ∀x ∃y Lxy

PHIL 110; Spring 2020; Lecture 23 5

Exercise

1. ∃y ∀x Lxy Prem 2. ∀x Lxi 1, EI

3. ∀x ∃y Lxy Supp/RA 4. ∃x ∃y Lxy 3, QN 5. ∃x ∀y Lxy 4, QN 6. ∀y Ljy 5, EI 7. Lji 6, UI 8. Lji 2, UI 9. ⊥ 7, 8 Conj

10. ∀x ∃y Lxy 3-9, RA, DN

PHIL 110; Spring 2020; Lecture 23 6

1 : I n t r o d u c i n g I d e n t i t y

Qualitative And Numerical Sameness

• Suppose I say that Ashni and Ben’s partners are “the same”. There are two things I might mean: • I might mean that Ashni and Ben are dating two people who are very

similar. (Perhaps their partners are twins.)

• I might mean that Ashni and Ben are dating the very same person – i.e. there is one person who is dating both of them.

• There is a similar ambiguity in the English word “identical”.

PHIL 110; Spring 2020; Lecture 23 8

Qualitative And Numerical Sameness

• Philosophers avoid confusion by distinguishing between “qualitative” and “numerical” identity.

• To say that x and y are qualitatively identical is to say that x and y are exactly alike (or at least very similar).

• To say that x and y are numerically identical is to say that x and y are not two things, but one.

• If x and y are not numerically identical, they are said to be “distinct”.

PHIL 110; Spring 2020; Lecture 23 9

Qualitative And Numerical Sameness

For example:

• Adrian Brody is qualitatively identical to John Locke (but they are not numerically identical).

PHIL 110; Spring 2020; Lecture 23 10

Qualitative And Numerical Sameness

For example:

• Adrian Brody is qualitatively identical to John Locke (but they are not numerically identical).

PHIL 110; Spring 2020; Lecture 23 11

Qualitative And Numerical Sameness

For example:

• Adrian Brody is qualitatively identical to John Locke (but they are not numerically identical).

• Eric Blair is numerically identical to George Orwell.

PHIL 110; Spring 2020; Lecture 23 12

The Identity Symbol: =

a) In mathematics, we use the symbol “=” to express propositions about numerical identity. For example:

b)  John Locke = Adrian Brody

c) Eric Blair = George Orwell

d) 7 + 5 = 12

e) 12 = 7 + 5

f)  12 = 7 + 7

g) 12 ≠ 7 + 7

PHIL 110; Spring 2020; Lecture 23 13

PHIL 110; Spring 2020; Lecture 23 14

a

d

c

b

e

PHIL 110; Spring 2020; Lecture 23 15

a

d

c

b

e

Everything is what it is, and not another thing.

2 : U s e s o f t h e I d e n t i t y S y m b o l

First Example

• How should we symbolize “Ashni loves Ben and someone else”?

• The statement means: Ashni loves Ben, and Ashni also loves someone who isn’t Ben.

• In symbols: (Lab & ∃x (Lax & x ≠ b))

PHIL 110; Spring 2020; Lecture 23 17

Second Example

• How about “Ashni loves Ben and nobody else”?

• This means: Ashni loves Ben, and it is not the case that there is somebody other than Ben who Ashni loves.

• In symbols: (Lab & ∃x(Lax & x≠b))

• Equivalently: (Lab & ∀x(Lax → x = b))

PHIL 110; Spring 2020; Lecture 23 18

Third Example

• How would we symbolize “There are (at least) two people dancing”?

• We might try: ∃x ∃y(Dx & Dy)

• But this doesn’t quite work!

• The correct approach: ∃x ∃y ((Dx & Dy) & x≠y)

PHIL 110; Spring 2020; Lecture 23 19

Fourth Example

• What about “There are (at least) three people dancing”?

• This is no good: ∃x ∃y ∃z ((Dx & Dy) & Dz)

• The correct symbolization is this:

∃x ∃y ∃z (Dx & Dy & Dz & x≠y & y≠z & z≠x)

(I’ve omitted some brackets to make the statement easier to read…)

PHIL 110; Spring 2020; Lecture 23 20

Domain of Quantification: People

b: Bob Dylan P: ____ is a poet.

r: Robert Zimmerman

d: Dylan Thomas

• Robert Zimmerman and Bob Dylan are the same person.

• Bob Dylan and Dylan Thomas are not the same person.

• There are at least two poets.

• Dylan Thomas is a poet, and there are no other poets.

PHIL 110; Spring 2020; Lecture 23 21

• Robert Zimmerman and Bob Dylan are the same person.

r = b

• Bob Dylan and Dylan Thomas are not the same person.

b ≠ d

• There are at least two poets.

∃x ∃y ((Dx & Dy) & x≠y)

• Dylan Thomas is a poet, and there are no other poets.

(Pd & ∀x(Px → x = d)

PHIL 110; Spring 2020; Lecture 23 22

3 : I n f e r e n c e R u l e s f o r I d e n t i t y

The Reflexivity Of Identity

• Here is an obvious fact about identity: ∀x x=x

• Richard Arthur allows you to assume this whenever you want. It should be labelled “Implicit Premise” or “Impl. Prem.” for short.

• Suppose for example that we wish to prove the following inference:

Premise: Da

Conclusion: ∃x(Dx & x=a)

PHIL 110; Spring 2020; Lecture 23 24

The Reflexivity Of Identity

1. Da Prem

2. ∀x x=x Impl. Prem

3. a=a 2, UI

4. (Da & a=a) 1, 3 Conj

5. ∃x(Dx & x=a) 4, EG

PHIL 110; Spring 2020; Lecture 23 25

The Substitution of Identicals

• Suppose you know that Ashni is in Vancouver. And suppose you know that Ashni is Mrs. Anand. Then obviously you can infer that Mrs. Anand is in Vancouver.

• Here is a mathematical example. Suppose you know that N is a prime number, and you know that N=K. Then you can infer that K is a prime number.

• These are examples of the “SI” rule.

PHIL 110; Spring 2020; Lecture 23 26

The Substitution of Identicals

• For example, suppose you are asked to prove the following inference:

Premise: a=b

Premise: Da

Premise: Sb

Conclusion: ∃x(Dx & Sx)

PHIL 110; Spring 2020; Lecture 23 27

The Substitution of Identicals

1. a=b Prem

2. Da Prem

3. Sb Prem

4. Db 1, 2 SI

5. (Sb & Db) 3, 4 Conj

6. ∃x(Dx & Sx) 5, EG

PHIL 110; Spring 2020; Lecture 23 28

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

(1) Premise: ∀x(Lax → b = x) Premise: b ≠ c Conclusion: Lac

(2) Premise: a = b Premise: b = c Premise: Da Premise: Sc Conclusion: ∃x(Dx & Sx)

(3) Premise: Da Premise: Db Premise: Sa Premise: Sb Conclusion: ∃x ∃y((Sx & Sy) & x ≠ y)

PHIL 110; Spring 2020; Lecture 23 30

(1) Premise: ∀x(Lax → b = x)

Premise: b ≠ c

Conclusion: Lac

1. ∀x(Lax → b = x) Premise

2. b ≠ c Premise

3. (Lac → b = c) 1, UI

4. Lac 2, 3 MT

PHIL 110; Spring 2020; Lecture 23 31

(2) Premise: a = b Premise: b = c Premise: Da Premise: Sc Conclusion: ∃x(Dx & Sx)

1. a = b Premise 2. b = c Premise 3. Da Premise 4. Sc Premise 5. Db 1, 3 SI 6. Dc 2, 5 SI 7. (Dc & Sc) 4, 6 Conj 8. ∃x(Dx & Sx) 7, EG

PHIL 110; Spring 2020; Lecture 23 32

(3) Premise: Da Premise: Db Premise: Sa Premise: Sb Conclusion: ∃x ∃y((Sx & Sy) & x ≠ y)

1. Da Premise 2. Db Premise 3. Sa Premise 4. Sb Premise

5. a = b Supp/RA 6. Db 1, 5 SI 7. ⊥ 2, 6 Conj

8. a ≠ b 5-7, RA 9. (Sa & Sb) 3, 4 Conj 10. ((Sa & Sb) & a ≠ b) 8, 9 Conj 11. ∃y((Sa & Sy) & a ≠ y) 10, EG 12. ∃x ∃y((Sx & Sy) & x ≠ y) 11, EG

PHIL 110; Spring 2020; Lecture 23 33