Philosophy assignment 10 questions

20: The EI Rule PHIL 110; Spring 2020; Tom Donaldson

A Quick Symbolization Exercise

Domain: Movies

C ____ is a comedy.

A ____ was directed by Ang Lee.

J ____ stars Jake Gyllenhaal.

(a) Ang Lee has directed a comedy.

(b) Every movie Ang Lee has directed is a comedy.

(c) No movie directed by Ang Lee is a comedy.

(d) Ang Lee directed a movie, not a comedy, which stars Jake Gyllenhaal.

PHIL 110; Spring 2020; Lecture 20 2

Natural Deduction Practice

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

Premise: Every SFU student lives in Vancouver or Burnaby.

Premise: Ahmed is an SFU student who doesn’t live in Burnaby.

Conclusion: There is an SFU student who lives in Vancouver.

PHIL 110; Spring 2020; Lecture 20 3

1: The EI Rule

The Problem of Shared Names

• Suppose there are two people called “Ashni”. Now consider:

Ashni is currently in Toronto.

Ashni is currently in Vancouver.

Therefore:

Ashni is currently in both Toronto and Vancouver.

• This inference presents a challenge for us. We can suppose that both premises are true: • One of the Ashnis is in Toronto, so the first premise is true. • One of the Ashnis is in Vancouver, so the second premise is true.

• The inference appears to be an instance of the conjunction rule.

• And yet the conclusion is false!

• Solution: In our symbolism, each name is only used once!

PHIL 110; Spring 2019; Lecture 18 5

Witnesses

• We’ve said that a universal generalization can be refuted by just one example - a “counterexample”. • For example, if someone claims that every bird can fly, we can refute them

by telling them about Pingu, the TV star.

• An existential generalization can be shown to be true using just one example – a “witness”. • Suppose someone asks whether the following statement is true: “Some

bird knows how to ice skate.”

• We can show that it is true by presenting Pingu as an example.

• Pingu is a witness to the statement “Some bird knows how to ice skate.”

PHIL 110; Spring 2020; Lecture 20 6

The EI Rule

• When we use the EI rule, we start with an existential generalization. We then introduce a name for an arbitrary witness.

• For example: • There are residents of Burnaby who play the piano. Let’s call one of them

“Smith” …

• Some SFU students are champion wrestlers. Let’s call one of them “Diana”. Then …

• We know that there are prime numbers greater than one million. Let N be one of them. Then …

PHIL 110; Spring 2020; Lecture 20 7

We know that someone broke in to the palace on Friday wearing muddy shoes. Let’s call the guy “Smith”. Now Smith must have come in through the garden, and we know from his footprints that he was wearing shoes with a heel, and …

PHIL 110; Spring 2020; Lecture 20 8

Claim: For all x and y, if x is rational and y is rational, then x+y is rational.

Proof: Suppose that u and v are arbitrary rational numbers.

Since u is rational, there are integers x and y such that 𝑢 = 𝑥

𝑦 .

Suppose that a and b are integers with 𝑢 = 𝑎

𝑏 .

Since v is rational, there are integers x and y such that v = 𝑥

𝑦 .

Suppose that c and d are integers with 𝑢 = 𝑐

𝑑 .

Then 𝑢 + 𝑣 = 𝑎

𝑏 +

𝑐

𝑑 =

𝑎𝑑

𝑏𝑑 +

𝑐𝑏

𝑏𝑑 =

𝑎𝑑+𝑐𝑏

𝑏𝑑

Therefore, u+v is rational. 

PHIL 110; Spring 2020; Lecture 20 9

Premise: ∃x Dx Someone is dancing.

Conclusion: Di i is dancing.

Premise: ∃x(Sx & Rx) Some singer is rich.

Conclusion: (Sj & Rj) j is a rich singer.

PHIL 110; Spring 2020; Lecture 20 10

Premise: ∃x(Sx & Rx)

Conclusion: ∃x Sx

(1) ∃x(Sx & Rx) Prem

(2) (Si & Ri) 1, EI

(3) Si 2, Simp

(4) ∃x Sx 3, EG

PHIL 110; Spring 2020; Lecture 20 11

From ∃x Φx, infer Φi, where i is an arbitrary individual name (one that has not occurred either in the symbolization of the argument or on any previous line of the proof.)

PHIL 110; Spring 2020; Lecture 20 12

Premise: ∃x Tx Premise: ∃x Vx Conclusion: ∃x(Tx & Vx)

(1) ∃x Tx Prem (2) ∃x Vx Prem (3) Ti 1, EI (4) Vi 2, EI (5) (Ti & Vi) 3, 4 Conj (6) ∃x(Tx & Vx) 5, EG

PHIL 110; Spring 2020; Lecture 20 13

Premise: ∃x Tx Premise: ∃x Vx Conclusion: ∃x(Tx & Vx)

(1) ∃x Tx Prem (2) ∃x Vx Prem (3) Ti 1, EI (4) Vi 2, EI (5) (Ti & Vi) 3, 4 Conj (6) ∃x(Tx & Vx) 5, EG

PHIL 110; Spring 2020; Lecture 20 14

5: An Exercise To Finish

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

Premise: ∀x(Bx → Fx) (Every bird can fly.)

Conclusion: ∃x(Bx & Fx) (It’s not true that some bird can’t fly.)

Hint: Use reductio ad absurdum.

PHIL 110; Spring 2020; Lecture 20 16