Philosophy assignment 10 questions

1 9 : I n t r o d u c i n g t h e E x i s t e n t i 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

A Quick Symbolization Exercise

Domain: Animals at a particular zoo.

M ____ is a mammal.

T ____ has a tail.

B ____ is brown.

(a) Every mammal has a tail.

(b) Not every mammal has a tail.

(c) No mammal has a tail.

(d) Only the mammals are brown.

PHIL 110; Spring 2020; Lecture 19 2

1 : I n t r o d u c i n g t h e E x i s t e n t i a l Q u a n t i f i e r

Domain: People at a certain party

D: “____ likes dancing.”

M: “____ likes muffins.”

S: “____ likes swimming.”

• Suppose that there are five people at the party: a, b, c, d, and e.

• Suppose we wish to symbolize the statement “Someone likes dancing”.

• We could write this: ((((Da  Db)  Dc)  Dd)  De)

• But what if the domain is very large?

• This is where the existential quantifier comes in!

PHIL 110; Spring 2020; Lecture 19 4

The Existential Quantifier ∃x

• There is at least one x such that …

• It’s true for some x that …

PHIL 110; Spring 2019; Lecture 17 5

English sentence Sentence in our symbolism

Someone likes dancing. ∃x Dx

Someone likes dancing but not muffins.

∃y (Dy & My)

There’s someone who either likes dancing, or likes both swimming and muffins.

∃z(Dz  (Mz & Sz))

Instances

• Like universal quantifications, existential quantifications have “instances”.

∃x (Mx & Dx) (An existential quantification…)

(Ma & Da) (… and one of its instances.)

• An existential generalization is true just in case one or more of its instances is true.1

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

PHIL 110; Spring 2020; Lecture 19 6

Instances

For example, supposing that the people at the party are Ashni, Ben, Chiara, Deshaun, and Emma, the following two statements have the same truth value:

∃x Mx

Someone likes muffins.

((((Ma  Mb)  Mc)  Md)  Me)

Either Ashni, Ben, Chiara, Deshaun, or Emma likes muffins.

PHIL 110; Spring 2020; Lecture 19 7

Symbolizing I-Statements

It is straightforward to symbolise I statements, using the existential quantifier:

1. There is at least one red fox. ∃x(Rx & Fx)

2. Some foxes are red. ∃x(Rx & Fx)

3. At least one bear lives in Vancouver. ∃x(Bx & Vx)

(You might object that “Some foxes are red” doesn’t mean quite the same thing as “There is at least one red fox.” Don’t worry – we’ll deal with this point later in the term!)

PHIL 110; Spring 2020; Lecture 19 8

Symbolizing O-Statements

It is straightforward to symbolise O statements, using the existential quantifier:

1. There is at least one snake that is not poisonous. ∃x(Sx & Px)

2. Some snakes are not poisonous. ∃x(Sx & Px)

3. Some philosophers are not atheists. ∃x(Px & Ax)

PHIL 110; Spring 2020; Lecture 19 9

A Symbolisation Exercise

Domain: The people at a certain party. D: ____ is dancing. B: ____ is drinking beer. F: ____ is having fun.

(1) Someone is having fun. (2) Some dancer is having fun. (3) At least one dancer is not having fun. (4) Someone is dancing and drinking beer, but not having fun.

∃x(Dx & (Bx & Fx))

PHIL 110; Spring 2020; Lecture 19 10

2 : T h e E G R u l e

The Existential Generalization Rule (EG)

• There are several inference rules associated with the existential quantifier.

• Today, we’ll just look at one of them.

• It’s very simple!

• The EG rule allows us to infer an existential generalization from any instance.

PHIL 110; Spring 2020; Lecture 19 12

The Existential Generalization Rule (EG)

Here are some examples:

Premise: (Da & Fa) (Ashni is dancing and having fun.)

Conclusion: ∃x(Fx & Fx) (Someone is dancing and having fun.)

Premise: (Db & Fb) (Ben is dancing, but not having fun.)

Conclusion: ∃z(Dz & Fz) (Someone is dancing, but not having fun.)

PHIL 110; Spring 2020; Lecture 19 13

Example

The following inference is valid. Establish this, by giving a natural deduction proof.

Premise: Da Ashni is dancing.

Premise: ∀x(Dx → Fx) Every dancer is having fun.

Conclusion: ∃x(Dx & Fx) Some dancer is having fun.

PHIL 110; Spring 2020; Lecture 19 14

1. Da Prem

2. ∀x(Dx → Fx) Prem

3. (Da → Fa) 2, UI

4. Fa 1, 3 MP.

5. (Da & Fa) 1, 4 Conj

6. ∃x(Dx & Fx) 5, EG

PHIL 110; Spring 2020; Lecture 19 15

Exercise

The following inference is valid. Establish this, by giving a natural deduction proof.

Premise: Da Ashni is dancing.

Premise: (Da → Db) If Ashni is dancing, so is Ben.

Premise: Fb Ben is having fun.

Premise: ∀x(Fx → Bx) Everyone who is having fun is drinking beer.

Conclusion: ∃x((Dx & Fx) & Bx) Someone is dancing, having fun, and drinking beer.

PHIL 110; Spring 2020; Lecture 19 16