Philosophy assignment 10 questions

2 1 : Q u a n t i f i e r N e g a t i o n 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

Announcements

• There is a tutorial handout for you to work on. I’ll upload answers tomorrow.

• If you have questions about the tutorial handout, you can post them at Sli.do, using the code S904. There’ll be a new code for next week’s handout.

• Using Sli.do, Duke asks: “I wanna ask what's going to be covered on the final exam? will it cover knowledge points before midterm 1 and midterm 2?”

• The answer to Duke’s question is that the final is cumulative – it will include everything that we’ve covered this term.

• The final online assignment will also go up tomorrow. Good luck with it!

PHIL 110; Spring 2020; Lecture 21 2

Exercise

Show that the following argument is valid, by providing a natural deduction proof:

Premise: ∃x Bx

Premise: ∀x (Bx → Mx)

Conclusion: ∃x Mx

PHIL 110; Spring 2020; Lecture 21 3

1 : T h e Q N R u l e

These two statements are equivalent:

• Not everyone is having a good time.

• Someone’s not having a good time.

In symbols:

• ∀x Gx

• ∃x Gx

PHIL 110; Spring 2020; Lecture 21 5

Similarly, these two statements are equivalent:

• It’s not true that someone is married.

• Everyone is unmarried.

In symbols:

• ∃x Mx

• ∀x Mx

PHIL 110; Spring 2020; Lecture 21 6

The Quantifier Negation Rule

(1) From ∀x Φx, derive ∃x Φx, and vice versa.

(2) From ∃x Φx, derive ∀x Φx, and vice versa.

PHIL 110; Spring 2020; Lecture 21 7

2 : T h e S q u a r e o f O p p o s i t i o n

The Square of Opposition

• We say that two statements are “contradictories” if it’s not possible for both to be true, and not possible for both to be false.

• You can show that p and q are contradictories by proving q from p, and vice versa.

• It’s a useful fact to remember that every A-statement is contradictory to the corresponding O statement, and …

• … every E statement is contradictory to the corresponding I statement.

• These points are traditionally represented on a diagram, the “square of opposition”.

• https://plato.stanford.edu/entries/square/

PHIL 110; Spring 2020; Lecture 21 9

A All A are B.

E No A are B.

I Some A are B.

O Some A are not B.

PHIL 110; Spring 2020; Lecture 21 10

Show that the following statements are equivalent, using natural deduction proofs:

Not all A are B.

Some A are not B.

PHIL 110; Spring 2020; Lecture 21 11

First Worked example

1. ∀x(Ax → Bx) Prem

2. ∃x (Ax → Bx) 1, QN

3. ∃x (Ax  Bx) 2, MI

4. ∃x (Ax & Bx) 3, DM

5. ∃x(Ax & Bx) 4, DN

PHIL 110; Spring 2020; Lecture 21 12

First Worked example

1. ∃x(Ax & Bx) Prem

2. ∃x (Ax & Bx) 1, DN

3. ∃x (Ax  Bx) 2, DM

4. ∃x (Ax → Bx) 3, MI

5. ∀x(Ax → Bx) 4, QN

PHIL 110; Spring 2020; Lecture 21 13

First Worked example

Show that the following statements are equivalent, using natural deduction proofs:

It is not true that some A are B.

No A are B.

PHIL 110; Spring 2020; Lecture 21 14

Second Worked example

Show that the following statements are equivalent, using natural deduction proofs:

It is not true that some A are B. ∃x(Ax & Bx)

No A are B. ∀x(Ax → Bx)

PHIL 110; Spring 2020; Lecture 21 15

Second Worked example

1. ∃x(Ax & Bx) Premise

2. ∀x (Ax & Bx) 1, QN

3. ∀x (Ax  Bx) 2, DM

4. ∀x(Ax → Bx) 3, MI

PHIL 110; Spring 2020; Lecture 21 16

Second Worked example

3 : E x e r c i s e s

An Exercise to Finish

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

Premise: ∃x Bx

Premise: ∃x (Bx & Mx)

Conclusion: ∃x Mx

PHIL 110; Spring 2020; Lecture 21 18

An Exercise to Finish

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

Premise: ∀x (Bx → Mx)

Premise: ∀x Mx

Conclusion: ∀x Bx

PHIL 110; Spring 2020; Lecture 21 19