10 Tenth and Eleventh Week

Question 1

Summarize the QE Rules

Answer 1

Question 2

UD: The people at a certain party
Lxy: x loves y.

Explain the difference in meaning between these two statements:
(a) ∀x ∃y Lxy
(b) ∃x ∀y Lxy

Answer 2

(a) means ‘everyone has someone that they love’.
(b) means ‘there’s this one person who loves everyone.’

Question 3

Explain the difference between qualitative and numerical identity.

Answer 3

x and y are ‘qualitatively identical’ if they are exactly alike, or at least very similar. (For example, you might open a box of a dozen hen’s eggs and say that they are qualitatively identical.)

x and y are ‘numerically identical’ if they are not two things, but one. (For example, Eric Blair is numerically identical to George Orwell.)

Question 4

In logic and mathematics, what symbol do we use to express numerical identity?

Answer 4

We use the symbol ‘=’.

For example, ‘7 + 5 = 12’.

Question 5

Describe the introduction rule for ‘=’.

Answer 5

Obviously, everything is identical to itself. So at any point in a natural deduction proof, we can write a = a, or u = u, or c = c, or whatever.

Question 6

Describe the elimination rule for ‘=’.

Answer 6

Question 7

UD: The people at a certain party.
Dx: x is dancing.

Symbolize the statement ‘At least two people are dancing’.

Answer 7

∃x ∃y [(Dx ^ Dy) ^ ~x = y]