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Sample Final

Note: Bring Blue Books to the Final!

Part I

Determine the truth value of the following sentences in the model given.

UD = {apple, banana, cherry, melon}
extension (A) = {banana, cherry, melon}
extension (K) = {cherry, banana}
extension (L) = {apple, melon}
extension (G) = {<apple, banana>, <apple, cherry>, <apple, melon>}
referent (a) = apple
referent (b) = banana
referent (c) = cherry
referent (m) = melon

1. (x) (Kx ~Lx)
2. (Gaa Ka) & (Gab Lb)
3. (Ac La) & (La Ac)
4. (x) Rax
5. (x) (Ax & Lx)
6. Lc & (x) (Kx v Lx)
7. (x) (y) (Gxy & xy)

Part II

Symbolize and give a proof for each of the arguments below. Give one symbolization key for 1-2, another for 3-5.

1. Every dachshund is a dog.
Every dog will have its day.
Therefore, every dachshund will have its day.

2. Every dog will have its day.
Something will not have its day.
Therefore, something is not a dog.

3. Either someone will laugh or someone will dance.
If someone laughs, Alice will dance.
Therefore, someone will dance.

4. Exactly one person will dance.
Alice will dance.
Miss Jones will dance.
Therefore, Alice is Miss Jones.

5. The dancer will laugh if and only if everyone will laugh.
No one will dance.
Therefore, not everyone will laugh.

[Hint: Questions 4 and 5 are there to remind you about quantity and definite descriptions. In addition to the particular examples, you should understand these things and how to symbolize them.]

Part III

Show each of the following by reasoning about possible models.

1. (x) (Bx & Cx) and (x) Bx & (x) Cx are quantificationally equivalent.

2. (x) Rxc (x) (y) Rxy

3. (x) Px (x) Px is a logical truth (tautology).

4. (A B) v (B A) is a logical truth (tautology) for all sentences A and B.

5. The following set of sentences is quantificationally inconsistent:

{ (x) (y) x=y, (x)Px, (x)~Px }

Part IV

Show each of the following by constructing models or interpretations.

1. (x) (Rax v Rxa) is quantificationally indeterminate.

2. (x) Rxx is quantificationally indeterminate.

3. (x) (y) (Rxy Ryx) is quantificationally indeterminate.

4. (x) (y) (Hx x=y) is quantificationally indeterminate.

5. (x) Rxa and (x) Rxa are not quantificationally equivalent.

6. (x) [Kx & (y) (Ky x=y)] and (x) (y) (Ky x=y) are not quantificationally equivalent.

7. Show that the following set is quantificationally consistent:

{(x)Rxx, (x) (y)Rxy}