PHIL 120A, Summer 2006
July 26
5 points
Use SI to prove (i.e. derive) the following sequent:
Fa ├ -Fa → (x)Fx
If you use a substitution-instance, state what has been substituted for what.
Answer:
1 (1) Fa A
1
(2) - - Fa 1 DN
1
(3) -Fa → (x)Fx 2 SI (S) 51
Sequent 51 -P ├ P → Q
-Fa has been substituted
for P, and (x)Fx has been
substituted for Q
P: -Fa , Q: (x)Fx is the
substitution for line 3
July 27
3 points
Prove the following sequent. If you use a substitution-instance, state what
has been substituted for what.
├ ($x)(Fx v -Fx)
Answer:
(1) Fa v -Fa TI (S) 44
(2) ($x)(Fx v -Fx) 1 EI
Sequent 44
├ P v -P
Fa has been substituted for P
P: Fa is the substitution for line 1
July
31
5 points
Complete the following proof:
Fa ├ -(x)-Fx
1 (1) Fa A
Answer:
2 (2) (x)-Fx A / RAA
2 (3) -Fa 2 UE
1,2 (4) Fa & -Fa 1,3
&I
1,2 (5) -(x)-Fx 2,4 RAA
(Assumption 2 is discharged on line 5)
Aug.
3
3 points
Consider the following assignments of predicates to predicate-letters:
F:
feeds (2-place predicate)
G: is generous
H: is a hamster
I: is an insect
Using those assignments, give two different correct English translations of the following wff:
(x)(Hx → ($y)((Gy & Iy) & Fyx))
Answer:
1. Every hamster is fed by a generous insect.
2. A generous insect feeds every hamster.
Note:
Variants of each sentence are fine, such as "All hamsters are fed by some
generous insect" for the first.
Notice that the given wff is not
ambiguous, but each of the English sentences is ambiguous. Thus, each of
the English sentences could be translated not only by the given wff, but also by this non-ambiguous wff:
($x)((Gx & Ix) & (y)(Hy →
Fxy))
This second wff implies the first one above, but not
vice versa. The second wff claims that some one
particular generous insect feeds every hamster, while the first wff claims that every hamster has some generous insect (not
necessarily the same one for each hamster) that feeds it.
Aug.
4
2 points
Consider the following assignments of predicate-letters and proper names:
m: Tom
H: hates (2-place predicate)
I: individualistic
G: girl
Using those assignments, translate the following English sentence into a wff:
Tom hates an individualistic girl.
Answer:
The quantifier in that sentence, ‘an,’ is ambiguous. There are two major correct answers, with variants also OK.
The sentence could mean that Tom hates every individualistic girl (compare, for example, ‘Tom loves a fast car,’ or ‘Tom loves an adventure’). In that case, the translation would be
(x)((Ix & Gx) → Hmx)
On the other hand, the sentence could mean that Tom hates some particular individualistic girl. In that case, the translation would be
($x)((Ix & Gx) & Hmx)
Aug.
8
4 points
Consider the following wff, which may be translated into English as "Everyone who feeds herself is fed by someone."
(x)(Fxx → ($y)Fyx)
This wff is a theorem. Complete the following proof of the sequent, '├ (x)(Fxx → ($y)Fyx)'.
1 (1) Faa A/CP ├ ($y)Fya
Answer:
1
(2) ($y)Fya
1 EI
1 (3) Faa
→ ($y)Fya
1,2 CP
(4) (x)(Fxx
→ ($y)Fyx) 3 UI
(Assumption 1 is discharged on line 3)
Aug.
10
5 points
Complete the following proof of the given sequent:
($x)x ≠ x ├ Q
1 (1) ($x)x ≠ x A
2 (2) a ≠ a A
/ EE
Answer:
2 (3) a
= a → Q 2 SI (S) 51
(4) a = a =I
2 (5) Q 3,4 MPP
1,2 (6) Q 1,2,5 EE
(Assumption 2 is discharged on line 6)
Substitution
for line 3:
Sequent 51 -P ├ P
→ Q
P: a = a
Q: Q
Aug.
14
2 points
Translate the following English sentence into a wff:
Some hyenas feed only innocent gorillas.
Answer:
There are several different correct answers. Here are a few.
|
($x)(Hx & (y)(Fxy → (Iy & Gy))) |
This says that some hyena is such that everything it feeds is an innocent gorilla. |
|
($x)(Hx & (y)((Fxy & Iy) → Gy)) |
This says that some hyena is such that every innocent thing it feeds is a gorilla. |
|
($x)(Hx & (y)((Fxy & Gy) → Iy)) |
This says that some hyena is such that every gorilla it feeds is innocent. |
Aug. 16
10 points
Fill in the following table about
symbols of the predicate calculus.
|
|
Answers |
|
List 3 proper names that we have used in class |
m n o |
|
List 3 arbitrary names that we have used in class |
a b c |
|
List 3 variables that we have used in class |
x y z |
|
List 5 predicate symbols that we have used in class |
F G H I = |
|
Give examples of 2 different kinds of quantifiers (which may use the same or different variables) |
(x) ($x) |
|
Give an example of an atomic sentence |
Fab |
|
Give an example of a formula of the predicate calculus that is not a wff of the predicate calculus, and also not a formula of the propositional calculus |
Fxy |
|
Give an example of a wff of the predicate calculus that is not a theorem, and also not a wff of the propositional calculus |
(x)Fx |
|
Give an example of a theorem of the predicate calculus that is not a wff of the propositional calculus |
(x)(Fx v –Fx) |
©2006 by Gabriela Remow
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