LPS/PHILOS 30 HOMEWORK 2
DUE MONDAY FEBRUARY 6 BY 5PM
Name:
Student ID number:
Note 1. It is permissible and you are encouraged to work in groups of 2-3. Please indicate
here the names of the other 1-2 students with whom you worked on this assignment:
Name 1:
Name 2:
Note 2. To complete this homework, please print it out and fill out the answers by hand
with pen or pencil. Then scan the homework and submit via canvas in the form of a single
scanned pdf. If homework is not turned in the form of a single scanned pdf, it will be marked
off 10%. If homework is turned in upside down, or sideways, it will be marked off 10%. The
reason for this is that we are using canvas to grade, and it slows down the grading process
tremendously if the homework is turned in as multiple separate pdfs, or turned in upside, or
sideways.
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(1) (10 points) Translate the following sentences into predicate logic. Preserve as much
of the structure as possible, and in each case, give the key. (For examples of problems
like this, see Gamut vol. 1 Exercise 1 p. 69 with solutions on pp. 241-242). It is
okay to assume that you are using a =Anne, b =Bill, and c =Claire in the key, so
that you do not have to explicitly say this in your key; although if you deviate from
this assumption, please indicate as much in your key.
(a) Anne trusts Claire and Claire trusts Bill. But Anne does not trust Bill.
(b) Anne is better at chess than Bill only if Anne is better at chess than Claire. But
Anne is not better at chess than Claire. So, Anne is not better at chess than
Bill.
(c) Anne trusts Bill if and only if Bill trusts Anne. But Bill does not trust Anne.
Therefore, Anne does not trust Bill.
(d) Anne admires Claire and Bill admires Claire. But Claire does not admire herself.
(e) Anne does not respect Bill and she does not trust Bill. Claire does not trust
Anne, but she respects Anne.
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(2) (10 points) This problem is the converse to the previous problem. For each of the
following sentences of predicate logic, choose a key and use the key to translate the
sentences into English. It is okay to assume that you are using a =Anne, b =Bill,
and c =Claire in the key, so that you do not have to explicitly say this in your key;
although if you deviate from this assumption, please indicate as much in your key.
(a) (Rac ∧ Rcb) → Rab
(b) ¬((Rab ∧ Rba) → Raa)
(c) ¬(Lba ↔ Lab).
(d) Rab → ¬Raa
(e) (F a → Lab) → F b
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(3) (10 points) Translate the following sentences into predicate logic. Preserve as much
of the structure as possible and give in each case the key. (For examples of problems
like this, see Gamut vol. 1 Exercise 2 p. 74 with solutions on p. 242).
(a) Some hockey players are baseball players. Every baseball player is patient.
Therefore, some hockey players are patient.
(b) All chess club members are diligent. Some karate club members are diligent.
(c) Every book club member is not humble. No book club members are karate club
members. But some karate club members are not humble.
(d) Some golf club members are baseball players. No baseball players are rash.
Therefore, some golf club members are not rash.
(e) No math club members are kind. But some chess club members are kind. But
all chess club members are math club members.
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(4) (10 points) Translate the following sentences into predicate logic. Preserve as much
of the structure as possible and give in each case the key. (For examples of problems
like this, see Gamut vol. 1 Exercise 5 pp. 82-83 with solutions on pp. 243-246).
(a) All students vote. Bill is a student. Therefore, Bill votes.
(b) If every student works hard, then Anne works hard. But Anne does not work
hard. Therefore, some students do not work hard.
(c) If Claire trusts Anne, then Claire trusts someone. If Claire trusts Anne, then
someone trusts Anne.
(d) No one respects Bill. Therefore, Anne does not respect Bill.
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(5) (10 points) Translate the following six sentences into predicate logic, using key Lxy =
x loathes y, and N z = z is nice. For help on problems like this, see Gamut vol. 1
Exercise 5 parts vii, viii, xxvi, xxviii on pp. 83, 243-246.
(a) Everyone loathes someone.
(b) Everyone loathes someone who is nice.
(c) Someone loathes everyone.
(d) Someone loathes everyone who is nice.
(e) Everyone who is not nice loathes someone who is nice.
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(6) (10 points) This problem is about sets and set notation. Determine whether the
following are true or false. You do not need to provide any justification for your
answer. You might consider reviewing Gamut vol. 1 § 3.5 pp. 83 ff prior to attempting
this problem. In this problem, the symbol N denotes the set of natural numbers
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .}.
(a) 4 ∈ {1, 2, 3, 5}
(b) {1, 2, 6, 7} = {1, 6, 3, 7}
(c) {5, 7, 6, 8} ⊆ {0, 1, 8, 4, 3, 5, 6, 7}
(d) {1, 6, 4, 3, 5, 2, 7} = {x ∈ N : 0 < x < 8}
(e) {x ∈ N : x > 3} = {x ∈ N : x ≥ 4}
(f) ∅ = {x ∈ N : 0 < x + x < x}
(g) ∅ = {x ∈ N : x + 5 = 10}
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(7) (10 points) Consider the language L which has individual constants p, a, s for “protagonist,” “antagonist” and “sidekick” respectively, as well as a unary predicate for U x
for “x has superpowers” and a binary predicate M xy for “x is more popular than y”.
(a) Consider the sentence: “The sidekick is more popular than the protagonist.”
Translate this sentence into predicate logic, using the language L as the key.
(b) Consider the sentence: “Someone who has superpowers is not more popular than
the sidekick.” Translate this sentence into predicate logic, using the language L
as the key.
(c) Consider the sentence “Someone who does not have superpowers is more popular
than the antagonist.” Translate this sentence into predicate logic, using the
language L as the key.
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(8) (10 points) Anne, Bill, and Claire are actors, all of whom act together in two distinct
television series.
In the television series Spiderman: The Lost Years, Anne is the antagonist, Bill
is the protagonist, and Claire is the sidekick. Bill has superpowers but Anne
and Claire do not. Further, Bill is more popular than Claire who in turn is more
popular than Anne.
In the television series Batman vs. Jessica Jones, Claire is the protagonist, Bill
is the antagonist, and Anne is the sidekick. Claire and Anne have superpowers
but Bill does not. Anne is more popular than Claire, who in turn is more popular
than Bill.
Fill in each cell of the table with “true” or “false”, depending on whether the sentence
in question is true or false in the model in question. You do not need to justify your
answers.
Hint: consider using the previous exercise to help you fill out the cells of the table.
Sentence
The sidekick is more popular
than the protagonist
Someone who has superpowers is
not more popular than the sidekick
Someone who does not have superpowers
is more popular than the antagonist
Spiderman: The Lost Years Batman vs. Jessica Jones
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(9) (10 points) This problem asks you to draw Venn diagram models like in Lecture 11
which make certain sentences true. The correct solution to the problem consists
merely in a digram, and you do not need to show or prove or say why the sentence
is true on the model.
(a) Draw a Venn diagram model in which the following sentence is true:
(∀ x Bx → Ax) ∧ ∀ x Ax → Cx)
(b) Draw a Venn diagram model in which the following sentence is true:
(∀ x Cx → Ax) ∧ (∃ x Bx ∧ Ax) ∧ ¬(∃ x Bx ∧ Cx)
(c) Draw a Venn diagram model in which the following sentence is true:
(∀ x Ax → (Bx ∨ Cx)) ∧ (∃ x Ax ∧ Bx) ∧ (∃ x Ax ∧ Cx).
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(10) (10 points) This problem asks you to draw arrow models like in Lecture 11 which
make certain sentences true. The correct solution to the problem consists merely in
a digram, and you do not need to show or prove or say why the sentence is true on
the model.
(a) Draw an arrow model with at least four points in which the following sentence
is true:
(∀ x ¬Rxx) ∧ (∀ x ∃ y Rxy)
(b) Draw an arrow model with at least four points in which the following sentence
is true:
(∃ y ∀ x Rxy) ∧ (∀ x ∀ y Rxy → Ryx)
(c) Draw an arrow model with at least four points in which the following sentence
is true:
∀ x ∀ y ∀ z ((Rxy ∧ Ryz) → Rxz)
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