Problem Set 1
Each Question is Worth 1 Mark
1. PL Translations
1.1.
Using the glossary:
A: Aristotle was an ancient Greek philosopher
B: Brentano was an influential German philosopher
C: Albert Camus was a French philosopher
D: John Dewey was an American philosopher
What is the best translation of: ((D ↔ ¬B) ∧ (C → A)) into english?
[1] John Dewey was an American philosopher if and only if Brentano was an influential German philosopher, and either Albert Camus was a French philosopher or
Aristotle was an ancient Greek philosopher.
[2] John Dewey was an American philosopher if and only if Brentano was an influential German philosopher, and if it is false that Albert Camus was a French
philosopher, then Aristotle was an ancient Greek philosopher.
[3] John Dewey was an American philosopher if and only if it is false that Brentano
was an influential German philosopher, and if Albert Camus was a French philosopher then Aristotle was an ancient Greek philosopher.
[4] John Dewey was an American philosopher if and only if Brentano was an influential German philosopher, and if Albert Camus was a French philosopher then
Aristotle was an ancient Greek philosopher.
1.2.
Using the glossary:
A: Aristotle was an ancient Greek philosopher
B: Brentano was an influential German philosopher
C: Albert Camus was a French philosopher
D: John Dewey was an American philosopher
1
What is the best translation of:
If Albert Camus was a French philosopher then John Dewey was an American philosopher, but Aristotle was an ancient Greek philosopher just in case it is false that Brentano
was an influential German philosopher.
Into PL?
[1] ((C ∧ (D ∧ (A ∧ ¬B)))
[2] ((C → D) ↔ ¬(A ∨ ¬B))
[3] ((C → D) ∧ (A ↔ ¬B))
[4] ((C ∧ D) ↔ (A ∨ ¬B))
1.3.
Using the glossary:
A: Axel is late for the show
B: Barry holds a note
C: Celine’s heart will go on
D: Dylan plays in the key of F
E: Elvis leaves the building
F: Freddie looks up to the skies
What is the best translation of: (((F → E) ∨ B) ↔ (C ∧ (D ∧ ¬A))) into english?
[1] Either it is the case that Freddie looks up to the skies if and only if Elvis leaves
the building, or it is the case that Barry holds a note, just in case Celine’s heart
will go on and Dylan plays in the key of F, but it is not the case that Axel is late
for the show.
[2] Either it is the case that if Freddie looks up to the skies then Elvis leaves the
building, or it is the case that Barry holds a note, just in case Celine’s heart will
go on and Dylan plays in the key of F, but it is not the case that Axel is late for
the show.
[3] Either it is the case that if Freddie looks up to the skies then Elvis leaves the
building, or it is the case that Barry holds a note, and Celine’s heart will go on and
Dylan plays in the key of F, but it is not the case that Axel is late for the show.
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[4] Either it is the case that if Freddie looks up to the skies then Elvis leaves the
building, or it is the case that Barry holds a note, just in case Celine’s heart will
go on and Dylan plays in the key of F, and Axel is late for the show.
1.4.
Using the glossary:
A: Axel is late for the show
B: Barry holds a note
C: Celine’s heart will go on
D: Dylan plays in the key of F
E: Elvis leaves the building
F: Freddie looks up to the skies
Either Freddie looks up to the skies or Barry holds a note just in case Celine’s heart will
go, but Elvis leaves the building and Axel is late for the show just in case it is false that
Celine’s heart will go on.
Into PL?
[1] (F ∨ ((B ↔ C) ∧ (E ∧ (A ↔ ¬C))))
[2] (F ∨ ((B ↔ C) ∨ (E ∧ (A ↔ ¬C))))
[3] (F ∨ ((B ↔ C) ∧ (E ∧ (A ∧ ¬C))))
[4] (F ∨ ((B → C) ∨ (E ∧ (A ↔ ¬C))))
1.5.
Using the glossary:
A: Axel is late for the show
B: Barry holds a note
C: Celine’s heart will go on
D: Dylan plays in the key of F
E: Elvis leaves the building
F: Freddie looks up to the skies
If Barry holds a note and Elvis leaves the building, but it is false that Celine’s heart will
go on, then it is false that Dylan plays in the key of F and either Axel is late for the show
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or Freddie looks up to the skies
Into PL?
[1] ((B ∧ (E ∧ ¬C)) → (¬D ∧ (A ∨ F)))
[2] ((B ∧ (E ∧ ¬C)) → (¬D ∧ (A ∧ ¬F)))
[3] ((¬B ∧ (¬E ∧ ¬C)) → (¬D ∧ (A ∨ F)))
[4] ((B ↔ (E ∧ ¬C)) → (¬D ∧ (A ∨ F)))
2. Testing Arguments
Translate the following arguments into PL and then assess them for validity.
2.1.
Einsteinium is a synthetic element. Fermium is a synthetic element. If either Einsteinium is a synthetic element or Fermium is a synthetic element, then the experiment
will be a success. So, the experiment will be a success.
[1] Valid
[2] Invalid
[3] Neither valid nor invalid (i.e., it is impossible to determine whether the argument
is valid or invalid)
[4] Both valid and invalid (i.e., there is a sense in which it is valid and a sense in
which it is invalid)
2.2.
Murder is Wrong. If Murder is wrong, then there are moral facts. If there are moral
facts then god exists. If God exists then murder is wrong. So, there are moral facts.
[1] Valid
[2] Invalid
[3] Neither valid nor invalid (i.e., it is impossible to determine whether the argument
is valid or invalid)
[4] Both valid and invalid (i.e., there is a sense in which it is valid and a sense in
which it is invalid)
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2.3.
It is impossible to change the past. It is impossible to change the laws of nature. If it
is impossible to change past and it is impossible to change the laws of nature, then it
follows that it is impossible that you could have done otherwise. Therefore, free will is
an illusion.
[1] Valid
[2] Invalid
[3] Neither valid nor invalid (i.e., it is impossible to determine whether the argument
is valid or invalid)
[4] Both valid and invalid (i.e., there is a sense in which it is valid and a sense in
which it is invalid)
2.4.
Either consciousness emerges from non-conscious matter or matter is conscious. It is
false that consciousness emerges from non-conscious matter. If matter is conscious,
then trees are conscious. Therefore, matter is conscious just in case trees are conscious.
[1] Valid
[2] Invalid
[3] Neither valid nor invalid (i.e., it is impossible to determine whether the argument
is valid or invalid)
[4] Both valid and invalid (i.e., there is a sense in which it is valid and a sense in
which it is invalid)
2.5.
Mary knows she has hands. If Mary knows she has hands, then it is false that Mary is
a brain in a vat. Mary knows she has hands if and only if there is an external world. If
there is an external world, then Mary knows she has hands. Either it is false that Mary
knows she has hands, or it is true that Mary is a brain in a vat. So, there is an external
world.
[1] Valid
[2] Invalid
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[3] Neither valid nor invalid (i.e., it is impossible to determine whether the argument
is valid or invalid)
[4] Both valid and invalid (i.e., there is a sense in which it is valid and a sense in
which it is invalid)
2.6.
If X is analytically equivalent to good, then the question “Is it true that X is good?”
is meaningless. The question “Is it true that X is good?” is not meaningless. X is
analytically equivalent to good, just in case there are moral facts. It is false that there
are moral facts. So, it is false that X is analytically equivalent to good.
[1] Valid
[2] Invalid
[3] Neither valid nor invalid (i.e., it is impossible to determine whether the argument
is valid or invalid)
[4] Both valid and invalid (i.e., there is a sense in which it is valid and a sense in
which it is invalid)
2.7.
Roses are Red. Violets are Blue. Sugar is Sweet. If Sugar is sweet then consuming
sugar-sweetened beverages can raise blood pressure. Roses are Red just in case Roses
have thorns. Either Violets are Blue or consuming sugar-sweetened beverages can raise
blood pressure. If consuming sugar-sweetened beverages can raise blood pressure, then
we ought to drink more water. So, we ought to drink more water.
[1] Valid
[2] Invalid
[3] Neither valid nor invalid (i.e., it is impossible to determine whether the argument
is valid or invalid)
[4] Both valid and invalid (i.e., there is a sense in which it is valid and a sense in
which it is invalid)
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2.8.
The universe is expanding. If the universe is expanding, then the universe had a beginning. If the universe had a beginning then the Big Bang theory is the most plausible
account of the origin of the universe. If the Big Bang theory is the most plausible account of the origin of the universe, then the Big Crunch is one possible scenario for the
ultimate fate of the universe. If the Big Crunch is one possible scenario for the ultimate
fate of the universe, then time is running out. Time is not running out. So, it is false that
the Big Bang theory is the most plausible account of the origin of the universe.
[1] Valid
[2] Invalid
[3] Neither valid nor invalid (i.e., it is impossible to determine whether the argument
is valid or invalid)
[4] Both valid and invalid (i.e., there is a sense in which it is valid and a sense in
which it is invalid)
2.9.
Either Aliens built the pyramids or the Ancient Egyptians built the pyramids. If Aliens
built the pyramids, then there would be evidence that the Aliens built the pyramids.
Ancient Egyptians built the pyramids just in case the evidence shows that that thousands
of labourers transported 170,000 tons of limestone along the River Nile in wooden boats.
So, Ancient Egyptians built the pyramids.
[1] Valid
[2] Invalid
[3] Neither valid nor invalid (i.e., it is impossible to determine whether the argument
is valid or invalid)
[4] Both valid and invalid (i.e., there is a sense in which it is valid and a sense in
which it is invalid)
2.10.
Kenya is a country in Africa. If Kenya is bordered by Tanzania to the South, then Kenya
is bordered by Uganda to the west. Kenya is a country in Africa just in case Kenya is
bordered by Tanzania to the South. Either Kenya is bordered by Tanzania to the South
or Kenya is bordered by Algeria to the North. So, Kenya is bordered by Tanzania to the
South just in case it is false that Kenya is a country in Africa.
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[1] Valid
[2] Invalid
[3] Neither valid nor invalid (i.e., it is impossible to determine whether the argument
is valid or invalid)
[4] Both valid and invalid (i.e., there is a sense in which it is valid and a sense in
which it is invalid)
3. PL Categorisations
Consider the truth-tables for the following two new connectives:
α
T
T
T
T
F
F
F
F
β
T
T
F
F
T
T
F
F
χ
T
F
T
F
T
F
T
F
χ
α ] β α β
T
T
T
F
F
F
F
F
F
F
F
F
F
F
F
T
‘]’ is a two-place connective: it is a function of the truth-values of two propositions.
‘’ is a three-place connective. It is a function of the truth-values of three propositions.
With that in mind, answer each of the following questions:
3.1.
Consider the following sentence of PL:
((¬(A ∧ ¬B) ∧ (A ] B)) ↔ ¬(B → ¬A))
Using a truth table, determine whether it is:
[1] A tautology.
[2] A contradiction.
[3] Neither a tautology nor a contradiction.
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3.2.
Consider the following sentence of PL:
C
((A ] C) → ¬((A B ) ∨ (¬B ∨ A)))
Using a truth table, determine whether it is:
[1] A tautology.
[2] A contradiction.
[3] Neither a tautology nor a contradiction.
3.3.
Consider the following sentence of PL:
C
((¬A ↔ (B → ¬C)) ] ((¬A ∨ C) ∨ (A B )))
Using a truth table, determine whether it is:
[1] A tautology.
[2] A contradiction.
[3] Neither a tautology nor a contradiction.
3.4.
Consider the following sentence of PL:
C
((¬B ] (¬C ∧ (A B ))) ↔ (¬B ↔ ¬C))
Using a truth table, determine whether it is:
[1] A tautology.
[2] A contradiction.
[3] Neither a tautology nor a contradiction.
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3.5.
Consider the following sentence of PL:
C
C
C
(((A B ) ↔ (A B )) ∨ ((A B ) → ¬(¬A ∧ B)))
Using a truth table, determine whether it is:
[1] A tautology.
[2] A contradiction.
[3] Neither a tautology nor a contradiction.
3.6.
Consider the following sentence of PL:
C
((((A B ) ∨ A) ] B) ∧ ((C ∧ ¬B) ∧ ¬(A → ¬B)))
Using a truth table, determine whether it is:
[1] A tautology.
[2] A contradiction.
[3] Neither a tautology nor a contradiction.
3.7.
Consider the following sentence of PL:
C
(¬(¬A ∨ (((A B ) ∧ A) ] (A ∧ ¬C))) ∨ ¬((¬C ↔ B) ∧ (¬A ∧ A)))
Using a truth table, determine whether it is:
[1] A tautology.
[2] A contradiction.
[3] Neither a tautology nor a contradiction.
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3.8.
Consider the following TWO sentences of PL:
C
((A ] C) ∨ (¬B ] ((A B ) → C)))
(C ∧ (¬A ↔ ¬(B ∨ A)))
Using a truth table, determine whether these two sentences are:
[1] Equivalent
[2] Contradictories
[3] Contraries
[4] None of the above
3.9.
Consider the following TWO sentences of PL:
C
((A B ) ∧ (A ∧ (¬B ] B)))
(A → ¬(C ∧ (C ∧ B)))
Using a truth table, determine whether these two sentences are:
[1] Equivalent
[2] Contradictories
[3] Contraries
[4] None of the above
3.10.
Consider the following TWO sentences of PL:
C
((A B ) ∧ (A → (¬A ] ¬B)))
((¬(¬A ] C) ∧ ¬A) ↔ B)
Using a truth table, determine whether these two sentences are:
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[1] Equivalent
[2] Contradictories
[3] Contraries
[4] None of the above
4. Truth Trees
In what follows, you will need to apply the following two rules to any instances of ‘r’,
C
e.g., (A r B ):
χ
((α rβ ))
α
β
χ
χ
¬(α rβ )
¬α
¬χ
¬β
4.1.
(C ↔ B)
C
(¬(A r B ) ∧ (A ∨ ¬B))
————
C
¬(¬A ∧ ¬(A r B ))
Using a truth tree, determine whether the argument is:
[1] Invalid, A = T, B = T, C = F
[2] Invalid, A = F, B = F, C = F
[3] Invalid, A = T, B = F, C = F
[4] Valid
4.2.
(A ∨ ¬¬(¬A ∧ C))
C
(C ↔ (A r B ))
————
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C
¬(A r B )
Using a truth tree, determine whether the argument is:
[1] Invalid, A = T, B = F, C = T
[2] Invalid, A = F, B = T, C = F
[3] Invalid, A = T, B = T, C = T
[4] Valid
4.3.
C
((A r B ) ∧ ¬(B ↔ D))
(¬(¬C ∨ B) ∧ (¬C ∨ D))
————
C
(A ∨ ¬(A r B ))
Using a truth tree, determine whether the argument is:
[1] Invalid, A = T, B = T, C = T, D = F
[2] Invalid, A = F, B = T, C = F, D = T
[3] Invalid, A = F, B = F, C = F, D = F
[4] Valid
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4.4.
¬(A ∧ (C ∨ ¬(¬A ∨ ¬D)))
(C ∨ (¬D ∨ ¬(A ↔ B)))
————
C
¬((A r B ) ∧ ¬(D ∧ A))
Using a truth tree, determine whether the argument is:
[1] Invalid, A = T, B = F, C = F, D = T
[2] Invalid, A = T, B = F, C = F, D = T
[3] Invalid, A = T, B = T, C = T, D = F
[4] Valid
4.5.
¬(A ∧ (C ∨ ¬(¬A ∨ ¬D)))
C
(D → ¬((¬A ∨ A) ∧ (A r B )))
(¬B ∨ (¬A ∨ D))
¬(A → B)
C
(¬A ∧ ¬(A r B ))
————
(C ∨ (¬D ∨ ¬(A ↔ B)))
Using a truth tree, determine whether the argument is:
[1] Invalid, A = F, B = T, C = T, D = F
[2] Invalid, A = F, B = T, C = F, D = T
[3] Invalid, A = F, B = T, C = T, D = F
[4] Valid
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5. Understanding Conditionals
IMPORTANT: In each of the questions below, consider every possible way
of flipping the cards in which you successfully rule it out that the conditional is false. There is some number of cards n such that at least n cards
are flipped over in every such possibility. The number n is the minimum
number of cards you would need to flip over in order to rule it out that a
given conditional is false. Note that all cards are flipped at the same time
and you only get one chance to flip the cards. Note also that this is not a
normal deck of cards. A diamond or a heart can be black and a spade
or a club can be red.
5.1.
Consider the following five cards. Each card has a suit on one side (diamond, heart,
spade, club) and a number on the other side. Now consider the following conditional:
(1C) If a card has a spade on one side, then it has a three on the other side.
r
♠
5
6
_
What is the minimum number of cards you would need to turn over in order to rule it
out that (1C) is false?
[1] 1 card
[2] 2 cards
[3] 3 cards
[4] 4 cards
[5] 5 cards
[6] 0 cards
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5.2.
Consider the following ten cards. Each card has a suit on one side (diamond, heart,
spade, club) and a number on the other side. Now consider the following pair of conditionals:
(1C) If a card has a three on one side, then it has a diamond on the other side.
(2C) If a card has a three on one side, then the card to the right of that card either has a
four on one side and a heart on the other side or a seven on one side and a spade
on the other side.
_
6
7
♠
r
♠
_
6
5
7
What is the minimum number of cards you would need to turn over in order to rule it
out that both conditionals are false?
[1] 1 card
[6] 6 cards
[2] 2 cards
[7] 7 cards
[3] 3 cards
[8] 8 cards
[4] 4 cards
[9] 9 cards
[5] 5 cards
[10] 0 cards
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5.3.
Consider the following ten cards. Each card has a suit on one side (diamond, heart,
spade, club) and a number on the other side. Now consider the following pair of conditionals:
(1C) If a card has a heart on one side, then it has a three on the other side.
(2C) If a card has a heart on one side, then the card to the left of that card either has a
six on one side or a seven on one side.
_
6
7
♠
5
♠
_
♠
5
7
What is the minimum number of cards you would need to turn over in order to rule it
out that both conditionals are false?
[1] 1 card
[6] 6 cards
[2] 2 cards
[7] 7 cards
[3] 3 cards
[8] 8 cards
[4] 4 cards
[9] 9 cards
[5] 5 cards
[10] 0 cards
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5.4.
Consider the following ten cards. Each card has a suit on one side (diamond, heart,
spade, club) and a number on the other side. Now consider the following conditional:
(1C) If a card – call it ‘C’ – has a heart on it, then the card to the left of C has a five on
it only if C has a four on its other side.
_
r
6
r
2
r
_
5
6
7
What is the minimum number of cards you would need to turn over in order to rule it
out that (1C) is false?
[1] 1 card
[6] 6 cards
[2] 2 cards
[7] 7 cards
[3] 3 cards
[8] 8 cards
[4] 4 cards
[9] 9 cards
[5] 5 cards
[10] 0 cards
5.5.
Consider the following ten cards. Each card has a suit on one side (diamond, heart,
spade, club) and a number on the other side. Now consider the following conditional:
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(1C) If a card – call it ‘C’ – has a heart on it, then the card to the right of C has a seven
on it only if the card to the left of C has a four on it.
r
6
_
r
2
7
_
r
_
6
What is the minimum number of cards you would need to turn over in order to rule it
out that (1C) is false?
[1] 1 card
[6] 6 cards
[2] 2 cards
[7] 7 cards
[3] 3 cards
[8] 8 cards
[4] 4 cards
[9] 9 cards
[5] 5 cards
[10] 0 cards
6. Section 6: Sad Sally
Sally has recently completed PHIL2002, but she wasn’t completely satisfied with propositional logic. Sally believes that there are valid arguments that cannot be expressed
using PL. So, Sally seeks to extend the logic. She does this by introducing three new
connectives: ‘?’, ‘’ and ‘^’. She defines these connectives via their truth-tables, which
are set out as follows:
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α
T
T
F
F
β α?β
T
F
F
F
T
F
F
T
αβ
F
F
T
T
α^β
T
T
F
F
6.1.
Sally wants to use truth trees, but current tree rules don’t apply to her new connectives.
Which of the following rules should she add so as to handle (α^β)?
(α^β)
A
[1]
(α^β)
×
(α^β)
B A
(α^β)
A
B
B
[4]
¬A
[3]
[4]
6.2.
And which rule should she add so as to handle ¬(α^β)?
¬(α^β)
¬A
[1]
¬(α^β)
¬(α^β)
A
B
× ×
B
[2]
[3]
Help Sally apply her new logic.
6.3.
((B ? G)^C)
(C ∧ ¬(¬CD))
(H → (G ∧ (¬A ∨ F)))
———————
(A ↔ ((A ? D) ∧ E))
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¬(α^β)
A
[4]
What are the values for A, B, C, D, E, F, G and H on an open branch of the truth tree for
this argument?
[1] Invalid, A = T, B = F, C = T, D = T, E = F, F=T, G = T, H = T
[2] Invalid, A = T, B = F, C = T, D = F, E = F, F=T, G = T, H = T
[3] Invalid, A = T, B = F, C = T, D = T, E = F, F=T, G = T, H = F
[4] Invalid, A = F, B = F, C = T, D = T, E = F, F=T, G = T ...

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