Philosophy assignment 10 questions
Question One (3 points)
Which of the following three inferences are valid? (There is no need to explain your answers.)
(a) Premise: Liu Yang spends two hours every day playing the violin.
Conclusion: Liu Yang wants to be a good violinist.
(b) Premise: If Liu Yang is in class, she is on campus.
Premise: Liu Yang is on campus.
Conclusion: Liu Yang is in class.
(c) Premise: If Liu Yang is in class, she is on campus.
Premise: Liu Yang is in class.
Conclusion: Liu Yang is on campus.
Question Two (2 points)
Consider the following argument:
If God is both omnipotent and loving, then His creatures never suffer. But it just isn’t true
that God’s creatures never suffer (just look around!) so it is not true that God is both
omnipotent and loving. But we know for sure that God is loving. That is certain.
Therefore, God is not omnipotent. The conventional wisdom is wrong on this point.
Identify two inference rules that are used in the argument.
Question Three (2 points)
Juan has shown that a certain inference is valid, using a truth table. Ella is trying to show that the
inference is valid by giving a natural deduction proof. Do you think it’s possible for Ella to find a proof of
the inference? Briefly explain your answer.
Question Four (2 points)
Zeynep is asked to prove the following statement:
The sum of the internal angles in a pentagon is 540°.
Zeynep responds by carefully drawing a number of pentagons, and measuring their internal angles. She
confirms that, in each case, the sum of the angles is 540°.
Has Zeynep proved the statement? Briefly explain your answer.
Question Five (10 points)
Consider the following argument:
Premise One: Either Ashni or Ben attended the party.
Premise Two: If Ashni attended the party, it was a great success.
Premise Three: Ben didn’t attend the party, if it wasn’t a great success.
Conclusion: The party was a great success.
Symbolize the argument, using the following abbreviations:
A: Ashni attended the party.
B: Ben attended the party.
S: The party was a great success.
Is the argument valid? Justify your answer in detail.
Question Six (7 points)
Exactly one of these two inferences is valid. Give a natural deduction proof of the valid inference.
(1) Premise: ∀x(Ax Bx)
Premise: ∃x Ax
Conclusion: ∃x Bx
(2) Premise: ∀x(Ax Bx)
Premise: ∃x Bx
Conclusion: ∃x Ax
Question Seven (2 points)
Identify an inference rule that is used in the mathematical proof written in this box:
Theorem For any whole numbers x and y, x2 – 4y ≠ 2.
Proof Suppose for contradiction that it is false that for any whole numbers x
and y, x2 – 4y ≠ 2.
Then there exist whole numbers x and y, where x2 – 4y = 2.
Let’s say that a and b are whole numbers, where a2 – 4b = 2
Then a2 = 2 + 4b, so a2 = 2(1 + 2b).
So a2 is even.
So a is even.
So for some whole number c, a = 2c.
Thus, (2c)2 – 4b =2, and so 4c2 – 4b = 2.
So 2c2 – 2b = 1, and so 2(c2 – b) = 1.
Now clearly 2(c2 – b) is even, so 1 is even.
But this is absurd: 1 is not an even number! Thus the proof is complete.
Question Eight (5 points)
Using the following symbols, symbolize the statements listed below.
Universe of Discourse: the people at a certain party
b Ben
c Chiara
Axy x admires y.
Mx x is a mathematician.
(1) Ben doesn’t admire Chiara, even though Chiara admires Ben.
(2) Ben and Chiara admire each other.
(3) Every mathematician admires Chiara.
(4) There are at least two mathematicians who admire Ben.
(5) There’s a mathematician who admires everyone.
Question Nine (2 points)
For this question, we will use the same symbols as in Question Eight. We will continue to assume that the
universe of discourse is the class of people at a certain party.
Here are three statements. Choose two of them to be premises, and one of them to be the conclusion, in
such a way that the resulting argument is valid:
(1) ∀x(Mx → ∃y Ayx)
(2) Mc
(3) ∀x Axc
There is no need to explain your answer.
Question Ten (10 points)
“Every statement is either true or false”.
Do you agree or disagree? Explain your answer.