King’s College London
UNIVERSITY OF LONDON
This paper is part of an examination of the College counting towards
the award of a degree. Examinations are governed by the College
Regulations under the authority of the Academic Board.
BA EXAMINATION
4AANB008
Methodology
SUMMER 2013
TIME ALLOWED: 2 HOURS
You should attempt to answer all questions. Each question carries
the same number of marks.
Avoid overlap in your answers.
DO NOT REMOVE THIS PAPER FROM THE EXAMINATION
ROOM
TURN OVER WHEN INSTRUCTED
2013 © King’s College London
1
4AANB008
1. Consider the set {London, Paris, Madrid, {Bonn, Rome}, {Paris,
Madrid}}
For each of the following seven items, say whether or not it is (a) a
member of this set (b) a subset of this set.
Paris
{Paris}
Bonn
{Bonn, Rome}
{Paris, Madrid}
{}
{London, Paris, Madrid, {Bonn, Rome }}
Is there a general rule telling you how many subsets a given set has?
2. What is the ‘axiom of comprehension’ of naïve set theory’? Which
condition did Bertrand Russell use to show that naïve set theory is
inconsistent? Explain carefully how he showed this.
3. What can you infer about a set if all its members can be paired up
one-to-one with the members of one of its proper subsets?
Can the rational numbers be paired up one-to-one with the natural
numbers? Explain the basis for your answer.
Can the real numbers be paired up one-to-one with the natural
numbers? Explain the basis for your answer.
4. Explain what is meant by a ‘synthetic a priori’ statement. Is there
any reason to suppose that the statement the angles in any triangle
add up to 180º is synthetic? Is there any reason to suppose that this
statement is a priori? Can it simultaneously be understood to be both
synthetic and a priori?
SEE NEXT PAGE
2
4AANB008
5. Which of these are ‘de re’ claims, and which ‘de dicto’? Also say of
each claim whether it is true or false. (Assume that species
membership is essential to animals.)
(a) (The inventor of the zip) (might not have invented the zip)
(b) (It might have been the case that) (the inventor of the zip did not
invent the zip)
(c) (It might have been the case that) (Julius did not invent the zip) –
[where it is stipulated that ‘Julius’ refers to whomever invented the
zip].
(d) (Necessarily) (the first mammal on the moon was human)
(e) (Necessarily) (Neil Armstrong was human)
6. What are degrees of belief?
Is there any reason why rational degrees of belief should satisfy the
axioms of probability?
What principle, if any, governs the relation of rational degrees of
belief to objective probabilities?
What principle, if any, governs the way you should rationally change
your degrees of belief in response to evidence?
SEE NEXT PAGE
3
4AANB008
7. Suppose that the probability of having heart disease (H), being male
(M), and being a smoker (S) are given by
Prob (H) = 10%
Prob (M) = 60%
Prob (S) = 40%
And suppose that
Prob (H and M) = 6%
Prob (H and S) = 8%
Prob (M and S) = 30%
For each pair of properties (H and M), (H and S), (M and S), say
whether the two properties are positively dependent, negatively
dependent or independent. For each of the pairs, work out the
conditional probability of the first given the second.
8. Suppose that you find that the probability of a stroke among
joggers is higher than in the population in general:
Prob(Stroke/Jogging) = 0.25 while Prob(Stroke) = 0.2. In the absence
of any further evidence, what does this indicate about the causal
connection between jogging and strokes?
But suppose that you then also find that:
Prob(Stroke/Jogging & Male) = 0.2 while Prob(Stroke/Male) = 0.3, and
that
Prob(Stroke/Jogging & Female) = 0.05 while Prob(Stroke/Female) =
0.1.
What does this now indicate about the causal relationships between
gender, jogging and strokes?
SEE NEXT PAGE
4
4AANB008
9. What is required for one sentence to be a syntactic consequence of
some other sentences in propositional logic? What is required for it to
be a semantic consequence of those sentences in propositional logic?
Use this to explain the notions of soundness and completeness for
propositional logic.
10. What does it mean to say that a theory for arithmetic is
incomplete? Why is it not a contradiction to say that Gödel’s Theorem
proves that a certain arithmetic sentence is unprovable, yet at the
same time proves that it is true?
FINAL PAGE
5