SECTION A
1 Attempt all parts of this question.
(a) Carefully define the following notions:
(i) truth-function
(ii) tautological entailment
(iii) material conditional
(iv) metalanguage
(b) Explain, with illustrations, what is meant by saying that a deductive
system is sound and complete. Give examples of deductive
systems for propositional logic that are:
(i) sound but not complete
(ii) complete but not sound
(c) What does it mean to say that any truth-function can be expressed
using just the connectives ∧, ∨ and ¬? Prove this.
2 Attempt all parts of this question. Through all parts of this question, take
the domain to be all people and use the following translation scheme:
‘j’ denotes Jenny
‘k’ denotes Karl
‘Mx’ expresses: x is miserable
‘Bx’ expresses: x is a Bolshevik
‘Lxy’ expresses: x loves y
(a) Translate the following into QL=, commenting on any difficulties.
(i) Karl loves Jenny if and only if the latter is not miserable.
(ii) Anyone who loves no miserable person loves neither Jenny
nor Karl.
(iii) Jenny is not miserable only if she is the only person loved
by Karl.
(iv) Precisely two Bolsheviks love Jenny and precisely one of
those two is miserable.
(v) No one who loves the only miserable person loves the only
Bolshevik.
(vi) If two Bolsheviks love Karl then they love one another.
[continuation of question 2 on page 3]
– 3 – PHT0/3
(b) Render the following arguments into QL=. Then use trees to show
that they are valid.
(i) Jenny is miserable but she loves exactly those people whom
Karl loves. Karl loves everyone who is miserable. Therefore
Jenny loves herself.
(ii) Karl is the only person who is not miserable, and he is a
Bolshevik. Therefore Karl is the only person who is both
Bolshevik and not miserable.
(iii) Karl is not a Bolshevik only if Jenny is miserable. Jenny is not
miserable if someone loves her. Karl loves no one who loves
any Bolshevik who loves Jenny. Karl loves only Jenny.
Therefore Karl does not love himself.
(iv) Everyone who is loved loves everyone. Exactly one person is
utterly unloved. Therefore there is exactly one person.
3 Explain what it is for a relation to be reflexive, symmetric and transitive.
We say that a relation R is negatively transitive if and only if
∀x∀y∀z ((¬Rxy ∧¬Ryz) ⊃ ¬Rxz). Of the following relations say which
are reflexive, symmetric, transitive and negatively transitive on the domain
of people. In each case give a brief explanation. In questions (a) – (e)
assume that all siblings share both parents. In question (i) assume that
Jane loves John, and John loves Jane, but nobody else loves either of
them.
(a) x is y’s father.
(b) x is a brother of y.
(c) x is y’s only sibling.
(d) x and y have no common ancestor.
(e) x is an ancestor of y.
(f) x loves y ≡ y loves x.
(g) x loves y ∨ y loves x.
(h) ∀z (x loves z ≡ z loves y).
(i) x loves John ≡ y loves Jane.
(j) A majority of people prefer x to y.
4 In probability, what is an event space? What is a field? What is conditional
probability? What is Bayes’s Theorem? Now solve the following problems.
In questions (b) and (c) please assume that there are exactly as many
boys as girls.
[TURN OVER for continuation of question 4]
– 4 – PHT0/3
(a) Johnny draws two socks at random from a drawer containing six
socks, all either black or white. His chance of drawing a pair of white
socks is 2/3. What is his chance of drawing a pair of black socks?
(b) Jane has two children. One is a boy. What is the probability that she
has two boys?
(c) Jane has two children. One is a boy who was born on a Monday.
What is the probability that she has two boys? If (b) and (c) have
different answers then explain briefly why.
(d) Coin A has a 1/2 chance of landing heads. Coin B has a 1/3 chance
of landing heads. Billy tosses one of them at random and it lands
tails. What is the probability that he has tossed coin A?
(e) 20% of a certain population takes an illegal drug. A test for this drug
gives the correct result 90% of the time whether or not the subject
has taken the drug. A random subject tests positive. What is the
probability that he has taken the drug?
SECTION B
5 How much of the meaning of ‘if ‘ does ‘⊃’ capture?
6 Does ‘The cat is on the moon’ mean the same as ‘There is one and only
one cat and it is on the moon’?
7 Can all necessary truths be known a priori?
8 Can the verification theory of meaning be given a formulation that is both
clear and defensible?