Each question has equal weight. A perfect answer would receive a notional 100
points. For Part A [formal questions] the number in square brackets after each
component of a question designates the number of points that a full and correct
answer to that component would merit.
SECTION A
1 (a) Carefully define the following:
(i) validity; [5]
(ii) tautological validity; [5]
(iii) a tautology. [5]
(b) What is the difference between ‘∴’, ‘╞ ‘ , and ‘⊃’? [15]
(c) Can an inference be made invalid by adding extra premises? Give
a reason for your answer. [15]
(d) The tree method for PL is sound and complete. Explain carefully
what this means. [15]
(e) The connectives {¬, ∨ } are expressively adequate. Explain what
this means and prove that they are. [40]
2 (a) Carefully define the following terms in the context of the syntax
and semantics of QL=:
(i) term; [5]
(ii) operator; [5]
(iii) scope; [10]
(iv) q-valuation; [10]
(v) extended q-valuation; [10]
(vi) q-validity. [10]
(b) Explain the semantics of the QL operators ∀ and ∃. Explain why
any inference of the form ¬∃ x F x, ∴ ∀x¬Fx is q-valid. [50]
– 3 – PHT0/3
3 (a) Let R be a relation. State the following conditions on R in the
language of QL =:
(i) R is reflexive; [5]
(ii) R is symmetric; [5]
(iii) R is transitive. [5]
(b) Let us say that a relation R is Euclidean if ∀x∀y∀z
((Rxy ∧ Rxz) ⊃ Ryz). Let us say that R’ is the converse of R if
∀x∀y(Rxy ≡R’yx).
Now translate the following arguments into QL = and show using trees
that they are valid.
(i) if R is reflexive and symmetric and transitive then R is
Euclidean; [15]
(ii) if R is symmetric and Euclidean and R’ is the converse of R
then R’ is symmetric and Euclidean; [15]
(iii) if R is reflexive and R’ is the converse of R then R’ is
reflexive. [15]
(c) Using the definitions in (b), give examples of relations that are:
(i) Euclidean and symmetric but not transitive; [10]
(ii) transitive and Euclidean but not symmetric; [10]
(iii) transitive and symmetric but not reflexive; [10]
(iv) not Euclidean but with a Euclidean converse. [10]
4 (a) You randomly draw two cards without replacement from a
standard pack containing 52 cards: 13 of each suit (spades, hearts,
diamonds, clubs) and no jokers. What is the probability that:
(i) the second is a spade? [5]
(ii) the first is a spade given that the second is not a spade?
[10]
(iii) the second is a club given that the first is a heart? [10]
(iv) the second is a diamond given that the first is a
diamond? [10]
(v) the second is a diamond given that the first is the king of
diamonds? [15]
[TURN OVER for continuation of question 4]
– 4 – PHT0/3
(b) Billy is on trial for a crime. The DNA test shows that somebody with
Billy’s DNA was at the scene of the crime. 0.01% of the UK population
shares Billy’s DNA. So, the prosecution argues, there is a 99.99%
probability that Billy was at the scene. Not so, says the defence: the UK
population is 60 million, so there are 6,000 people with Billy’s DNA, so the
probability that Billy was at the scene is 1/6000. Who is right? [50]
SECTION B
5 Does ‘The largest odd number is greater than 100’ have a truth value?
6 Can there be synthetic necessary truths?
7 Explain (a) the type/token distinction and what is meant by ‘an indexical
expression’ and (b) the relevance of these notions to logic.
8 What are the ‘paradoxes of material implication’? How would you set about
trying to resolve the concerns they might raise?