Each question has equal weight. A perfect answer would receive a notional ƥƤƤ points.
For Section 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.
ňĹķʼnĽŃł ĵ
ƥ. Attempt both parts of this question.
(a) Each of the following claims is either true or false:
(i) ⊢ ((¬B → ¬A) → A) → A [ƥƩ]
(ii) ¬E → ¬¬F,¬G → ¬E ⊢ G → ¬F [ƥƩ]
(iii) (C ∨ ¬A) → (B ∧ D),C → ¬B,¬A → ¬D ⊢ A ∧ ¬C [ƥƩ]
(iv) ¬(T → S), (S ↔ (T ↔ U)) ⊢ S ∧ ¬(T ↔ U) [ƥƩ]
(v) C ↔ B, B ∨ D, (C ∨ D) → A ⊢ A [ƥƩ]
(vi) P ↔ ¬Q,Q ↔ ¬R ⊢ P ↔ R [ƥƩ]
For eachtrue claim, showthat it istrue by providing a suitable formal proof
(using the proof-system described in forallx). For each false claim,
show that it is false by providing a suitable truth-table.
(b) Explain the difference between the meanings of ‘⊢’ and ‘⊧’. Explain, with
reference to one of the false claims in part (a), why we are licensed in inferring its falsity from the truth table that you provided. [ƥƤ]
Ʀ. Attempt all parts of this question.
(a) Using the following symbolisation key
domain: all people
Nx: x is a ninja
Bxy: x is behind y
Sxy: x can see y
a: Akira
symbolise each of the following sentences as best you can in FOL. Comment on your translations where appropriate, in particular highlighting
any difficulties in symbolisation. [ƪƤ]
(i) Everyone behind Akira is a ninja.
(ii) If Akira cannot see someone, that person doesn’t exist.
(iii) Only a ninja standing behind Akira is invisible to him.
(iv) Akira is invisible to everyone except ninjas.
(v) If one ninja stands behind another, the latter can see the former.
(vi) The ninja Akira cannot see is behind him
Ʀ
(vii) No one can see the ninja behind Akira, not even that ninja herself
(viii) Behind every ninja Akira cannot see, there is another ninja Akira
cannot see.
(ix) For every ninja Akira cannot see, there aretwo more ninjas Akira cannot see.
(x) Akira can see each of the three ninjas.
(b) Formalise these arguments, and then use the proof-system described in
forallx to show that they are valid [ƨƤ]
(i) If Akira cannot see someone, that person doesn’t exist. The ninja is
behind Akira. So Akira can see the ninja.
(ii) There is at least one ninja whom Akira cannot see. Behind every ninja
Akira cannot see, there is another ninja Akira cannot see. So there are
at least two ninjas whom Akira cannot see.
Ƨ. Attempt all parts of this question.
(a) Define the following set-theoretic notions: union, intersection, subset,
power set, Cartesian product. [ƥƤ]
(b) Each of the following statements is either true or false. In each case say
which and explain briefly why: [ƨƩ]
(i) For any sets A, B and C, if A ⊆ B and B ⊆ C then A ⊆ C.
(ii) For any sets A and B, if A ⊆ B then A ⊆ ℘(B).
(iii) For any sets A and B, if A ∈ ℘(B) then ℘(A) ⊆ ℘(B).
(iv) For any sets A, B and C, (A ∩ B) × (B ∩ C) ⊆ A × C.
(v) For any set A, ℘(A) ∈ ℘(℘(A)).
(c) Define the following notions from the logic of relations: reflexive, symmetric, transitive, equivalence relation. [ƥƤ]
(d) Call a relationRnegativelytransitive if∀x∀y∀z((¬Rxy∧¬Ryz) → ¬Rxz).
Give examples of relations with each of the following properties: [ƧƩ]
(i) equivalence, but not negatively transitive
(ii) reflexive and negatively transitive, but not symmetric
(iii) reflexive and symmetric, but not transitive
(iv) transitive and symmetric, but not reflexive
(v) transitive, but neither symmetric nor negatively transitive
ƨ. Attempt all parts of this question.
(a) Define the terms: field, event space, conditional probability. [ƥƤ]
(b) You are held captive inthe Bayesian Republic of Zembla. The gaoler places
two bullets in consecutive chambers in a six-chambered revolver, spins the
wheel and takes a shot at you. You are in luck: the chamber was empty!
Ƨ
The gaoler then decides he will take a second shot. However, he decides
to let you choose between the following options: either he will fire from
the next chamber, or he will spin the chamber again and then fire. Which
should you choose and why? [ƥƩ]
(c) One octopus in every ƥƤƤ is psychic. Psychic octopi have perfect knowledge of the results of future football tournaments. Non-psychic octopi can
only guess randomly. Paul, a randomly chosen octopus, correctly predicts
the winner of the next (football) World Cup, from the ƧƦ teams that qualified. What is the probability that Paul is psychic? [ƦƩ]
(d) I choose three different numbers, at random, from the (whole) numbers
between ƥ and ƨ (inclusive). What is the probability that my choices are
all in increasing (not necessarily consecutive) numerical order? [ƥƩ]
(e) I choose three different numbers, at random, from the (whole) numbers
between ƥ and ƥƤ (inclusive). What is the probability that they are in increasing (not necessarily consecutive) numerical order? [ƧƩ]
ňĹķʼnĽŃł Ķ
Ʃ. Is ‘the’ a quantifier phrase?
ƪ. Are there any contingent a priori truths?
ƫ. ‘“If Rome is in the US, then it is in Europe” is not simply unassertible; it is also
unbelievable. Hence the paradoxes of material implication cannot be explained
away using the theory of implicatures.’ Discuss.
Ƭ. Which should be considered primary in a philosophical account of meaning:
sentence-meaning or speaker-meaning?