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. This is a question about TFL. Attempt all parts of this question.
(a) Explain what these three sentences mean, and explain the differences between them:
• A ⊧ C
• A ⊢ C
• A → C [ƥƩ]
(b) State and prove the Disjunctive Normal Form Theorem. [ƩƤ]
(c) Explain what it means to say, of some connectives, that they are jointly
expressively adequate. Show that ‘∧’ and ‘¬’ are jointly expressively adequate. You may rely upon your answer to part (b). [ƥƩ]
(d) Are the connectives ‘∧’, ‘∨’, ‘→’ and ‘↔’ jointly expressively adequate? Explain your answer. [ƦƤ]
Ʀ. Attempt all parts of this question. You must use the proof system from the
course textbook, forallx: Cambridge Version.
(a) Show each of the following: [ƨƤ]
(i) ⊢ (P → Q) ∨ (Q → P)
(ii) ¬(P ↔ Q),¬P ⊢ Q
(b) Show each of the following: [ƪƤ]
(i) ∃x(Fx ∨ Gx) ⊢ ∃xFx ∨ ∃xGx
(ii) ∀x(Fx → ∀yRxy),∀x(Gx → ∀zRxz),∀x(∀wRxw → (Fx ∧ Gx)) ⊢
∀x(Fx ↔ Gx)
(iii) ∀x∃yRxy, ∃x∀y x = y ⊢ ∃y∀xRxy
Ƨ. Attempt all parts of this question.
(a) Using the following symbolization key
domain: all physical objects
Mx: x is a mug
Rx: x is red
Tx: x is a table
Bxy: x belongs to y
a: Alice
Ʀ
symbolize each of the following sentences as best you can in FOL. If any
sentences are ambiguous, or cannot be symbolized very well in FOL, explain why. [ƪƩ]
(i) Every mug belonging to Alice is red.
(ii) The table is red.
(iii) Alice’s mug is red.
(iv) Alice’s mug does not exist.
(v) Two mugs are on the table.
(vi) If the mug belongs to anyone, it belongs to Alice.
(vii) None of the mugs on the table is Alice’s.
(viii) Every mug is on exactly one table, and on every table there is exactly
one mug.
(b) Show that each of the following claims iswrong. You may assume the conventionsfor representing interpretations described inthe coursetextbook,
forallx: Cambridge Version. [ƧƩ]
(i) Fa,¬Ga, Fb,¬Gb,¬Fc,Gc ⊧ ∀x(Fx ↔ ¬Gx)
(ii) ∀x(Fx → ∃y(Gy ∧ Rxy ∧ ∀z((Gz ∧ Rxz) → y = z))) ⊧
∀x(Gx → ∃y(Fy ∧ Ryx ∧ ∀z((Fz ∧ Rzx) → y = z)))
(iii) ∃x∀y¬Ryx,∀x¬Rxx,∀x∃yRxy ⊧ ∃x∃y(¬x = y ∧ ∃z(Rxz ∧ Ryz))
ƨ. Attempt all parts of this question.
(a) Write down the axiom of extensionality. Then, using the standard notation, define the set-theoretic notions of: union, intersection, subset,
proper subset and power set. [ƥƤ]
(b) Give examples for each of the following: [ƥƤ]
(i) Three non-empty sets, A, B, and C, such that none of A∩B, B∩C and
A ∩ C is empty, but such that (A ∩ B) ∩ C is empty
(ii) Two different non-empty sets, A and B, such that ℘(A) ∪ ℘(B) =
℘(A ∪ B)
(c) Give examples for each of the following: [ƦƩ]
(i) a set whose intersection with its power set is not empty
(ii) a set whose intersection with the power set of its power set is not
empty
(iii) a non-empty set that is a subset of the power set of one of its members.
(d) Write down the axioms of probability. Explain conditional probability.
[ƥƤ]
(e) There are two equally probable hypotheses: either Bryce baked exactly ƥƤ
cupcakes today, or Bryce baked exactly ƥƤƤ cupcakes today. In either case,
Bryce piped unique numbers onto them: between ƥ and ƥƤ, if there are ex-
Ƨ
actly ƥƤ cupcakes, or between ƥ and ƥƤƤ, if there are exactly ƥƤƤ cupcakes.
Bryce hands you a cupcake, with the number ƭ piped onto it. How probable is it, now, that Bryce baked exactly ƥƤƤ cupcakes today? Explain your
reasoning, highlighting any assumptions that you have made. [ƥƩ]
(f) Attempt both parts of this question. [ƧƤ]
(i) You aretossing a fair six-sided die. What isthe probabilitythat it lands
ƪ on each of the first three tosses? What is the probability that it lands
ƥ, then Ʀ, then Ʃ?
(ii) Mr Corleone always chooses the same national lottery numbers.
They came up in three successive lotteries, and now Mr Corleone is
rich. But the lottery organizers are suing him for fixing the lottery.
Mr Corleone’s defence lawyer says: ‘It is no more nor less likely, that
these numbers should come up three times in a row, than that any
other sequence of numbers should come up; so why should it be special grounds for suspicion?’ Is there anything wrong with the lawyer’s
argument? Carefully explain your answer.
Ʃ. Attempt all parts of this question.
(a) Explain what it means to say that a relation is [Ʃ]
(i) reflexive
(ii) symmetric
(iii) transitive
(b) Say that a relation Rxyis Euclidean iff ∀x∀y∀z((Rxy ∧ Rxz) → Ryz). For
each ofthe following relations onthe domain of all people (living or dead),
say whether the relation is reflexive, whether it is symmetric, whether it is
transitive, and whether it is Euclidean. In each case that the relation fails
to have one of these properties, briefly explain your answer. [ƩƤ]
(i) x and y have the same surname
(ii) x and y have the same surname or the same first name
(iii) x loves y only if y loves x
(iv) x loves y iff y loves x
(v) x is Winston Churchill or y is Bertrand Russell
(vi) x is Winston Churchill iff y is Bertrand Russell
(c) Give examples of relations with the following properties. In each case, be
careful to specify the domain. [ƨƩ]
(i) Reflexive and transitive but not symmetric
(ii) Euclidean and transitive but not reflexive
(iii) Symmetric and transitive but not reflexive
(iv) Reflexive and symmetric but neither transitive nor Euclidean
(v) Neither reflexive, symmetric, transitive nor Euclidean
ƨ
ňĹķʼnĽŃł Ķ
ƪ. Does Russell’s theory of definite descriptions provide a correct method of eliminating definition descriptions from all contexts in which they occur, from some
contexts, or from none at all?
ƫ. What, if anything, do the paradoxes of material implication tell us about the
meaning of ‘if…, then…’ in natural language?
Ƭ. Are mathematical truths synthetic a priori?
ƭ. Can meaning be explained in terms of intention?