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SECTION A
6AANB031
1. Use the propositional tableau proof systems to prove the following three
formulas
(a) 3(2X ⊃ 3Y ) ⊃ 3(X ⊃ Y ) in the system S4,
(b) 2(2X ⊃ 2Y ) ∨ 2(2Y ⊃ 2X) in the system S5,
(c) 2(3X ⊃ Y ) ≡ 2(X ⊃ 2Y ) in the system S5.
Use the constant domain tableau system to determine whether (d) is
valid on all K models with constant domain
(d) (3(∀x)A(x) ∧ 2(∃x)B(x)) ⊃ (∃x)3(A(x) ∧ B(x)).
Use the variable domain tableau system to determine whether (e) is
valid on all K models with varying domain
(e) (3(∀x)A(x) ∧ 2(∃x)B(x)) ⊃ 3(∃x)(A(x) ∧ B(x)).
2. (a) Is the following valid in T: 3(X ∧ Y ) ≡ (3P ∧ 3Y )? Justify your
answer.
(b) Show that the characteristic thesis of the Brouwerian system, X ⊃
3X, is not valid in S4.
(c) (a) Provide a model to show that (X ⊃ 3X) is not valid in
T.
(b) Provide a model that shows that (3X ⊃ X) is not valid
in S4.
3. (a) Let ‘CX’ abbreviate ‘It is contingently true that X’. Define CX
in terms of one of the standard modal operators and explain why
CX ⊃ 2CX is unacceptable.
(b) What are the modal systems S4 and S5? Under what conditions
on the accessibility relation on a model M for propositional modal
logic is M a model of (a) S4, (b) S5?
Show that 32X ⊃ 2X is true at every possible world of any S5
model but false at some possible world of some S4 model.
TURN OVER
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6AANB031
SECTION B
4. Explain what it means for a tableau of propositional modal logic to be
satisfiable and prove the following facts:
(a) a closed tableau is not satisfiable,
(b) applying the tableau extension rules for σ (X∧Y ), σ (X ⊃ Y ), σ 3X,
and σ 2X to a satisfiable tableau results in a satisfiable tableau,
(c) applying the B-tableau extension rule to tableau satisfiable in a
symmetric frame results in a tableau still satisfiable in that frame.
5. Let (G, R) be the following frame: G = {0, 1, 2, 3 . . .}, the set of natural
numbers, and nRm holds if and only if n is a smaller number than
m. Let (G, R, `) be a model over (G, R). In such a model, ‘n ` X’,
means that natural number n has some property PX , that is, the model
interprets X as a specific property of natural numbers (eg., “n is smaller
than a given number a”, “n is a multiple of 3”, “n is 4, 7 or 18”, etc.).
(a) For each of the following formulas give a property PX of natural
numbers such that the formula is valid on the model (G, R, `)
(b) (i) X ⊃ 23X,
(ii) X ⊃ 2X,
(ii) 2(X ⊃ 2¬X).
(c) Argue that (3X ∧ 3Y ) ⊃ 3((3X ∧ Y ) ∨ (X ∧ 3Y ) ∨ (X ∧ Y )) is
valid on the frame (G, R).
(d) Let nQm mean that number n is larger than number m. Give a
formula that is valid on the frame (G, Q) but not on (G, R).
TURN OVER
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6AANB031
6. In the quantified modal logic of Fitting & Mendelsohn:
“Some constants don’t designate; some constants are rigid designators;
some rigid designators designate existents and some don’t; some constants are nonrigid; some of the constants that are nonrigid designate
non existents; some of the constants that are nonrigid designate in other
worlds but not in this one.”
Give examples of this variety and explain how the semantics treats the
constants in each case.
7. (a) Give the syntax and semantics of predicate abstraction in formulas
of modal predicate logic.
(b) Formalize rigidity of a constant a using predicate abstraction and
explain your formalisation.
(c) Give a formalisation of the premises and conclusion in the language of quantified modal logic with predicate abstraction such
that the following argument is valid:
Ken knows that the Morning Star is the Evening Star.
Ken doesn’t know that the Morning Star is Venus
Ken doesn’t know that the Evening Star is Venus.
END OF PAPER
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