King s College London

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Derrick Gardner
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1 King s College London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority of the Academic Board. MA EXAMINATION MODAL LOGIC Examination Period 2 (May/June 2016) TIME ALLOWED: TWO HOURS ANSWER TWO QUESTIONS, ONE FROM EACH SECTION NB Candidates should avoid overlap in their answers ANSWER EACH QUESTION ON A NEW PAGE OF YOUR ANSWER BOOK AND WRITE ITS NUMBER IN THE SPACE PROVIDED. DO NOT REMOVE THIS PAPER FROM THE EXAMINATION ROOM TURN OVER WHEN INSTRUCTED 2016 King s College London

2 SECTION A 1. (a) Is the following valid in T: (p q) ( p q)? Justify your answer. (b) Show that the characteristic thesis of the Brouwerian system, p P, is not valid in S4. (c) i. Provide a model to show that ( p p) is not valid in T. ii. Provide a model that shows that ( p p) is not valid in S4. (d) Provide an S5 model in which xp x x P x is false at some world. (e) Explain why, if domains are allowed to vary freely, the converse of the Barcan Formula is not valid even in T. 2. Let three frames (G 1, R 1 ), (G 2, R 2 ), (G 3, R 3 ), be defined by G 1 = {a}, R 1 = { a, a } G 2 = {a, b}, R 2 = { a, b, b, a, b, b } G 3 = {a, b}, R 3 = { a, a, a, b, b, b } ( x, y in R i means xr i y). (a) For each of the three frames determine which of the following formulas are valid on the frame. i. P P ii. P P iii. P P 2

3 iv. P P v. P P Moreover, if one of the formulas φ is not valid on frame (G i, R i ), give a world x in G i and a forcing relation between G i and {P } such that x φ. (b) Show that a frame (G, R) is reflexive if and only if every formula of the form P P is valid in it. 3. Use the propositional tableau proof systems to prove the following three formulas (a) (p p) in the T system, (b) ( p q) ( p q) in the system S4, (c) (( ( p p) ( p p)) in the system S5. Use the constant domain tableau system to determine whether (d) is valid on all K models with constant domain (d) ( ( x)a(x) ( x)b(x)) ( x) (A(x) B(x)). Use the variable domain tableau system to determine whether (e) is valid on all K models with varying domain (e) ( ( x)a(x) ( x)b(x)) ( x)(a(x) B(x)). 3

4 SECTION B 4. Given the following facts a closed tableau is not satisfiable, applying tableau extension rules to a satisfiable tableau results in a satisfiable tableau, an open tableau, in which all rules that can be applied have been applied, is satisfiable, if X has no L-proof, then there is a saturated tableau for X with an open branch, An open branch of a saturated tableau for X can be identified with a model with a world that makes X true. Show the following (a) The Tableau Method for Propositional Modal Logic is sound and complete. (b) If a tableau for P is satisfiable, then so is the tableau that results from applying the tableau extension rule for. (Of course, you may not now assume the second bullet point above). 5. 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. 4

5 6. (a) Describe Predicate Abstraction and explain what problem in quantified modal logic (without abstraction) it is meant to solve. (b) Formulate De Re and De Dicto readings of the following sentences in the language of quantified modal logic with predicate abstraction and indicate which of the readings is the more likely one: i. The mayor of London will be a Labour member ii. The number of planets is necessarily odd. (c) Give a formalisation of the premisses 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. 5 FINAL PAGE

This paper is also taken by Combined Studies Students. Optional Subject (i): Set Theory and Further Logic

UNIVERSITY OF LONDON BA EXAMINATION for Internal Students This paper is also taken by Combined Studies Students PHILOSOPHY Optional Subject (i): Set Theory and Further Logic Answer THREE questions, at