Do part (ii) please.
Question 1: (15 marks) In this question we will walk through, step by step, the argu- ment showing that if I contains the modal formula (OA O A, then RS is conver- gent in the sense that IfRrA and RETA* then there exists a such that RSA and RSA*O To do this we need to show that given (1) O-' CA and (ii) 0A* CT, that there exists a complete and consistent set WE such that RAO and RA*O. Consider the set where: 0 =0- AUD-'A* If this set is consistent, then there must be a complete E-consistent set extending it, and we can let that set be our . Part (i): Show that if is a complete consistent set extending that we will have both R?AO and RATO All that remains to be done, then, is to show that is consistent. Suppose, for a contradiction, that it isn't. Then we must have that - to I. So we must have OD,...,ODn E A and OS.,..., OSm E A* for which D.A...ADn, S.A....Smte I So by the deduction theorem (Proposition 3.36(4) we have Si A... A Smte (DA ...ADn) +1, and so by (Proposition 3.36(5)) and the fact that (A + 1) +-A is a tautological instance we have SA... ASmte -(DA... AD) So by Lemma 4.6 it follows that O(SA...Asm) t-( DA... ADn) Part (ii): From the above it follows that: 1. OD A...ADNEA 2. OS, A... AS) E A* 3. 0-( DA...AD) EA Explain why. 1 So, as -(D,A...ADn) E A* it follows that we must have 00-(D.A...ADn) E T, and so by the fact that DA OVA E I, that 00-(DA...ADn) ET. So by the fact that 0-15 C Ait follows that we must have (-(DA...ADn) A. But (as you will show) this means that A is inconsistent. Question 1: (15 marks) In this question we will walk through, step by step, the argu- ment showing that if I contains the modal formula (OA O A, then RS is conver- gent in the sense that IfRrA and RETA* then there exists a such that RSA and RSA*O To do this we need to show that given (1) O-' CA and (ii) 0A* CT, that there exists a complete and consistent set WE such that RAO and RA*O. Consider the set where: 0 =0- AUD-'A* If this set is consistent, then there must be a complete E-consistent set extending it, and we can let that set be our . Part (i): Show that if is a complete consistent set extending that we will have both R?AO and RATO All that remains to be done, then, is to show that is consistent. Suppose, for a contradiction, that it isn't. Then we must have that - to I. So we must have OD,...,ODn E A and OS.,..., OSm E A* for which D.A...ADn, S.A....Smte I So by the deduction theorem (Proposition 3.36(4) we have Si A... A Smte (DA ...ADn) +1, and so by (Proposition 3.36(5)) and the fact that (A + 1) +-A is a tautological instance we have SA... ASmte -(DA... AD) So by Lemma 4.6 it follows that O(SA...Asm) t-( DA... ADn) Part (ii): From the above it follows that: 1. OD A...ADNEA 2. OS, A... AS) E A* 3. 0-( DA...AD) EA Explain why. 1 So, as -(D,A...ADn) E A* it follows that we must have 00-(D.A...ADn) E T, and so by the fact that DA OVA E I, that 00-(DA...ADn) ET. So by the fact that 0-15 C Ait follows that we must have (-(DA...ADn) A. But (as you will show) this means that A is inconsistent
