There is a lack of consistency regarding symbols for modal operators. Hughes and Cresswell use a different symbolization schema than many and a comment about L & M may provide clarity for reader unfamiliar with their notation.
They use L for the necessity operator.
They use M for the possibility operator.
So for those not familiar with this scheme: L = ◻ and M = ◇
I just want to see if I got this correct.
Show that the following wff is valid in every seating arrangement: Mp -> (Lq->Mq).
Here's my argument:
Assume player A sees at least one player B that raises his hand for p, and thus also A raises his hand for Mp.
Assume player A doesn't raise his hand for Lq ->Mq, but he raises his hand for Lq but not for Mq.
Then player A sees all other players raising their hands for q, and thus A raises his hand for Mq, contradiction.
Another question from the same book.
exercise 1.2: show that in any seating arrangement in which there's a player who cannot see himself Lp->p isn't valid.
My answer: Suppose A cannot see himself. Let p be on every player's sheet except A's. Suppose that A raises his hand for Lp->p then he must raise his hand for p, since he raised his hand for Lp, but he cannot see everyone's sheet (only his own), so he should keep his hand down for p, and we get that Lp->p isn't valid.
Is my reasoning ok or not?