Other things the same, an increase in the price level causes the real value of the dollar to fall in the market for foreign-currency exchange.

An increase in the price level actually causes the real value of the dollar to rise in the domestic economy, but in the context of foreign-currency exchange, it makes the dollar depreciate (or fall in value) against foreign currencies, not the real value of the dollar itself. The statement as phrased confuses the concepts of real value in domestic terms with exchange value in international terms.