What happens when a country's central bank significantly reduces the available currency? The central bank of Iraq has significantly reduced the note count from 55 Trillion to 5 trillion.  

Logic would dictate that when a supply is reduced the value of the item that has been reduced will go up. 

Is this incorrect thinking? And if so, what is the correct way to look at this?
According to standard monetary equation:

$$MV=PY$$

Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ real output. And where the‘value’ of money is inversely proportional to the price level $P$, we can rearrange the above for $P$ to see how it changes when parameters change and from that we can infer what happens to the value of money:

$$P=\frac{MV}{Y}$$

Hence if the money supply goes down and velocity and output stays the same the $P$ will decrease and money become worth more. Important caveat is that both velocity and output might not stay constant. If real output decreases faster than the decrease in money supply then the price level would still increase and value of money decrease. Also, in more complex models expectations play big role. If the change in money supply is expected to be reversed in near future it might not affect price levels and the relationship might not hold at 0 lower bound (see the influential paper from Krugman on this here).