Darklands Puzzle #22

A grim iron door blocks your way. Above the door is carved:
ALL STATEMENTS ARE TRUE.
On the door are embossed four statements:

1. If the silver key does not open the door, neither does the copper key.
2. If the copper key does not open the door, neither does the silver key.
3. If the gold key opens the door, so does exactly one other key.
4. If exactly two keys open the door, one of them is gold.

Four keys hang nearby. You ponder, then choose...
...the gold key.
...the silver key.
...the copper key.
...the lead key.

(Source: Darklands cluebook by MicroProse Software)

Consider first the possibility of statement #3: "if the gold key opens the door, so does exactly one other key". This statement can be connected with statement #4: "if exactly two keys open the door, one of them is gold". The other key must be lead because exactly one other key can join gold. Statements #1 and #2 make it clear that silver and copper must both be able to open the door, or neither of them do. So it is possible that the pair of gold and lead keys can open the door. The preceding discussion should make it clear that no other pairs are possible.

But that possibility is only that: a possibility. There may be others. If one and only one key opens the door, it can only be lead. Silver and copper must be paired. Gold also requires a further key (which was previously established as lead).

The possibility that three keys open the door is also available, in which case these keys must be silver, copper and lead. Gold is excluded per statement #3. Four keys are also impossible according to the same statement.

All told, there are three possibilities. Which of them is true is underdetermined, but fortunately lead is the common factor among all of them, so one should choose the lead key.