The fourth and final tale is the most baffling of all, and it illustrates a logical principle of basic importance.

The suitor of the last story passed all three tests and happily claimed Portia III as his bride. They had many children, great-grandchildren, etc.

Several generations later a descendant was born in America who looked so much like the ancestral portraits that she was named Portia Nth-henceforth to be referred to as "Portia." When this Portia grew to young womanhood she was both clever and beautiful-just like all the other Portias. In addition, she was highly vivacious and a bit on the mischievous side. She also decided to select her husband by the casket method (which was somewhat of an anomaly in modern New York, but let that pass).

The test she used appeared simple enough; she had only two caskets, silver and gold, in one of which was Portia's portrait. The lids bore the following inscriptions:

Gold: THE PORTRAIT IS NOT IN HERE

Silver: EXACTLY ONE OF THESE TWO STATEMENTS IS TRUE


Which casket would you choose? Well, the suitor reasoned as follows. If the statement on the silver casket is true, then it is the case that exactly one of the two statements is true.

This means that the statement on the gold casket must be false. On the other hand, suppose the statement on the silver casket is false. Then it is not the case that exactly one of the statements is true; this means that the statements are either both true or both false. They can't both be true (under the assumption that the second is false), hence they are both false. Therefore again, the statement on the gold casket is false. So regardless of whether the statement on the silver casket is true or false, the statement on the gold casket must be false. Therefore the portrait must be in the gold casket.

So the suitor triumphantly exclaimed, "The portrait must be in the gold casket" and opened the lid. To his utter horror the gold casket was empty! The suitor was stunned and claimed that Portia had deceived him. "I don't stoop to deceptions," laughed Portia, and with a haughty, triumphant, and disdainful air opened the silver casket. Sure enough, the portrait was there.

Now, what on earth went wrong with the suitor's reasoning?

"Well, well!" said Portia, evidently enjoying the situation enormously, "so your reason didn't do you much good, did it? However, you seem like a very attractive young man, so I think I'll give you another chance. I really shouldn't do this, but I will! In fact, I'll forget the last test and give you a simpler one in which your chances of winning me will be two out of three rather than one out of two. It resembles one of the tests given by my ancestor Portia III. Now surely you should be able to pass this one!"

So saying, she led the suitor into another room in which there were three caskets-gold, silver, and lead.

Portia explained that one of them contained a dagger and the other two were empty. To win her, the suitor merely need choose one of the empty ones. The inscriptions on the caskets read as follows:

Gold: THE DAGGER IS IN THIS CASKET

Silver: THIS CASKET IS EMPTY

Lead: AT MOST ONE OF THESE THREE STATEMENTS IS TRUE

(Compare this problem with the first test of Portia III! Doesn't it seem to be exactly the same problem?)

Well, the suitor reasoned very carefully this time as follows: Suppose statement (3) is true. Then both other state;nents must be false-in particular (2) is false, so the dagger is then in the silver casket. On the other hand, if (3) is false, then there must be at least two true statements present, hence (1) must be one of them, so in this case the dagger is in the gold casket. In either case the lead casket is empty.

So the suitor chose the lead casket, opened the lid, and to his horror, there was the dagger! Laughingly, Portia opened the other two caskets and they were empty!

I'm sure the reader will be happy to hear that Portia married her suitor anyhow. (She had decided this long before the tests, and merely used the tests to tease him a little) . But this still leaves unanswered the question: What was wrong with the suitor's reasoning?

SOLUTION

The suitor should have realized that without any information given about the truth or falsity of any of the sentences, nor any information given about the relation of their truthvalues, the sentences could say anything, and the object (portrait or dagger, as the case may be) could be anywhere.

Good heavens, I can take any number of caskets that I please and put an object in one of them and then write any inscriptions at all on the lids; these sentences won't convey any information whatsoever. So Portia was not really lying; all she said was that the object in question was in one of the boxes, and in each case it really was.

The situation would have been very different with any of the previous Portia stories, if the object had not been where the suitor figured it out to be; in this case one of the old Portias would have had to have made a false statement somewhere along the line (as we will soon see).

Another way to look at the matter is that the suitor's error was to assume that each of the statements was either true or false. Let us look more carefully at the first test of Portia Nth, using two caskets. The statement on the gold casket, "The portrait is not in here," is certainly either true or false, since either the portrait is in the gold casket or it isn't. It happened to be true, as a matter of fact, since Portia had actually placed the portrait in the silver casket. Now, given that Portia did put the portrait in the silver casket, was the statement on the silver casket true or false? It couldn't be either one without getting into a paradox!

Suppose it were true. Then exactly one of the statements is true, but since the first statement (on the gold casket) is true, then this statement is false. So if it is true, it is false.

On the other hand, suppose this statement on the silver casket is false. Then the first is true, the second is false, which means that exactly one of the statements is true, which is what this statement asserts, hence it would have to be true! Thus either assumption, that the statement is true or is false, leads to a contradiction.

It will be instructive to compare this test with the second test given by Portia III, which also used just two caskets. The gold casket said the same thing as the gold of the problem, "The portrait is not in here," but the silver casket, instead of saying "Exactly one of these two statements is true," said "Exactly one of these two caskets was fashioned by Bellini." Now, the reader may wonder what significant difference there is between these two statements, given that Bellini inscribed only true statements and Cellini only false ones. Well, the difference, though subtle, is basic. The statement, "Exactly one of these two caskets was fashioned by Bellini" is a statement which must be true or false; it is a historic statement about the physical world: either it is or it is not the case that Bellini made exactly one of the two caskets. Suppose, in the Portia III problem, that the portrait had been found to be in the silver casket instead of the cold casket. What would you conclude: that the statement on the silver casket was neither true nor false? That would be the wrong conclusion! The statement, as I have pointed out, really is either true or false. The correct conclusion to draw is that if the portrait had been in the silver casket, then Portia In would have been lying in saying what she did about Bellini and Cellini. By contrast, the modern Portia could place the portrait in the silver casket without having lied, since she said nothing about the truth-values of the statements.

The whole question of the truth-values of statements which refer to their own truth-values is a subtle and basic aspect of modern logic and will be dealt with again in later chapters.