70. _______________________
The suitor should have realized that without any informa tion given about the truth or falsity of any of the sentences, nor any information given about the relation of their truth values, 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 a t the matter i s 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
The suitor should have realized that without any informa tion given about the truth or falsity of any of the sentences, nor any information given about the relation of their truth values, 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 a t the matter i s 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 state ments 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 state ments, giventhatBelliniinscribed onlytrue 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 statementwhich 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.
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 state ments 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 state ments, giventhatBelliniinscribed onlytrue 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 statementwhich 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.
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