CONCLUDING REMARKS
I hope I have given the reader some feeling for why these modal systems are so interesting. We have seen that they can be interpreted both internally (self-referentially) and externally (as applying to provability in other mathematical systems), as well as to reasoning processes—both for naturally intelligent beings (some humans and other animals) and for artificially intelligent mechanisms (such as computers). What applications this may have in the field of psychology is something that might be worth further investigation.
It is a happy turn of fate that the field of modal logic, which historically arose out of purely philosophical interests, should have turned out to be so important today in proof theory and computer science—this by virtue of the theorems of Gödel and Löb and of the work of those who have subsequently looked at proof theory from a modal-theoretical viewpoint. And now even those philosophers who in the past have taken a dim view of the significance of modal logic are forced to realize its mathematical importance.
The past philosophical opposition to modal logic has been grounded roughly in three quite different (and incompatible) beliefs: First, there are those who believe that everything true is necessarily true, and hence that there is no difference between truth and necessary truth. Second, there are those who believe that nothing is necessarily true, and hence that for any proposition p, the proposition Np (p is necessarily true) is simply false! And third, there are those who claim that the words “necessarily true” convey no meaning whatsoever. And so each of these philosophical types has rejected modal logic on his own grounds. Indeed, one well-known philosopher is reputed to have suggested that modern modal logic was conceived in sin. To which Boolos has aptly replied: “If modern modal logic was conceived in sin, then it has been redeemed through Gödliness.”
ALSO BY RAYMOND SMULLYAN
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First Order Logic
The Tao Is Silent
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This Book Needs No Title
The Chess Mysteries of Sherlock Holmes
The Chess Mysteries of the Arabian Knights
The Lady or the Tiger?
Alice in Puzzle-Land
5000 B.C.
To Mock a Mockingbird
Forever Undecided Page 21
