Paradoxical Puzzles of Truth

1. Some Puzzles About Truth (and belief, knowledge, opinion, and lying)

2. Puzzle #1: the postcard paradox • Consider the following sentences: 1) the following sentence is true: 2) the preceding sentence is false. • (examine the index card now circulating) • Q: are these sentences true or false?

3. Puzzle #2: the liar paradox • Is the following proposition true or false? This proposition is false • • If every proposition is either true or false then this proposition will be either true or false • • If it is true, then it is true that it is false; so it must be both true and false • • If it is false, then it is false that it is false; so it must be true; so it must be both true and false • • So in both cases it is both true and false, which is impossible

4. Puzzle #3: On the island of knights and knaves • On the island of Knights and Knaves, every inhabitant is either a knight or a knave. Knights always tell the truth. Knaves never tell the truth; any sentence uttered by a knave is false. A stranger came to the island and encountered three inhabitants, A, B, and C. He asked A, "Are you a knight, or a knave?" A mumbled an answer that the stranger could not understand. The stranger then asked B, "What did he say?" B replied, "A said that there is exactly one knight among us." Then C burst out, "Don't believe B, he is lying!" What are B and C? • One day I went to the island of knights and knaves and encountered an inhabitant who said, "Either I am a knave or else two plus two equals five." What should you conclude?

5. Puzzle #4: the ‘well-named’ ‘ill-named’ paradox • Have you ever noticed that some people are very well named? Martin Short is, after all, rather short. I once met a realtor named 'Isolde Haus' and a preacher named 'Mike Pentacost'. Just recently, I received a letter from an evolutionary biologist named 'Steve Darwin'. • Let's call everyone else 'ill-named'. Some people who are ill-named are rather spectacularly ill-named. For example, Tiny Tim is really rather large. Most of us are ill-named in a less interesting way, though. In any case, let's just agree to call everyone who isn't well-named 'ill-named'. • I used to play a game of classifying everyone I met as well named if their name is, somehow, particularly appropriate for them and ill named if it is not. I quit playing this game when a new neighbor moved in next door. His name is John Ill-named. Is he well named, or ill named? (due to Raymond Smullyan)

6. Puzzle #5: how to prove anything • Let A be any arbitrary sentence, and let B be the sentence "If this sentence is true, then A is true". Suppose B is, in fact, true. Then, according to B, A is true. Thus, we have established that if B is true, then A is true. But this is exactly what B asserts! Thus, B must be true, from which it follows (by B) that A must be true. Hence, all sentences are true! (due to M. H. Lob)

7. Thinking about the puzzles

8. Puzzle #6: the lottery paradox • Imagine a fair lottery with a thousand tickets in it.  • Each ticket is so unlikely to win that we are justified in believing that it will lose.  • So we can infer that no ticket will win.  • Yet we know that some ticket will win.

9. Puzzle #7: the preface paradox • Authors are justified in believing everything in their books.  • Some preface their book by claiming that, given human frailty, they are sure that errors remain, errors for which they take complete responsibility.  • But then they justifiably believe both that everything in the book is true, and that something in it is false. • Q: does this paradox look familiar? 

10. Puzzle #8: the knowability paradox • The following two claims seem eminently rasonable: a) some truths are not known, and b) any truth is knowable.  • Since the first claim is a truth, it must be knowable.  • From these claims it follows that it is possible that there is some particular truth that is known to be true and known not to be true.