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

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?

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

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?

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)

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)

Thinking about the puzzles

Puzzle #6: the lottery paradox  Imagine a fair lottery with a thousand tickets in it.  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.  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.  So we can infer that no ticket will win.  Yet we know that some ticket will win.

Puzzle #7: the preface paradox  Authors are justified in believing everything in their books.  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.  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?  Q: does this paradox look familiar?

Puzzle #8: the knowability paradox  The following two claims seem eminently rasonable: a) some truths are not known, and b) any truth is knowable.  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.  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.