Some Puzzles About Truth and belief knowledge opinion

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 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’ ‘illnamed’ 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. 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.

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? }

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. }