JR'S Free Thought Pages
No Gods  ~ No Masters   

Below are four Cards, with a letter on one side and a number on the other side.  Which, and only which cards must be turned over in order to determine the truth of the following statement?

     "If there is a vowel on one side then there is an even number on the other side." 

For the solution check the bottom of this page.....

(1) Liar Paradoxes:  (i)  "I am lying."       (Is the sentence true or false) 

                                 The statement inside this box is false.

Ralph: "What Clyde is about to say is false."      Clyde: "What Ralph has just spoken is true."

"Nothing on this page is true."

Take a blank piece of paper, and on one side write : The statement on the other side of this paper is true and on the other side write: The statement on the other side of this paper is false.

(ii) Liar Puzzle: In attempting to reach your destination you come to a fork in the road, but only one path of which will take you there. You meet two people here, one who always tells the truth (Tom the Truth Teller) and one who always lies (Larry the Liar). You are allowed one, and only one, question. What should it be? Check the bottom of this page for the solution...

(2) Russell's Paradox:  There exists a barber in a town who shaves those citizens and only those citizens who do not shave themselves. Does he shave himself?       See the discussion at the bottom of the page.

(3) Groucho Marx once said "I refuse to join any club that would have me as a member!"

(4) When Raymond Smullyan was asked "Do you believe in astrology?", he replied "I'm a Gemini and Geminis don't believe in astrology!" 

(5) Proof of the existence of the Invisible Flying Green Unicorn (IFGU) - or the metaphysical entity of your choice.

Statement 1: The Invisible Flying Green Unicorn exists.                   

 Statement 2: Both of these statements are false.                    

One must only examine statement 2. Assume Statement 2 true OR assume statement 2 false.   In either case Statement 1 is true. 

(6) Proof of the existence of Invisible Purple Goblins on the far side of Neptune .....or once again, the metaphysical entity of your choice (perhaps Tooth Fairy?)               

   If this statement is true, then there exist Invisible Purple Goblins on the far side of Neptune. 

(7) Two M .C. Escher Drawings (Visual Paradoxes)

A fiction writer who revels in paradox is Jorge Luis Borges (1899-1986). Check him out!

My Favorite Circular Argument (Has this ever happened to you on your doorstep Saturday morning?):  True Believer : "God exists!"  Skeptic: "But how do you know God exists?" True Believer: "It says so right here in the Bible." Skeptic: "But how do you know what the Bible says  is true?" True believer: "It's the word of God!"

Theist: "Do you believe in God"?  SuperTheist: "No - I believe in something much greater!"

Solution to Skepticism Quotient:   "A" and "7". You must turn over "A" in order to confirm the hypothesis and turn over " 7" to test for a possible disconfirmation of the hypothesis.

Comments: Most people will have no problem in realizing "A" must be turned over. This simply confirms the conditional "If P, then Q". Turning over "7" is frequently missed because it involves counter-evidence to the hypothesis, something few people are inclined to look for, especially when it involves a belief to which  they are emotionally attached.

The disposition to look for falsifications of one's hypotheses or beliefs is important to any scientist and Charles Darwin was a good example of a scientist possessing this valuable intellectual virtue. It's interesting to note that Darwin's theories on evolution and natural selection are continually being refined and confirmed, particularly with the recent advancements in genetics. These refinements and confirmations are above all the direct result of this intellectual trait.

 Analysis: Turning over "7" will test the hypothesis "If not Q, then not P" which is logically equivalent to "If P, then Q". For example, consider the conditional : "If the sun shines,  Ralph plays tennis". If one assumes the truth of this statement then one must accept the truth of : "If  John does not play tennis, then the sun does not shine". Logicians refer to the latter statement as the contrapositive of the former and is a crucial logical mechanism in initiating indirect proofs.

Those who feel that "B" must be turned over are choosing an irrelevancy to the hypothesis. It is irrelevant whether there is an even or an odd number on the reverse side of "B", since "B" is not a vowel. They are committing  what logicians call "the fallacy of denying the antecedent" - "If not P, then not P". If one accepts the truth of "If P, then Q", it does not follow that "If not P, then not Q".

Those who feel that "4" must be turned over are also choosing an irrelevancy to the hypothesis. It is irrelevant whether there is a vowel or a consonant on the reverse side of "4". They are committing what logicians call "the fallacy of "affirming the consequent" - If Q, then P". If one accepts the truth of "If P, then Q", it does not necessarily follow that "If Q, then P".

To help you with this analysis, use the example "If the sun shines, Ralph plays tennis" or any other conditional that is consistent with your experience. For an additional challenge, try to answer this one: "If there is an even number on one side, then there is a vowel on the other side" OR "If there is not an even number on one side then there is not a vowel on the other side".

Solution to Liar Puzzle: Direct the question to either Tom Or Larry and ask the question: Which direction would the other man give? Then proceed to take the opposite route. Why?

Russell's "Barber Paradox": Suppose the barber does shave himself. Then, since he shaves only those who do not shave themselves, it follows that he does not shave himself - an internal contradiction. Suppose, on the other hand, he does not shave himself. Then, since he shaves all those who do not shave themselves, it follows that he does shave himself - another contradiction. Hence, if our barber shaves himself, then he doesn't and if he does not shave himself, then he does. How do we avoid these logical snares and how do you explain this riddle? It would appear that such a barber cannot exist. A paradox of this genre essentially destroyed approximately ten years work by Bertrand Russell and A.N. Whitehead. Their magnum opus, Principia Mathematica, was essentially an attempt to derive mathematics from the rules of logic. When the paradox was revealed to Russell  by Frege, he  was devastated by the revelation. Russell felt that if there was any certainty to be had in the Universe, it would surely be found in Mathematics. Here is Russell in his own words

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.                                                - Bertrand Russell

Can You detect the flaw in this Proof?

                     For Home: