Phil 2511: Paradoxes

Lecture 2:  More Paradoxes


Our aim in this course is to try to solve some paradoxes.  But a necessary preliminary to that is to familiarize ourselves with a variety of paradoxes, in the hope that we shall find some interconnections and similarities that might give us some clues as to solve them


The Paradox of the Preface [Clark, Course text: p.144]

It is quite usual for authors of books to apologize, in the preface for errors in the text.  If the author has written books before, he will know perfectly well that it is impossible to avoid at least one error of some sort or other, so it is quite rational for him to make this modest announcement.  Yet he would not have made any assertion in the text that he didn't believe to be true.  But then he is being inconsistent -- he believes that each assertion in the text is true, yet believes that at least one of them is false.

(D.C. Makinson, `The Paradox of the Preface', Analysis 25(1965), pp.205-207.


Kavka's Toxin[Clark, Course text: p.195]

An eccentric billionaire presents you with a vial of toxin that, if you drink it, will make you painfully ill for a day, but will not threaten your life or have any lasting effects.  He offers you one million dollars if, at midnight tonight, you intend to drink the toxin tomorrow afternoon.  You will be paid the money just for having that intention.  If, after midnight, you change your mind, and decide not to drink the stuff, you'll still get the million dollars.  Since you know this is the deal, you know that you won't drink it after having succeeded in forming the intention at midnight which qualifies you for the million.  But how can you intend to do something that you know you will not do?

(G. Kavka, `The Toxin Puzzle', Analysis 43 (1983), p.507.)


Roy Sorensen, in `A Strengthened Prediction Paradox', Philosophical Quarterly 36 (1986) argues for a family connection between Kavka's puzzle and the Surprise Examination.  In a very short paper, I try to show that there is an interesting connection between these two and Newcomb’s Paradox:  Laurence Goldstein `Examining Boxing and Toxin’, Analysis 63 (2003), pp.242-4.


Newcomb’s Paradox [Clark, Course text: p.125; see also Sainsbury’s book Paradoxes for a good chapter on this and other paradoxes of rationality]

A TV game show.  There are two boxes, one opaque, the other transparent.  In the latter you can see $10,000.  Your choice:  You can either take the content of both boxes, or just the content of the opaque box.  The catch:  Someone has put $1,000,000 in the opaque box if he has predicted that you will choose to pick only the opaque box; if he has predicted that you’ll choose to take the content of both boxes, he’ll have put nothing in the opaque box – and this predictor has been extremely accurate in his past predictions.



Grelling's Paradox [Clark, Course text: p.80]

The predicate `is short' is true of the adjective `short', i.e., the statement `"short" is short' is true.  Let's say that a predicate is autological if it has this property of being true of the corresponding adjective.  We shall say that a predicate is heterological if it does not have this property.  So, for example, `monosyllabic' is heterological because `monosyllabic' is not monosyllabic.  So, in general `x' is heterological if and only if `x' is not x.  Now check whether the predicate`heterological' is itself heterological, by substituting `heterological' for`x' in the above equivalence.  We obtain the contradiction `heterological' is heterological if and only if `heterological' is not heterological.

(Note: My own views on some of the lessons to be learned from Grelling’s paradox are contained in `Categories of Linguistic Aspects and Grelling's Paradox', Linguistics and Philosophy 4 (1981), pp.405-421 and `Linguistic Aspects, Meaninglessness and Paradox: A Rejoinder to John David Stone', Linguistics and Philosophy 4 (1982), pp. 579-592 and `Farewell to Grelling’, Analysis 63 (2003), pp.31-2.


The Barber of Alcala

Logicians tell of a village barber who shaves all those villagers - and only those - who do not shave themselves.  The question of the barber's own toilet holds a certain fascination for the logical mind.  For it has been agreed that the barber shaves any villager, x, if and only if x does not shave himself; hence when we let x be the barber, we conclude that he shaves himself if and only if he does not.

(W.V. Quine, `Russell's Paradox and Others', The Technology Review, November, 1941, pp.16-17)


Russell's Paradox [Clark, Course text: p.168]

Let R be the class of all non-self-membered classes, i.e. a class x is member of R if and only if x is not a member of x.  In symbols, x Î R ~(x Î x)


We can answer the question of whether R itself is a member of R by substituting `R' for `x' in the above equivalence.  The result is


           R Î R ~(R Î R)


and this entails a contradiction.


You should be able to see that there is a structural similarity between these two paradoxes and the Grelling Paradox we mentioned last time.  When paradoxes have a common structure, it is reasonable to suppose that they will have a common solution.



Take a coin and designate one side `Yes' and the other `No'.  Think of a question for which you would very much like an answer, e.g. `Will I find the girl of my dreams?'.  Toss the coin and note the answer it gives.  But how can we tell whether this answer is true or false?  Easy.  Ask the question `Will your present answer have the same truth value as your previous answer?', flip the coin and note the response (`Yes' or `No').  If the second response is `Yes' then MINIAC's answer to the first question was true; if the second response is `No' then the answer to the first question was false, so now you know for sure whether you will find the girl of your dreams.  Proof: (I'll just do the proof for the case where the second answer is `No.')  Suppose the second answer is `No'.  This answer must be true or false.  If it's true then the answer to the first question is false.  But, if the answer (`No') to the second question is false, then the truth-value of the second question must be the same as that of the first, so, again the answer to the first question is false.  Therefore, if the second answer is `No' we have proved that, whether or not that answer is true, the answer given by MINIAC to the first question must be false.  By similar reasoning we can prove that, if MINIAC answers `Yes' to the second question its answer to the first question must have been true.  What's paradoxical here, of course, is that one cannot guarantee to get correct answers to momentous questions merely from two flips of a coin.

(Reference: T. Storer, `MINIAC: World's Smallest Electronic Brain', Analysis 22 (1961-2), pp.151-152.)



Berry's Paradox [Clark, Course text: p.14]

The number of syllables in the English names of finite integers tends to increase as the integers grow larger.  Hence the names of some integers must consist of at least nineteen syllables, and among these there must be a least.  Hence `The least integer not nameable in fewer than nineteen syllables' must denote a definite integer; in fact, it denotes 111,777.  But `The least integer not nameable in fewer than nineteen syllables' is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables!

(Source: B. Russell and A.N. Whitehead, Principia Mathematica, Vol.1 (Cambridge, Cambridge University Press, 1913), p.61). Note: This is not a paradox essentially about the definability of numbers, for we can contrive variants such as `The shortest description not describable in less than twenty syllables'.


Some Variants on the Liar

The discovery of the Liar is, by tradition, credited to Eubulides of Miletus, a student of Euclides who founded the Megarian school of logicians (see Diogenes Laertius 2.108).  A performative version (as we should now call it) involving a man who swears that he will break his oath is discussed by Aristotle in De Sophisticis Elenchis 180a32ff.  The earliest extant version of the Liar (one that does not actually quite capture its true paradoxicality) is in Cicero's Academica II, 96.  It is not clear whether the mediaeval logicians became acquainted with the paradox through a ninth century compilation of Cicero's philosophical works or by some other route, or whether they discovered them afresh. (For some conjectures, see de Rijk). A. Rustow's Der Lügner (Leipzig, Teubner, 1910) is a history of the Liar. Two collections of essays ed. R.L. Martin (1970, 1984) are devoted to this paradox.


The starkest presentation of the Liar that I have encountered is due to E. Teensma, The Paradoxes (Assen, Van Gorcum, 1969):



    3 = 3



                                    3 = 3




`The sentence in the square is true' is verified by verifying that 3 = 3.



The sentence in the rectangle is not true



The sentence in the rectangle is not true




`The sentence in the rectangle is true' is verified by verifying that the sentence in the rectangle is not true.


Another Liar variant involves indirect self-reference.  Take a T-shirt.  Print on the front `The sentence on the back is false’, and print on the back `The sentence on the front is true’.


We need to distinguish paradoxes from merely mistaken arguments, where, though the mistake lies deep, we succeed in exposing it and the problem disappears.  Paradoxes are essentially unsolved (though they may not be unsolvable).  One should distinguish a paradox from a result which is merely surprising.  There are many results in mathematics which are surprising – for example that the square root 2 cannot be expressed in the form m/n (where m and n are integers) or that there is no greatest prime number.  The standard proof, in each of these cases, starts with plausible assumptions, but the reasoning shows these assumptions to be untenable, so we accept the negations of these assumptions as true. The following are two interesting examples of non-paradoxes:


The Three Salesmen (a non-paradox)

`Consider the familiar story of three salesmen at a convention, who decide to stay the night at the same hotel. To save money, they agree to share a room. Since the room costs $30, each contributes $10. Later on, however, the desk clerk discovers that he should have charged the salesmen only $25 for the room and thus sends the bellhop up to their room with the $5 change. The bellhop, deciding that it is too difficult to split $5 three ways, returns only $3, pocketing the remaining $2. Now, since each salesman originally contributed $10 and subsequently received $1 back, we can conclude that each spent only $9 for the room. Thus the three salesmen spent $27 for the room, and the bellhop received $2; this makes $29. But they originally started with $30. So what happened to the other dollar?’

(Source: Charles Chihara, Ontology and the Vicious Circle Principle


The Monty Hall Dilemma (a non-paradox)

`Suppose you’re on a game show, and you’re given the choice of three doors;  Behind one door is a car; behind the others, goats.  You pick a door, say #1, and the host, who knows what’s behind the other doors, opens another door, say #3, which has a goat.  He then says to you `Do you want to pick door #2?’  Is it to your advantage to make the switch?’

Most people answer `No’.  But they are wrong. The correct answer is that, after the contestant has made his choice, on 2/3 of the occasions, there will be a car behind one of the two remaining doors. That figure is not changed by Monty opening one of the doors, so, by choosing the remaining door, the contestant has a 2/3 chance of winning the car whereas initially his chances were 1/3.  So, although, in actual practice, contestants who did change their minds tended to win the car twice as often as contestants who did not, there is no paradox here once you have understood  the probabilistic reasoning.