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 [
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[
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
(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 [
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 [
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)
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.)
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'.
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
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’.
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