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LAURENCE GOLDSTEIN
Before you start thinking that this is a paper on environmental issues, let me say that what it is really about is how one may dispose of certain paradoxes by refusing, for good reason, to answer the questions they pose. That may sound a little enigmatic, so let us start with the most straightforward example, a paradox that Mark Sainsbury calls `The Infallible Seducer (Sainsbury 1995:148). In his words:
An unsuccessful wooer was advised to ask his beloved the following two questions:
Will you answer this question in the same way that you will answer the next?
Will you sleep with me?
If she keeps her word, she must answer `Yes to the second question whatever she has answered to the first.
It seems clear that, if the woman in question is honest and wants to avoid seduction, then she should refuse to answer the first question. Can she refuse to answer it? Of course she can, and anyone with a grain of sense would, on being confronted with question (1), refuse to answer it. An answer gifts a blank cheque to a possibly unscrupulous questioner.
Refusal to answer is often claimed to be the proper policy with regard to loaded or complex questions. Thus, any man who has never beaten her should refuse to be trapped into answering the question `Have you stopped beating your wife yes or no?. This case is not quite clear-cut, because, so it could be argued, the mans proper, truthful answer is `No or, just to forestall any possible misunderstanding `No in order to stop one must have started, and I never started beating her. However, there are other questions to which, clearly, neither `Yes nor `No is the correct answer. If I explain to you that I abhor gratuitous violence but applaud violence perpetrated in the pursuit of a just cause, then I should be right to refuse to answer your impatient question `Just tell me straight, yes or no, do you applaud violence?. I am not obliged to fall into a trap that has been set for me. Similarly, we should refuse to allow ourselves to be bullied by Kripke (1979) into answering his question `Does Pierre or does he not believe that London is beautiful?. If we know Pierre to believe to be beautiful those parts of London photographs of which he has seen in a book, but to be not beautiful those parts that he has actually seen in the raw (he doesnt realize that they are all parts of the same town), then it would be incorrect to answer Kripke with a `Yes he does or a `No, he does not (Goldstein 1993). (If we think that Pierre has inferred that what he saw in his picture book was representative of the whole of London and also that what he saw in the raw was representative of the whole city, then we can answer both `Yes, he does and `No, he does not to Kripkes question -- under these circumstances, poor inductive logic has led Pierre to two opposite conclusions.)
The Fallacy of False Choice is committed by someone who presents a number of alternatives as exhaustive when, in fact, further options exist. `Yes or `No may not be exhaustive alternatives as answers to a question; perhaps a qualified `Yes or a qualified `No would be the right answer; sometimes refusal to answer `Yes or `No is the correct response. Does the Barber of Alcala shave himself? Thats a loaded question that we should refuse to answer, since simple logic tells us that its presupposition that there is such a barber who shaves all and only those who do not shave themselves is false. Similarly, we should refuse to answer the question of whether the Russell class contains itself, since the same logic tells us that there is no such class.
In the Infallible Seducer case, questions occurred within the paradox, but it is easy to find counterparts that do not feature this feature:
The next statement has the same truth-value as this one.
Pigs can fly.
Here the question arises as to whether (3) is true or false. If we pick either of these options, we commit ourselves to the truth of (4). We do not wish to be so committed, so the proper course is to refuse to answer the question. This is not a sullen, unreasonable refusal, but a principled one. Equally, one should refuse, on principle, to ascribe a truth-value to
The next statement is false.
In the absence of a statement following it, (5) has a meaning (we can understand it and translate it into Polish, and it is in virtue of its meaning what it does that we recognise it to have the `blank cheque property), but it does not have a content, for the content of (5) is dependent on there being a (contentful) statement following it. And if (5), given a context in which no statement succeeds it, has no content, it has no truth-value, so we are correct to refuse to ascribe it one.
Just as refusing to answer `Yes to a question is not tantamount to answering `No, so refusing to ascribe the value `true is not tantamount to ascribing `false. Hence (obviously) refusing to ascribe a truth-value is not tantamount to ascribing both falsity and truth nor to ascribing `not true and not false. While to do either of the latter is to make a claim, a refusal is, as Oswald Hanfling rightly observes, merely an abstention. To deny that a sentence-token occurring in a particular context is true is by no means the same as asserting that, in that occurrence, it is false. A token of (5), occurring in splendid isolation, is not false; but, in refusing to call it false, we are not eo ipso calling it true.
When a regular, contentful statement occurs immediately after a token of (5) then we can certainly say that that token of (5), so used, has a truth-value; it has the truth-value opposite to that of the statement that succeeds it. Suppose, however, that, on a given occasion, the sentence succeeding (5) is
The previous statement is false.
Here we fail to supply a content, and, although there is a consistent set of truth-values that we could assign (`true to the first, `false to the second or vice-versa), we should be unwilling to do so, first, because what is without content is without truth-value, second, because there can be no rationale for ascribing opposite truth-values when (5) and (6) are entirely symmetrical. This is exactly Buridans take on his Sophism 8 (Hughes 1982: 51 54).
A comparison may be of some help here. In algebra, simultaneous linear equations can be put in a canonical form
x = f(y)
y = g(x)
and can normally be solved, since the values of x and y are healthily interdependent. In the simplest type of case (Case A), one of the functions is a `constant function, for example:
x = 3y + 2
y = 5
In Case B, there is again healthy interdependency. An example:
x = 3y + 2
y = x/2 3
But sometimes the interdependency relation is sick (Case C), for example:
x = 3y + 2
y = x/3 2/3
(since the second equation is merely a rewriting of the first), and sometimes it is a fatality (Case D), for example
x = 3y + 2
y = x/3 6
Compare Cases A-D with cases of pairs of statements. Case A corresponds to a situation in which the first statement in the pair refers to the second, and the second is a well-behaved, grounded, statement such as `Pigs can fly. A pair of statements bearing comparison to Case B would be:
The next statement is false
The previous statement is true and pigs can fly
Case C corresponds to `truth-teller pairs of statements. In the example given, x and y can take any values subject only to the constraint that the value of x is 2 more than thrice that of y, and in the pairing of (5) and (6) above, the statements (if it is correct so to call them) can take any value subject only to the constraint that the value of one is opposite to that of the other.
The `fatal case D compares, of course, to paradoxical pairs such as
The next statement is false
The previous statement is true
We can make the comparison even sharper by interpreting the variables x as standing for `the truth-value of (7), and y as standing for `the truth-value of (8), where the truth-values `true and `false are interpreted as the numerical values 0 and 1 respectively, with `+1 given a Boolean interpretation (`+ functions on {0,1} like ordinary arithmetic addition, except that +1+1 = 0) and which thus may be interpreted as a predication of falsity. Since (7) is `The next statement [viz. (8)] is false, the truth-value of (7) is the truth-value of `The next statement [viz. (8)] is false. The algebra of the (7), (8) pair is thus
x = y + 1
y = x + 0
-- self-evidently fatal.
The image of going round in circles when truth-evaluating (7) and (8) can be captured by a graphic representation:
+1
(7) (8)
+0
This circle floats free in a contentless void. But evaluative circles can be rooted in terra firma grounded -- if at least one of the sentences on the circumference can be truth-evaluated, as in Case A. For example
(10) (11)
(9) (12)
(13)
Suppose that (9) (12) are all tokens of `The next (clockwise) statement is false, and that (13) is a token of `Pigs can fly. Here (9), (11) and (13) are false, (10) and (12) are true.
A striking example of a free-floating, contentless circle is
(14)
where (14) is `The next (clockwise) statement is false. This is, of course, a version of the standard Liar. The corresponding equation is `x = x + 1, the fatal contour of which can be expressed by saying that there is no number that can satisfy x. And similarly, when we set up the Liar, by saying `Let L be the statement L is false, the proper response is that no such stipulation can be made, that there is no statement for L to be. Whatever truth-value L has, `L is false has the opposite truth-value so, by the Principle of the Indiscernibility of Identicals, L and `L is false cannot be identified. Equally, we cannot let S be the statement `S is not true (the Strengthened Liar). Clearly S cannot be true and it cannot be false, so we can refuse to ascribe it either value. What has no truth-value is no statement, so S fails to state that it is not true. We can state of the sentence S that it fails to yield a statement and hence is not (the sort of thing that could be) true.
The University of Hong Kong
Pokfulam Road, Hong Kong, China SAR
HYPERLINK "mailto:laurence@hkusua.hku.hk" laurence@hkusua.hku.hk
References
Goldstein, L. 1993. The fallacy of the simple question. Analysis 53:178 81.
Hanfling, O. 2001. What is wrong with Sorites arguments? Analysis 61: 29 35.
Hughes, G.E. 1982. John Buridan on Self-Reference: Chapter Eight of Buridans Sophismata. Cambridge: Cambridge University Press.
Kripke, S. 1979. A puzzle about belief. In Meaning and Use, ed. A. Margalit, 239 83. Dordrecht:Reidel.
Richard, M. 2000. On an argument of Williamsons. Analysis 60: 213 17.
Sainsbury, R.M. 1995. Paradoxes (Second Edition). Cambridge: Cambridge University Press.
Storer, T. 1961. MINIAC: Worlds smallest electronic brain. Analysis 22: 151 52.
As Sainsbury points out (1995: 148), a generalisation of this paradox produces MINIAC (Storer 1961).
What we have here is a simplified version of Buridans Sophism 10. See Hughes 1982: 57 58.
See Hanfling 2001: 35. I am grateful to Hanfling (private correspondence) for sometimes reminding me of what one would ordinarily say something one is likely to forget when operating in philosophical mode. The point that `there is a difference between denying a sentence and affirming.its negation is made by Richard 2000:214.
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