Phil 2511: Paradoxes

Lecture 11   A New Cassationism

 

  1. Let us start with a paradox that Mark Sainsbury (course text, p.148) calls `The Infallible Seducer’ .   In his words:

 

An unsuccessful wooer was advised to ask his beloved the following two questions:

(1)    Will you answer this question in the same way that you will answer the next?

(2)    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.

[This paradox should immediately remind you of MINIAC, and note J.L. Mackie’s demonstration, in Truth, Probability and Paradox, that MINIAC is a version of the Liar]

2.      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.

3.      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?’  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?  That’s 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.

4.      In the Infallible Seducer case, questions occurred within the paradox, but it is easy to find counterparts that do not feature this feature:

 

(3)    The next statement has the same truth-value as this one.

(4)    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

 

(5)    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 Chinese, 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.

  1. 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 (`What is wrong with Sorites arguments?’ Analysis 61 (2000), pp. 29 – 35).  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.
  2. 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

 

(6)    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 essentially Buridan’s take on his Sophism 8 (see G.E. Hughes, John Buridan on Self-Reference, pp. 51-4).

7.  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

 

(7)    The next statement is false

(8)    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.

8.  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)                                  

 

 

                                     (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.  Equally, we cannot let S be the statement `S is not true’ (the Strengthened Liar).  Clearly it 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.

 

 

 

 

Note:  If the above seems to you more like a scholarly paper than a set of lecture notes, the reason is that it is a scholarly paper – published recently in Analysis (July 2002).  I hope that it will give you some idea of a modern version of cassationism.  It would be foolish and arrogant of me to think that the views set out will be the last word on the Liar: like all other philosophical papers this one now faces the tribunal of critical examination, and it may turned out to be deficient – perhaps, for example, the algebraic analogy is too weak – or fatally flawed.