Phil 2511: Paradoxes
The Fibonacci series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…. has many beautiful and interesting properties. It is an infinite series of numbers in which each term is the sum of its two immediate predecessors: Fib[i] = Fib[i-2] + Fib[i-1]. Yablo’s Paradox (1993) consists of an infinite sequence of sentences, each saying of all its successors that they are not true. I hope to show that there is an interesting connection between the series and the sequence. There is nothing paradoxical about the series, but the sequence is paradoxical, and I want to claim that, once the connection is made, the paradox is swiftly solved.
Each member of the Fibonacci series `remembers’, and sums, its immediate two predecessors. Series constructed on the same general principle, but with better memories, are invariably less interesting than Fibonacci’s original. In the `total recall’ series, where each member sums all of its predecessors, the first two members are identical and any member after that is simply twice its predecessor. One can also have `soothsayer’ or `forward-looking’ variants of Fibonacci. For example, the series in which each member is the sum of its immediate two successors is just the Fibonacci series in reverse.
One way to manufacture an extremely boring variant of Fibonacci is to go Boolean, using a binary arithmetic in which addition (`+’) is defined by the following matrix:
x y x + y
1 1 0
1 0 1
0 1 1
0 0 0
The Boolean Fibonacci, in which each member is the Boolean sum (as defined above) of its immediate two predecessors, is:
1
1
0
1
1
0
.
.
.
.
Unlike Fibonacci’s series, or the related `Golden String’, this series is repetitive — it is an infinite repetition of a three-digit pattern. [For information on the Fibonacci numbers, the Golden Section and the Golden String, consult Ron Knott's impressive web site http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html]
Classical logic is two-valued. Negating a proposition yields a value opposite to that of the original proposition. It is usual to illustrate this by means of a truth-table, with the two values abbreviated as T and F, but, for present purposes, it will be useful to use the binary digits 0 and 1 respectively. The resulting table looks like this:
|
val (p) |
val (~p) |
|
0 |
1 |
|
1 |
0 |
Thus val (~p) = 1- val (p)
Classical disjunction is represented in the following table:
|
val (p) |
val (q) |
val (pvq) |
|
0 |
0 |
0 |
|
0 |
1 |
0 |
|
1 |
0 |
0 |
|
1 |
1 |
1 |
As should be obvious, classical disjunction translates as Boolean multiplication,
val (p v q) = val (p) x val (q). All of the classical truth-functional connectives can be interpreted as Boolean functions.
Since classical semantics has this arithmetic interpretation, we might entertain the thought that there is some counterpart in classical logic to the Boolean Fibonacci. Consider, then, an infinite sequence of statements, each one of which reads `Both of the next two statements are not true’. We shall call this the Fibbernacci sequence. Let us use the abbreviation `Si’ for the name of the ith statement in the sequence. Now construct a value table for the sequence (I show only the first five rows):
|
S1 |
0 |
|
1 |
|
|
S2 |
1 |
0 |
1 |
0 |
|
S3 |
1 |
1 |
0 |
0 |
|
S4 |
0 |
1 |
1 |
* |
|
S5 |
1 |
0 |
1 |
|
The first point to note is that, where any sentence Si has the value 0 (i.e., `true’) this requires that both Si+1 and Si+2 are not true; both have value 1. Further, since, under this assumption, Si+1 is not true, and what it says is that each of the statements succeeding it are not true, it follows that at least one of those statements must be true, and since, as we have already worked out, Si+2 is not true, it must be Si+3 that is true. And now this reasoning repeats: since Si+3 is true, the two statements following it are untrue etc.. Thus, it is easy to trace the consequence of any statement in the sequence being true. If some statement in the sequence (say S1) is not true, then (as shown) we have to consider three possibilities, shown by the three branches. However, the third branch is not a live option since, as we have already shown, a statement with value 0 must be followed by two with value 1. Therefore there are just three columns in the table, and each displays the `011’ pattern. So the Fibbernacci has a Boolean Fibonacci evaluation table.
Let us turn now to Yablo’s sequence of sentences. It is a total foresight sequence, since each of the constituent sentences is about all of the subsequent ones. Each statement Yk in the Yablo sequence says `For all integers n>k, Yn is not true’. The Yablo sequence is paradoxical. This is best seen by drawing a value table:
|
Y1 |
0 |
1 |
|
Y2 |
1 |
. |
|
Y3 |
1 . |
. |
|
Y4 |
1 . |
. |
|
. |
1 . |
. |
|
. |
1 . |
0 |
|
. |
1 . |
|
|
. |
1 0 |
|
|
. |
1 * |
|
|
. |
1 |
|
If Y1 has value 0 (true) then all subsequent statements in the sequence must be not true. In particular, Y2 is not true. But since Y2 says that all statements subsequent to it are not true, at least one of them must be true, and that is inconsistent with the result already obtained that, under the assumption that the first statement in the sequence is true, all of the rest must be not true. Hence, by reductio, Y1 is not true. However, if Y1 is not true, that means that at least one statement in the sequence must be true. But now we can reason about that statement’s truth and derive a contradiction in just the same way as we reasoned to a contradiction from the assumption that Y1 was true. Hence, the assumption that Y1 is not true equally cannot be sustained.
We talk about Yablo’s Paradox, but it should be pointed out that all we have so far shown is that Y1 cannot be true and also that Y1 cannot be not true. Now there would be a paradox only if we had independent grounds for saying that Y1 must be true or false. There certainly seem to be such grounds, because all sentences in the Yablo sequence are well-formed, and each is intelligible — each says that all subsequent statements in the sequence are not true. It is because we understand the sentences in the sequence that we can reason about them, infer contradiction and so on. However, let us go back to the Fibonacci series to see if it can throw some light on our problem.
Earlier on, we pretended to give a proper characterisation of the Fibonacci series. But our characterisation was incomplete; all we said was that Fib[i] = Fib[i-2] + Fib[i-1]. Clearly we cannot assign any numerical values to any member of the series unless we first specify Fib[1] and Fib[2], or at least first stipulate a value for at least two members of the series. For different stipulations, you obviously get different series; without such stipulation, no particular series but only a generic one is indicated. Without the relevant stipulation, the series is ungrounded. Now, the Yablo series is a counterpart of the total foresight variant of the reverse Fibonacci — any member of the series purports to say something about all the subsequent members. The truth-value of any member of this sequence is grounded on the truth-values of subsequent members, since it asserts the untruth of each of them. But the truth-value of each of these is, in turn, grounded on the truth-values of members subsequent to it. An infinite series of this sort has no last member (just as you cannot have the last stage at the end of an infinite process), so a fortiori there is no truth-value to be ascribed to its last member. And since the truth-value of any statement in the series is inherited from the truth-values of subsequent members, no member of the Yablo sequence has a truth-value (or each has the value GAP), just as no member of our underspecified (generic) Fibonacci has a numerical value.
One could write out the standard Fibonacci series in the following way:
1
1
The sum of the two preceding numbers
The sum of the two preceding numbers
The sum of the two preceding numbers
etc.
Each token noun-phrase in this series designates a definite number (and each designates a number different from that designated by the others). But, in the generic Fibonacci
The sum of the two preceding numbers
The sum of the two preceding numbers
The sum of the two preceding numbers
etc.
each of the noun-phrases fails to designate any number. And, if one writes out the Yablo sequence as follows:
Each following statement is untrue
Each following statement is untrue
Each following statement is untrue
etc.
we can likewise say that each of those token sentences fails to make a statement — fails to have a truth-value. Note that the failure to have a value (or to have the value GAP) is not a consequence of the fact that one can derive a contradiction from the Yablo sequence. No member of the infinite sequence
The next statement is true
The next statement is true
The next statement is true
etc.
has a truth-value, even though one could consistently ascribe the value `true’ (or the value `false’) to all of them.