A number of commentators, including the philosopher-logician G.
Spencer Brown and the anthropologist-systems theorist Gregory Bateson,
reframed variants of the Liar’s paradox as it might apply to real world
phenomena. Instead of being stymied by the undecidability of the logic or
the semantic ambiguity, they focused on the very process of analyzing these
relationships. The reason these forms lead to undecidable results is that
each time they are interpreted it changes the context in which they must be
interpreted, and so one must inevitably alternate between true and false,
included and excluded, consistent and inconsistent, etc. So, although there
is no fixed logical, thus synchronic, status of the matter, the process of
following these implicit injunctions results in a predictable pattern
across time. In logic, the statement “if true, then false” is a
contradiction. In space and time, “if on, then off” is an oscillation.
Gregory Bateson likened this to a simple electric buzzer, such as the bell
in old ringer telephones. The basic design involves a circuit that includes
an electromagnet which when supplied with current attracts a metal bar
which pulls it away from an electric contact that thereby breaks the
circuit cutting off the electricity to the electromagnet which allows the
metal bar to spring back into position where the electric contact re-closes
the circuit re-energizing the electromagnet, and so on. The resulting
on-off-on-off activity is what produces a buzzing sound, or if attached to
a small mallet can repeatedly ring a bell.
Consider another variant of incompletability: the concept of imaginary
number. The classic formulation involves trying to determine the square
root of a negative number. The relationship of this to the liar’s paradox
and the buzzer can be illustrated by stepping through stages of solving the
equation *i *x* i* = *-1*. Dividing both sides by *i* produces *i* = *-1/i,
*and then substituting the value of *i *one gets* i = -1/-1/i *and then
again* i = -1/-1/-1/i *and so forth, indefinitely. With each substitution
the value alternates from negative to positive and cannot be resolved (like
the true/false of the liar’s paradox and the on/off of the buzzer). But if
we ignore this irresolvability and just explore the properties of this
representation of an irresolvable value, as have mathematicians for
centuries, it can be shown that *i* can be treated as a form of unity and
subject to all the same mathematical principles as can 1 and all the real
numbers derived from it. So *i *+ *i* = 2*i* and *i* - 2*i* = *-**i* and so
on. Interestingly, 0 x *i* = 0 X 1 = 0, so we can conceive of the real
number line and the imaginary number line as two dimensions intersecting at
0, the origin. Ignoring the many uses of such a relationship (such as the
use of complex numbers with a real and imaginary component) we can see that
this also has an open-ended consequence. This is because the very same
logic can be used with respect to the imaginary number line. We can thus
assign *j *x *j = -i *to generate a third dimension that is orthogonal to
the first two and also intersecting at the origin. Indeed, this can be done
again and again, without completion; increasing dimensionality without end
(though by convention we can at any point restrict this operation in order
to use multiple levels of imaginaries for a particular application, there
is no intrinsic principal forcing such a restriction).
One could, of course, introduce a rule that simply restricts such
operations altogether, somewhat parallel to Bertrand Russell’s proposed
restriction on logical type violation. But mathematicians have discovered
that the concept of imaginary number is remarkably useful, without which
some of the most powerful mathematical tools would never have been
discovered. And, similarly, we could discount Gödel’s discovery because we
can’t see how it makes sense in some interpretations of semiosis. On the
other hand, like G. Spencer Brown, Doug Hofstadter, and many others,
thinking outside of the box a bit when considering these apparent dilemmas
might lead to other useful insights. So I’m not so willing to brand the
Liar, Gödel, and all of their kin as useless nonsense. It’s not a bug, it’s
a feature.
On Mon, May 2, 2016 at 2:19 PM, Maxine Sheets-Johnstone <m...@uoregon.edu>
wrote:
> Many thanks for your comments, Lou and Bruno. I read and pondered,
> and finally concluded that the paths taken by each of you exceed
> my competencies. I subsequently sent your comments to Professor
> Johnstone—-I trust this is acceptable—asking him if he would care to
> respond with a brief sketch of the reasoning undergirding his critique,
> which remains anchored in Gödel’s theorem, not in the writings of others
> about Gödel’s theorem. Herewith his reply:
>
> ********
> Since no one commented on the reasoning supporting the conclusions reached
> in the two cited articles, let me attempt to sketch the crux of the case
> presented.
>
> The Liar Paradox contains an important lesson about meaning. A statement
> that says of itself that it is false, gives rise to a paradox: if true, it
> must be false, and if false, it must be true. Something has to be amiss
> here. In fact, what is wrong is the statement in question is not a
> statement at all; it is a pseudo-statement, something that looks like a
> statement but is incomplete or vacuous. Like the pseudo-statement that
> merely says of itself that it is true, it says nothing. Since such
> self-referential truth-evaluations say nothing, they are neither true nor
> false. Indeed, the predicates ‘true’ and ‘false’ can only be meaningfully
> applied to what is already a meaningful whole, one that already says
> something.
>
> The so-called Strengthened Liar Paradox features a pseudo-statement that
> says of itself that it is neither true nor false. It is paradoxical in that
> it apparently says something that is true while saying that what it says it
> is not true. However, the paradox dissolves when one realizes that it says
> something that is apparently true only because it is neither true nor
> false. However, if it is neither true nor false, it is consequently not a
> statement, and hence it says nothing. Since it says nothing, it cannot say
> something that is true. The reason why it appears to say something true is
> that one and the same string of words may be used to make either of two
> declarations, one a pseudo-statement, the other a true statement, depending
> on how the words refer.
>
> Consider the following example. Suppose we give the name ‘Joe’ to what I
> am saying, and what I am saying is that Joe is neither true nor false. When
> I say it, it is a pseudo-statement that is neither true nor false; when you
> say it, it is a statement that is true. The sentence leads a double life,
> as it were, in that it may be used to make two different statements
> depending on who says it. A similar situation can also arise with a Liar
> sentence: if the liar says that what he says is false, then he is saying
> nothing; if I say that what he says is false, then I am making a false
> statement about his pseudo-statement.
>
> This may look like a silly peculiarity of spoken language, one best
> ignored in formal logic, but it is ultimately what is wrong with the Gödel
> sentence that plays a key role in Gödel’s Incompleteness Theorem. That
> sentence is a string of symbols deemed well-formed according to the
> formation rules of the system used by Gödel, but which, on the intended
> interpretation of the system, is ambiguous: the sentence has two different
> interpretations, a self-referential truth-evaluation that is neither true
> nor false or a true statement about that self-referential statement. In
> such a system, Gödel’s conclusion holds. However, it is a mistake to
> conclude that no possible formalization of Arithmetic can be complete. In a
> formal system that distinguishes between the two possible readings of the
> Gödel sentence (an operation that would considerably complicate the
> system), such would no longer be the case.
> ********
>
> Cheers,
> Maxine
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>
--
Professor Terrence W. Deacon
University of California, Berkeley
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