Socrates' Paradox

Socrates' Paradox

Truth Telling Series, Part the Umpteenth

By John Taylor; 5 January, 2006

This series on truth telling began when I came across a startling fact
in the recent primer, "Introducing Logic." Here I learned how an
ancient paradox known as the Liar's Paradox has acted as a bombshell
blowing apart all attempts to form an entirely consistent logical
system. In some form or other it has successfully foiled all attempts
by logicians throughout history, from the ancient Pre-Socratics to
Leibniz to Frege to Russell, right down to system makers of the
present day. You can put this paradox in one sentence, easily
understood even by the meanest non-logician like myself. So without
further ado, here is Epimenides Paradox, otherwise known as the Liar's
Paradox:

"I am telling you a lie."

This is a self-immolating declaration, an intellectual Black Hole. It
cannot be true unless it is false or false unless it is true.
Pre-Socratic philosophers used this along with various other paradoxes
for pious ends, to demonstrate that reason is not a reliable stepping
stone to the gods. Cynics and skeptics use it for less pious ends, to
prove that nothing ain't worth nothing, even if it is free. I had no
idea that such a simple trick could be so devastating throughout the
ages. It is astonishing to think that language or logical systems are
like precarious houses of cards brought to the ground by admitting
even one such self contradictory assertion. Every logician tries to
kick it out the door but perforce they have to let it in, and that is
the beginning of the end.

Some denizens of the 20th Century speculated that the Liar's Paradox
may be so perplexingly self-contradictory because it is
self-referential. It flags itself, as it were, then hits itself in the
face with the flag. Computer scientist Richard Hofstedter, for
instance, held aphoristically that "self-reference is the soul of
paradox." If so, it would be a phenomenon of prime interest to
scholars of the Baha'i writings, since most of Baha'u'llah's Writings
are self referential. Indeed He confides at one point that if it were
up to Him He would write about nothing else but the greatness of His
station, He would extol that only to the end of time. This stands to
reason, the highest truth is about itself only; if anything can and
should be self-referential speech it would be the speech of the
Supreme Being. But in the last century logicians demonstrated that the
Liars Paradox need not be self referential. Consider these two
sentences:

1. Sentence two is a lie.
2. Sentence one is a lie.

Taken together, they constitute a form of the Liars Paradox, I am
told, even though neither sentence refers to itself. The same goes
with three sentences in the same pattern,

1. Sentence two is a lie.
2. Sentence three is a lie.

3. Sentence one is a lie.

You can imagine any number of sentences accusing one another of lying,
right up to infinity. Taken together, they constitute an instance of
the liars paradox, without self-reference. Which leads to the question
whether it is the act of accusation of falsity itself that is the soul
of paradox. Consider these sentences:

1. Sentence two is true.
2. Sentence three is true.
...
499. Sentence 500 is true.
500. Sentence one is a lie.

One accusation of lying, then, will make any number of dependent
assertions of truth into a collective Liar's Paradox. Could it be that
it is the act of contradiction itself that makes the Liar's Paradox
paradoxical? Every sentence is an implicit assertion of truth. In the
front of every sentence that you possibly name you can tack on this
phrase: "It is the case that..." If a sentence denies that implicit
prefix, it nullifies itself, it nullifies language, and it violates
the implicit contract that any speaker has with any listener that both
seek truth. Are disunity, denial and contention, then, the soul of
paradox? As soon as experts disagree over a given issue, they nullify
themselves and their authority. Their expertise is a Liar's Paradox.
This has been recognized since ancient times. Worse, as James
Surowiecki in "The Wisdom of Crowds" points out, individual experts do
not known when they are making a mistake, thus further limiting their
wisdom in the face of a group.

Be that as it may, it is possible to distill a paradox into two words,
known as an oxymoron; examples are "screaming silence," "military
intelligence," and "lonely crowd." About a century ago Bertrand
Russell and set theory were tripping up on the Barber's Paradox. The
Barber of Seville shaves all men who do not shave themselves. The
paradox comes when you ask whether he shaves himself or not. At around
the same time linguist Kurt Grelling noticed that paradox can be
distilled even smaller than an oxymoron, he squeezed it into a single
word, "heterology." This became known as the Grelling's Paradox, or
the paradox of heterologicality. Here is how "A Dictionary of
Philosophy" explains it:

"Some words have the same property as that which they name: for
example, short is a short word and polysyllabic has many syllables.
These words are called homological or autological. In contrast,
heterological words such as `useless' or `monosyllabic' are not
instances of the properties they name. The paradox arises when
considering the word `heterological' itself; if it is heterological,
then it does not instantiate its meaning -- but this is what
`heterological' means, therefore the word is homological. Conversely,
if `heterological' is autological, then it must have the
characteristic of applying to itself and therefore heterological is
heterological."

Grelling was not the first to distill paradox into a single word,
however. When the oracle declared that there was no man wiser than
Socrates, it in effect proved the paradoxical nature of the word
"wisdom." Socrates knew that he was not wise and set out to prove the
oracle wrong by examining those in the know. He discovered that every
expert he visited thought he knew things when he really did not. Plato
describes how Socrates summed up this discovery in his Apology,

"So I left him, saying to myself, as I went away: Well, although I do
not suppose that either of us knows anything really beautiful and
good, I am better off than he is - for he knows nothing, and thinks
that he knows. I neither know nor think that I know."

Socrates knew that he did not know wisdom, and that made him wise. The
Wise Person's Paradox, as it were, or perhaps the Truth Teller's
Paradox. Anybody who knows how effective the liar's paradox demolishes
language would never venture to say, "This is true," for that in
itself would be an intentional lie. So Socrates dealt with the Liar's
Paradox by accepting the limits of reason. This did not make him
popular with those who professed to know, the worldly wise.

"This investigation has led to my having many enemies of the worst and
most dangerous kind, and has given occasion also to many calumnies,
and I am called wise, for my hearers always imagine that I myself
possess the wisdom which I find wanting in others: but the truth is, O
men of Athens, that God only is wise; and in this oracle he means to
say that the wisdom of men is little or nothing; he is not speaking of
Socrates, he is only using my name as an illustration, as if he said,
He, O men, is the wisest, who, like Socrates, knows that his wisdom is
in truth worth nothing."

--
John Taylor

badijet@gmail.com