[FOM] Dialetheism, strong negation, and paradox

Sandy Hodges SandyHodges at attbi.com
Mon Nov 25 12:49:12 EST 2002

I recently posted concerning the difficulty in a dialethic system, of
expressing certain knowledge states; my example concerned the borderline
blueness of a teacup.  The problem does not arise in a system that has a
strong negation (such as those of da Costa).    In those systems without
a strong negation, the problem is real; it was pointed out by Terrence
Parsons in his "Assertion, denial and the liar paradox" Journal of
Philosophical Logic, 1984, 13:137–152.   A related problem occurs with
truth-gap theories, again only with those that lack a strong negation.
But why would anyone not want to adopt a strong negation operator?
One answer concerns the treatment of the Liar paradox in these systems;
if you have only choice negation, you can formulate only a weak version
of the Liar, which these systems can handle in a quite satisfactory way,
but with strong negation, one can formulate a stronger Liar.  With the
stronger Liar, one can still ascribe it a truth status, and provide a
general algorithm for ascribing truth statuses to sentences, but this
algorithm is less satisfactory.

Here is a Liar formulated with choice negation:

Sentence 1.   ~ True(Sentence 1)

The truth status ascribed to sentence 1 in a dialethic system is that it
is both true and false.   Given a status of "both" for sentence 1, the
status of "True(Sentence 1)" can be found using the truth table for
"True(x)":  it has the status "both".   Then the status of  "~
True(Sentence 1)" can be found using the truth table for choice
negation: the status is again "both".     Thus when the status "both" is
ascribed to sentence 1, and the truth tables applied, the ascribed
status is confirmed.  And "both" is the only status that, if ascribed,
is confirmed.   This gives a good reason to think the ascription is
correct.    Now contrast this situation with a Liar formulated using
strong negation:

Sentence 2.   ~s True(Sentence 2)

If we ascribe the status "both true and false" to sentence 2, then
"True(Sentence 2)" gets status "both", but "~s True(Sentence 2)" gets
status "false."    There is no status we can ascribe that gets
confirmed.    So our only option is to say that it is precisely the fact
that no status is confirmed for sentence 2, that makes it a paradox; we
say it is both true and false *because* it is a paradox.    Thus, in our
conclusion, we call sentence 2, "both."    We have truth tables.   But
when we work out the status of sentence 2 according to our own
conclusion, using those tables, the tables say that sentence 2 has
status "false."    Nonetheless we stick to our conclusion that sentence
2 is "both."   So we have a system which has truth tables, but
paradoxical sentences get a status of "both" imposed on them,
over-riding what the truth tables say.   A system where a paradox status
over-rides the truth tables, will seem to many to be 'ad hoc' and
implausible.   But others will say that strong negation is a perfectly
ordinary concept, and dealing with the paradoxes by dropping this
concept is to avoid the paradoxes, rather than to come to grips with
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The situation is similar when we consider "truth-gap" theories; with
only choice negation, the status of "neither true nor false" for
sentence 1 is confirms itself.   With strong negation, there is no
status for sentence 2 that confirms itself, so "neither" has to be
imposed as a paradox status, over-riding the truth tables.
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Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.

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