[FOM] Blue or not blue

[Matt Insall]

Sandy Hodges wrote:
``My question is, is there any way, in a paraconsistent logic, for me to
assert that the saucer is not blue, and also not in the state of being
both blue and not blue?


In particular, does

~ Blue(saucer) & ~ ( Blue(saucer) & ~ Blue(saucer) )

have this meaning?''





I have not done much with paraconsistent logics or dialethetics, but it
seems to me that you are right in using ``~ Blue(saucer) & ~ ( Blue(saucer)
& ~ Blue(saucer) )'' to convey this meaning.  But you also asked in your
post about communicating to an audience, and it seems to me that in
communicating to an audience, one wants to balance complexity with
readability.  If your language includes (potential) measurements of levels
of blueness, you should be able to alert your audience more quickly that the
saucer is ``not blue at all'', while the cup ``has a blue tint to it''.
Maybe this begs your question, for it is likely that in any formal language
you use, there will be situations like the one you describe, in which the
language has not enough predicates or their interpretations have not been
established so as to make a (brief) communication of some particular
(sorites-type) phenomenon at hand that is expressible in natural language.

Thus, to solve the teacup and saucer problem itself, I would say to extend
the language, by adding some ``modified blueness'' predicates, and some
axioms that relate them to the ``blueness'' predicate.  One of these might
be a ``not at all blue'' predicate.  Another might be ``somewhat blue''.
Then add the axioms

somewhat_blue(x)--->(blue(x)&~blue(x))

somewhat_blue(x)--->blue(x)

not_at_all_blue(x)--->~blue(x)

and

not_at_all_blue(x)--->~(blue(x)&~blue(x)).

When your audience has learned these axioms, using ``somewhat_blue'' for the
teacup will, for the audience, not lead them to the same conclusions as
would using ``not_at_all_blue''.  In particular, ``not_at_all_blue''
eliminates the possibility that the teacup has some blueness to it, and for
some audiences is more pleasant to read or hear from a speaker than

``~ Blue(saucer) & ~ ( Blue(saucer) & ~ Blue(saucer) )'',

and yet, I have not specified that this new predicate means exactly

``~ Blue(saucer) & ~ ( Blue(saucer) & ~ Blue(saucer) )'',

merely that your description in the smaller language is implied by my
breifer predicate.  You may wish to emphasize the point by adding the axiom

not_blue_at_all(x)--->~somewhat_blue(x).

While this may be strictly redundant (I have not checked), for an audience
that uses both formal and natural language, I should think it would help
clarify the intended meaning of the new predicates, and improve the
readability of the output (if work is being done on a computer).  But to
completely specify the new predicates via an equivalence with statements
like what you described may be what you want, and I would suggest doing so,
in spite of the obvious redundancy, because of the reduction in the number
of symbols required to convey the intended meaning.  Then, in any
computation, if

``~ Blue(saucer) & ~ ( Blue(saucer) & ~ Blue(saucer) )''

appears, it is more pleasant to the audience to report

``not_blue_at_all(saucer)''.

However, blindly doing so for every sorites-type situation you come across
would lead to an unruly set of axioms and language extensions, so in making
these extensions, some cost-benefit analysis needs to be made, to help
determine which strings of the form

``~ P(x) & ~ ( P(x) & ~ P(x) )''

to replace with new predicates that are more expressive to an audience that
knows a particular natural language,a s well as the given formal language.





Matt Insall
>From a chilly Rolla