[FOM] A question about dialetheism and sorites

[Joao Marcos]

> > And this sentence is equivalent   (in da  Costa's   logic  C_{1},
> > for example)  of asserting a "strong negation"  ~s Blue(saucer),
> > having all properties of classical negation. So  you'd be  removing
> > any inexactness about the blueness of the  saucer, and  assuring
> > your audience that it is definitely not the case that the saucer is
> > blue.
>
> I looked up one of da Costa's papers, but not apparently the right
> one.   If you can say that a saucer is definitely blue, and say
> another is blue and not blue, and say a third is definitely not blue,
> could there be a saucer that was borderline in color between two of
> these three states?   For example could a saucer that is borderline
> between (definitely not blue) and (blue and not blue) be both?

   NO.  If something is [definitely not blue], in the sense mentioned by
Carnielli above and used by da Costa in several papers, than this
something is simply [not blue], in the classical sense.  Well, the
matter is in fact a bit more complicated, as you can surely have "strong
negations" as the one mentioned above inside some weak paraconsistent
logics, and still some of these negations might still not have "all
properties of classical negation", or might just not "behave
consistently"...
   Now, this whole matter might also depend, of course, on how you
define your "definitely (not)" operator(s) --in case you have them in
your language, to start with.  If you are willing to mean by this
"definitely", in the metalanguage, something like I thought you meant
before, that is, that the saucer is "not blue, and also not in the state
of being both blue and not blue" (sic), then I can hardly see anyway how
could you still allow for a "borderline"!!

   Note that this is *not* to say that paraconsistent logics will in
general have just three "logical values", namely [definitely true],
[definitely false], and --as you wish to put it-- [both true and false].
Indeed, *most* paraconsistent logics are not even finite-valued!  (This
is just the same phenomenon that happens in the case of intuitionistic
and most intermediate and normal modal logics.)  One can say thus that
there are, in general, several degrees of paradoxicality available,
along with the two usual classical values.

> Allen Hazen suggested I look at Graham Priest's Logic of Paradox.
> The primary claim made by this paper is:
>
> Claim (1): Some sentences are both true and false.
>
> But in his "Concluding self-referential postscript" he says (the
> equivalent of):
>
> It is not the case that some sentences are both true and false.
>
> This contradiction is not a retraction, however: rather it is an
> example of the very claim that (1) makes.    Thus, whatever
> Graham Priest may say, he is in no way committed not to say
> the exact opposite in another
> place.    No doubt there are some claims he has no intention on
> contradicting, but there seems to be no mechanism for indicating which
> these are.    In particular, if he wished to claim that a certain
> saucer was definitely not blue, it would not help in his system
> to say:
>
> The saucer is not blue and "The saucer is blue" is not true.

   How do you mean that he can "say" this sentence in his system?

> It would not help because he is using "weak" truth, so that if "The
> saucer is blue" is both true and false, then "'The saucer is blue' is
> not true" is both true and false.
>
> At one point in his paper he makes use of the concepts "true only" and
> "false only."  "true only" means true without also being false.    He
> may perhaps wish these concepts to be "strong," so that
>
> "'The saucer is blue' is true only"
>
> will be false (and false only) if "The saucer is blue" is both true
> and false.    He does not provide an analysis of sentences using "true
> only" however.     Priest claims that the Tarskian bi-conditional:
>
> True(a)  iff  A
>
> [where "a" names a sentence, and "A" stands for the content of that
> sentence]
> must be true (true only, I think he means) even is A is paradoxical.
> This is a somewhat remarkable claim, since he doesn't think even modus
> ponens is valid when the premises are paradoxical.    But the more
> interesting claim would be that:
>
> True-only(a)   iff   A
>
> is true only (or even that it is true) for paradoxical A's.    I guess
> that he would not claim this.

   There is no way of linguistically expressing, *inside* Priest's LP (a
logic previously presented several years before in Asenjo 1966), that
some proposition is "true-only" or "false-only".  This incredible
debility of LP has made it into a preferred target for several
philosophers, I cite only Batens, 1990 (see references below).  Priest
himself tried to recover from some such criticisms in a paper from 1991,
but he will still insist somehow on using LP "both at the logical and
the metalogical level", a proposal which not many people seem to have
agreed on, this far.  Thus, you will in fact find very few
paraconsistent logicians willing to agree on the manoeuver sketched by
Priest in the above mentioned "Concluding self-referential
postscript"...  Walt Whitman liked to contradict himself, but this is
not a universally agreed feeling.

   The failure of modus ponens does not seem to be so serious.  It is in
fact easy to upgrade LP (conservatively extending its
conjunction-disjunction-negation fragment) into logics with "fully
classical" implications, once you take implication as just another
primitive connective, abandoning once and for all the idea that it
should always be defined in terms of disjunction and negation
(inter-definition of connectives often fail, anyway, in the case of
other non-classical fragments of classical logic).  Check for instance
the three-valued logic Pac, in Avron 1991 and elsewhere.

   Now, to upgrade LP in order to be able to express "strong negation",
"classical negation", "definitely (not)", "really false" (a bottom
constant), "(in)consistency", etc, you will have to transform this logic
into something completely diverse, with completely different properties.
In fact, one of the most fundamental ideas of da Costa was exactly that
one should be allowed to "recover" classical reasoning inside some
paraconsistent logics.  There are of course several direct ways of doing
this, none of them being available neither in LP (as Priest had planned)
nor in Pac.  To be able to do such in a strong way (by way of a
grammatical translation, for instance, as in correspondence theory) one
had better make a further extension of these logics, adding some of the
operators mentioned in quotes at the beginning of this paragraph.  This
is feasible and simple, as you will see below.

   One particularly interesting and comprehensive approach seems to stem
from the addition of a "consistency" operator, as *consistency* is
exactly the property that a paraconsistent logic should lack, in order
to avoid *explosiveness*.  This idea gives origin to the previously
mentioned "Logics of Formal Inconsistency", a particularly interesting
subclass of paraconsistent logics, and extends the forerunning work of
da Costa, 1963, 1974, etc.  The idea is explored in extensive detail in
our Carnielli & Marcos 2002 paper.  Such framework allows
*inconsistency* to be expressed and tolerated, and its very notion to be
sharply separated from the notion of *contradictoriness*.  Many more
details about this and about in fact most other things I have written
above can be found in this last self-contained paper.

> Priest compares his "both true and false" system with other "neither
> true nor false" ones.    But if the comparison is to be made, then the
> mapping between them should be
>
> Priest           Belnap e.g.
> true only <----> true
> both true and false <-----> neither true nor false
> false only <-----> false
>
> with this mapping, the systems are not so different; the main
> difference
> is that Priest is willing to assert paradoxical statements (since
> after all they are true, even if they are also false).

   I am not sure about your example.  The logic usually referred to as
"Belnap's logic" is a four-valued logic, containing all the values
[true], [false], [both], and [neither].  Perhaps here you mean "Kleene's
(1952) logic", whose matrices in fact coincide with those of LP?

   A diversion.  I always feel that there is something fishy behind this
reading "both true and false".  Is it just the same thing to say that
"both a proposition p and its negation are true" and to say that "a
proposition p is both true and false"??  In practice, indeed, that seems
to result in just a preferred way of *reading* things, and while I have
found some people who do not "understand" the first reading, I have
found much more people that do not "agree" with the second reading.

   If we restrict ourselves to the first reading in the above paragraph,
though, as any practicing many-valued logician would, the difference
between LP and Kl are that the former selects TWO *designated* values
(the "true" values), while the latter selects only ONE designated value.
The proposed "mapping" is thus pretty obvious, but then it unfortunately
does not inform much:  the selection of designated values changes the
whole story, and it usually changes a given logic into a *completely
different* one.  (Check the set of theorems or of inferences, to see
what I mean.  Kl, for instance, validates disjunctive syllogism and
thus, if you wish, modus ponens, but there is no single theorem in such
logic.  LP, on the other hand, cannot validate disjunctive syllogism,
but offers a whole plethora of theorems.)

> For myself I would no more care to assert:
>
> This sentence is not true.
>
> than I would:
>
> The moon is made of blue cheese.

   Do you intend to say that both sentences should receive the value
"false"?  Do you see this as a reasonable solution in the case of the
first "Liar sentence"?

   Best, Joao Marcos



References:
A. Avron, "Natural 3-valued logics", The Journal of Symbolic Logic,
56:276--294, 1991.

D. Batens, "Against global paraconsistency", Studies on Soviet Thought,
39:209--229, 1990.

W. A. Carnielli, and J. Marcos, "A taxonomy of C-systems", in:
Paraconsistency -- The logical way to the inconsistent.  Proceedings of
the II World Congress on Paraconsistency (WCP'2000), pp. 1-94.  Marcel
Dekker, 2002.
Preprint available at: http://www.cle.unicamp.br/e-prints/abstract_5.htm

N. C. A. da Costa,  Sistemas Formais Inconsistentes, Thesis, UFPR,
Brazil, 1963.  Curitiba: Editora UFPR, 68p, 1993.

N. C. A. da Costa, "On the theory of inconsistent formal systems", Notre
Dame Journal of Formal Logic, 15:497-510, 1974.

G. Priest, "Minimally inconsistent LP", Studia Logica, 50:321--331,
1991.



PS: There is a discussion list on paraconsistency at
http://groups.yahoo.com/group/paraconsistency/.  I have noticed that not
much has been happening in it, but perhaps people over there will have
different opinions on the above matters and might be also willing to
discuss them with you.

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