[FOM] "Hidden" contradictions

Sam Sanders sasander at cage.ugent.be
Fri Aug 23 04:14:34 EDT 2013

(now in plain text)

Dear Joao,

As you know, "Eppur si muove" is used to mean "it doesn't matter what you believe; these are the facts".

In this context, belief does matter greatly:  In paraconsistent logic,  the meaning of the logical connectives is 
different than in classical logic (e.g. negation is weaker than classically, as far as I know) so as to avoid the principle of explosion (from A and NOT A, follows B).

For most logicians/mathematicians, the meaning connectives have in classical logic, represents THE meaning of the logical connectives.  One could indeed change
that meaning (e.g. the BHK interpretation or paraconsistent logic), but this would come across as unnatural to most.  

Classical logic is (whether you like it or not) the "intended" model of logic for most people (like ZFC is the intended foundation).

Perhaps what Tarski meant was that we would still regard a provable inconsistency (in a non-trivial inconsistent system)
as false, because most of us are so used to thinking classically.

To use an analogy:  we know that, in the context of intuitionistic mathematics, all total R->R functions are continuous.  
Still, most of us regard the existence of discontinuous total functions as evident, despite the "counter model" provided
by intuitionistic mathematics.  This too, I would say, is a product of our classical upbringing.        

In conclusion, let me quote a contemporary:  "Why can't we all just get along?".  What I mean is that paraconsistent logic (as well as constructive math)
has its place in the world, but the latter is dominated by classical logic.  



On 23 Aug 2013, at 08:07, Joao Marcos <botocudo at gmail.com> wrote:

> Mark Steiner wrote:
> >
> > Are there any historical examples in which inconsistent systems actually
> > yielded false theorems that could have made "bridges fall down" without
> > anybody noticing the inconsistency?
> Timothy Y. Chow wrote:
> >
> > I think that there are serious problems with the way this question is
> > phrased.
> >
> > For a start, it's not clear what you mean by "inconsistent systems
> > actually yielding false theorems."
> In fact, it does not seem to be entirely obvious even what is _meant_
> by the expression "false theorem".  If by "theorem" you mean a
> *provable formula*, as customary, a system to which a *sound*
> semantics is associated cannot have a "false theorem".
> The underlying philosophical attitude is epitomized in the
> following passage by Alfred Tarski:
>   "I do not think that our attitude towards an inconsistent theory
>   would change even if we decided for some reason to weaken
>   our system of logic so as to deprive ourselves of the possibility
>   of deriving every sentence from any two contradictory sentences.
>   It seems to me that the real reason of our attitude is a different
>   one: We know (if only intuitively) that an inconsistent theory
>   must contain false sentences; and we are not inclined to regard
>   as acceptable any theory which has been shown to contain such
>   sentences."
> The problem with such "intuition" is that it is simply wrong.  In
> usual *non-trivial inconsistent systems*, provable inconsistencies are
> just not outright "false".  One clearly sees, at any rate, that Tarski
> would not be willing to go along with paraconsistent logics.
> Eppur si muove.
> JM
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