[FOM] "Hidden" contradictions

Joao Marcos botocudo at gmail.com
Fri Aug 23 02:07:17 EDT 2013

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

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.
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