[FOM] Intutionistic "negation"

Arnon Avron aa at tau.ac.il
Wed Nov 9 02:44:03 EST 2005

```This time I got a lot of reactions to my questions. Among the reactions
the one of Hazen provides full answers. But even
his reply left me with the feeling that very often when intuitionists
talk about "decidable" or "undecidable" propositions or predicates
they have in mind exactly what ordinary mathematicians have in mind
(although they will not admit it) and sometimes (like in the case
of the counterexample from intuitionistic analysis) - their
official notion of decidability and undecidability (a side remark:
in my question about decidability I meant discrete structures -
I dont think that even ordinary mathematician would take {x|x>0}
as a decidable predicate on R!).

I cannot (and there is no need to) react to all replies. To explain the
point I am trying to make I chose to respond to parts of two.

On Tue, Nov 08, 2005 at 06:54:53PM -0500, Neil Tennant wrote:

>  An intuitionist can assert "it is not the case that
> every statement is either true or false" without being committed to the
> (strictly classical) conclusion "some statement is neither true nor
> false". Indeed (as pointed out by Mark van Atten, the intuitionist will
> also assert "no statement is neither true nor false".

A question: when your intuitionist asserts  "it is not the case that
every statement is either true or false" does the word "not" used there
has (as I suspect)  its *real* (classical!) meaning, or is that
intuitionist claiming that he has a procedure carrying each
proof of "every statement is either true or false" to an absurdity?
In the second case - what is this procedure? (and once again:
what is an absurdity?).

I repeat what I have already written: all the people, including
declared intuitionists, understand and *use* classical negation
all the time, and cannot avoid doing so. But intuitionists pretend
that real negation is meaningless and refuse
to include it in their *official* language.

On Fri, 4 Nov 2005, giovanni sambin wrote:

> When I explain this to students I say: the intuitionist can
> understand what the classicist says, including his proofs,
> but not conversely (unless one adds an extra modality).

First of all: when you wrote here "not conversely" - did you mean
(as I am sure) the *real* (classical "not"), or did you mean to
claim that you have a procedure that carries any proof
that classicist can understand what the intuitionist says
to a proof of absurdity, and if so - can you describe to me
this procedure? (and by the way: what is an absurdity?)

The intuitionists do not (or pretend not to) understand classical
negation.  In fact, as Hazen has
proved in his reply to my questions, not even a very modest counterpart
of classical negation is definable in the intuitionist language.
Moreover: it is well known that intuitionists cannot conservatively add
classical negation to their language. On the other hand classical
mathematicians do understand the intuitionists' connectives and
quantifiers, and when they feel they need them (which is rarely
the case), they can conservatively add them
to their language (Usually, as you have pointed out yourself,
using modalities).

> In conclusion, an intuitionist says that the classicist has no
> real disjunction (as well as no real existential quantifier),
> but only their negative counterpart.

Indeed the language of classical first-order logic does not
include constructive disjunction and an existential quantifier.
But, again: we can (and sometimes do) add them in case we need them
(exactly as we add the quantifiers to our basic propsitional
language, and nobody says that someone who uses classical
propositional logic "does not understand" the quantifiers).
Intuitionists, in contrast, neither have nor can add real
negation to their language. The best they can do is to add
Nelson's "strong negation" (indeed, Nelson's
connective deserves being classified as "negation" much more
then the so-called "intuitionistic negation" which is simply
not a negation!).

One final remark: by "this" in the first citation above
you meant the double-negation
interpretation, even though I was trying my best to formulate my
question in a way that would make it clear that the double-negation
interpretation is *not* an answer (in fact, in the original message
from October 22 where I discussed in geneal the question of comparing
logics, I explicitly emphasized that the double-negation interpretation
does not provide an answer to my question!).

Arnon Avron

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