[FOM] Misuse of standard terminology

[Harvey Friedman]

On Sat, Aug 31, 2013 at 6:30 AM, Arnon Avron <aa at tau.ac.il> wrote:

>
>
> 1) Would you join me in my conviction that intuitionists are
>    misusing (or even "usurping") the standard notion of negation
>    (and the standard \neg symbol for it) by calling "negation"
>    something that fails to respect the double-negation law
>    and excluded middle (note the latter has been considered as one
>    of the three basic laws of thoughts for centuries!)
>

I agree with this.

   Note that I am NOT (real `not'!) saying that `intuitionist
>    negation' is useless. I am just protesting against the fact
>    that they do not call it not* (according to your suggestion).
>    Worse than that: they pretend that the real negation
>    is meaningless...
>

The idea that ordinary classical negation is meaningless is extremely
difficult to defend.

>
> 2) Ironically, the main argument of the relevance school of
>    Anderson and Belnap (and many before them) against classical
>    logic is that classical logicians are misusing (and actually
>    usurping) the notion of `if-then' or `implication' when
>    they take  `material implication' as their implication
>    (and call it an implication). They usually bring q->(p->p)
>    as an example of an invalid implication that is valid
>    for the classical ifthen*. I believe that a better
>    example is given by the classical tautology:
>
>    (A/\B->C)->((A->C)\/(B->C))
>
>    according to which (if we take -> as meaning `if-then'),
>    if C is implied by the conjunction of A and B then
>    it is already implied by one of them alone - something
>    which is absurd.
>
>    So would you agree that our textbooks on logic are misusing
>    the notion of `if-then'?
>

Textbooks in logic use material not, and, or , ifthen , iff, and for good
historical (and other) reasons. That works for f.o.m. Non material ifthen,
iff, should have some different notation. Logicians generally do that with
fancier arrows and fancier double arrows, connected with provability or
semantic notions.

An interesting issue is to say just how and why the material and, or, not,
ifthen, iff work so well for f.o.m. Because they do, they have the right to
continue to be used without modification.

I have always attempted to explain this by saying that "we want to have a
purely truth functional way of formalizing mathematics (and more)". So
there is a unique way of treating ifthen truth functionally that works for
mathematical reasoning. And that unique way is given by the usual truth
table.

A device that has not been standardized - is to use different typefaces for
different notions. Exactly which typefaces should be used for what is not
(yet) standard.

Harvey Friedman
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