# [FOM] Misuse of standard terminology

Arnon Avron aa at tau.ac.il
Sat Aug 31 06:30:33 EDT 2013

I am quite sympathetic with most of what Harvey Friedman wrote
in his posting from  Wed, Aug 28, 2013. However, I suspect that I
might be *too* sympathetic in his eyes. So I wonder, Harvey,
how consistent you are ready to be with the following statements:

> The misuse of standard terminology with fixed meanings going back decades
> or even centuries or even longer, is an interesting phenomena.
...

> What is completely illegitimate is to usurp the usual symbols and
> terminology. The symbols and terminology must be altered in order to
> avoid any possible confusion. E.g., one might use and*, or*, ifthen*,
> iff*, Boolean equivalence*, to avoid confusion.

So here are two cases in point:

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!)

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...

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'?

To prevent misunderstandings, I would like at this point  to clarify
my view about the last question. So although I have done much work
on non-classical logics, I have always used classical logic in that
work, and I am 100% classical logician. However, I do think that
classical logicians and books are using bad terminology
when they present the `if-then' notion as a propositional connective
(a mistake done by the relevantists too, and as far as I understand
also by Friedman in his posting, when he spoke about "ifthen*").
Actually, the use of `if-then' in mathematical texts always
means some combination of universal quantifiers with `material implication',
not just one propositional connective. (I don't want to dwell
on this any further at this posting, because this is not
its main subject.)

Arnon Avron