[FOM] Why inclusive disjunction?

Dennis E. Hamilton dennis.hamilton at acm.org
Sat Jan 13 01:42:11 EST 2007

```Very well, here's more:

In Dabney House at Caltech (in 1957 at any rate) the student waiters would
come around after the main course and ask, "Coffee or tea?"  The smart-aleck
who said "yes" would receive a cup of coffee with a tea bag soaking in it.

Since there are a large number of ways where the (classical) connectives are
interdefinable given one or two as primitive (or one and a constant), the
only technical basis for using vel so much seems to be the appeal of duality
and of convention.  For Whitehead and Russell (at *1), the choice appears to
be for the convenience of moving from ~ and v directly to the conditional,
taken as ~p v q by definition.  They also concede that you have to start
somewhere and convenience is as good a guide as any.

In the lengthy introduction to the second edition of Principia Mathematica,
great attention is given to the use of the Sheffer stroke as a single
primitive, but they are not so thrilled about it as to rewrite the first
parts of Principia Mathematica.  I think the lack of a convenient
interpretation in English would have deterred them as well.

It seems to me that the abstraction of symbolic logic away from ordinary
language is a permanent condition.  It is not very fruitful to try matching
elements of symbolic logic to natural language words as if they were
compatibly formal.  Such an effort strikes me as damaging to both ordinary
language and symbolic logic.

- Dennis

PS: The identification of exclusive-or with "different" or "not equal" and
equivalence (or biconditional) with "equality" (in truth-value semantics) is
also interesting and, in the case of Boolean logic, something that seems to
have those two stand apart from the ~, v, and . trio. As I recall, Peano
used "=" where we would use equivalence now, and he used it for equality as
well.

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
Max Weiss
Sent: Friday, January 12, 2007 17:45
To: Foundations of Mathematics
Subject: Re: [FOM] Why inclusive disjunction?

On Jan 11, 2007, at 1:08 PM, Neil Tennant wrote:

> Back in 1975 there was a conference in Canterbury, Kent, where the
> last
> stage of the breakfast cafetaria line had a sign saying "Cereal or
> orange
> juice". I put both a glass of (rather synthetic-tasting) OJ and a
> bowl of
> (rather tired-looking) All Bran flakes on my tray, and the check-
> upbraided me. My appeal to inclusive "or", and the authority of
> several
> amused logicians in the line behind me, was to no avail. Exclusive
> "or"
> prevailed. I think I opted for the juice.

This reminds me of an observation due to Ray Jennings:

"You can have coffee or you can have tea"

entails

"you can have coffee".

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```