Or not -- free choice as an otiosity worth having

by JLS
for the GC

Galen saw that, too, like Grice. In his "Primary logic",
subtitled "Instruments for a dialogue between the two cultures"
(Herefordshire, 1997), Michele Malatesta quotes from S. Kieffer, as they consider
Galen's struggles with complex Greek propositions involving many disjunctions.
Galen's example, in Greek, was:

"Dion either is walking or is sitting or he is lying down or is running or
is standing still."

Malatesta thought that Lukasiewicz should prove Galen wrong, in that Galen
was failing to reciprocate the triadicity (and beyond) of 'or'. But
apparently Mau dissuaded him:

"Hier liegen logistisch gesehen zwei n-adische Wahrheitswertfunktoren vor,
wir koennen fuer die symbolische Darstellung NICHT DIE DYADISCHE
FUNKTOREN, die Kontravelenz J under
der Schefferschen funktor D verwenden. Es wird ein komplexer Ausdruck mit
verschiedenen dyadische Funktoren venoetigt."

compared to which Galen's becomes a veritable piece of *cake*.

The problem with Galen was not so much his sentences -- polydisjunctive --
but his explanation of them. The fact that he was using Hellenistic Greek
surely didn't help. On the other hand, S. Kieffer is being a bit naive when
he writes as he comments on the extant parchments by Galen, "This
passage," -- 'if you can read it' -- "illustrates the obvious difficulty Galen is
having in trying to express, as
Galen apparently desires, a mighty complex logical situation without a
proper symbology." (slightly edited).

It all starts rather simply. Malatesta provides the following illustration
of what Galen is after. Malatesta's example is, as I say, simple enough.
He calls it 'tetradic connective':

It is spring, or it is summer, or it is autumn, or it is winter.

"This is a tetradic connective." -- but see below Suber for its reduction
to a common-or-garden dyadic one -- and cfr. Grice 1989:iv on the Sheffer
stroke (or actually 'one of the strokes').

Malatesta notes: "We do not know if the Stoics had inference schemata in
which polyadic connectives of logic product appeared. But the truth table we
[can] draw from the Gellius quotation [in Attic Nights -- In the I Love
Loeb collection] is correct and corresponds to modern mathematical patterns."
Leaving Gellius behind, Malatesta goes on to analyse Galen's "Isagoge
dialektike" as, indeed, the first treatise to distinguish three usages for 'or':

vel --- p v q
aut --- p w q
'battle' p / q --- Galen's way of expressing 'contrast'. "It is raining"
is in battle with "it is not raining". When there's 1/3, the battle is not
complete (teleia) but incomplete. Cfr. elsewhere, "Brief history of
negation".

The Hellenistic terms that Galen used for all this are schematised by
Malatesta -- as he compares it with Lukasiewicz's and Church's notations:

paradiezegumena (paradisjunction) --------------- vel --- "p v q".
Matrix 1110.
diezegumnena (exclusive disjunction) ------------ aut ---"p w q". Matrix
1001
paraplesion diezeugmnenon (quasidisjunctive) ---------"p | q:. Matrix
0110

Galen, as studied by Malatesta, goes on to consider each not just as
dyadic but polyadic (tetradic and above). He considers inclusive disjunction and
its implicature as exclusive.

"For clear and concise exposition," Galen writes, "nothing prevents our
calling disjunctives [these] propositions and quasi-disjunctives [the
others]. Let there be no quibble over to say quasi-disjunctive. But in
some propositions it is possible for MORE THAN ONE or for _all_ the
members to be true. Some call paradisjunctives the propositions of this sort,
since the disjunctives have one member only true, whether
they be composed of two simple propositions or of more than two."

Galen then, feeling poetic, tries to provide a 'molecule' out of the
'atoms', and comes up with:

"Dion either is walking, or he is sitting, or he is lying down, or he is
running, or he is standing still."
----p q r
s t

In symbols

p v q v r v s v t

Galen comments, as foreshadowing Grice, that that above sounds _like_ a
mouthful ("be cooperative").

Yet, he adds, "whenever a proposition is composed in this way," Galen
says, "any one member is in incomplete battle [opposition] with each of the
other members."

"But taken all together they are in COMPLETE battle [opposition] with one
another."

And this is so, "since it is necessary that one of them must be true and
the others not."

His Hellenistic style is too charming to miss:

"epeidepter anagkaion estin en men huparkhein to en autois oukh'
huparkhein ta alla."

Galen, Inst. Log., V, 2."

Malatesta goes on to note that scholars now regard that the polyadic
disjunctive (complete battle, teleia makhe) and quasi-disjunctive (elippes
makhe) are not THAT simple extensions of non-equivalence (Lukasiewicz's J) and
non-product (Lukasiewicz's D). He refers to a commentary on Galen by J. Mau
(Berlin 1960) and to his own "Foundations of the Probability Calculus" in
his edited Metalogicon, 1989, for a proof. Also: Stakelum, "Galen and the
logic of the proposition" (Rome, 1940)

It is at this point that Malatesta brings in the reference to tetradic and
beyond by Mau, when Mau writes: "Hier liegen logistisch gesehen zwei
n-adische Wahrheitswertfunktoren vor, wir koennen fuer die symbolische
Darstellung NICHT DIE DYADISCHE FUNKTOREN, die Kontravelenz J under der
Schefferschen funktor D verwenden. Es wird ein komplexer Ausdruck mit
verschiedenen dyadische Funktoren venoetigt."

Maltesta goes on to provide many-valued truth-tables and the irreverence
by Kieffer: "This passage in Galen illustrates the difficulty of trying to
express a pretty complex logical situation without a proper
symbology."

But the fact remains that, out of the blue, but not really, since he was
considering issues of what comes to be called 'free choice' ('any' losing
implicatures in negated free-choice polar items), Galen distinguishes the
inferences with dyadic from those with connectives which are triadic, or
tetradic, or etc. But surely Galen's complications should be resolved, we trust,
by introducing, as Malatesta does, the idea of polyadic (or n-adic, as I
prefer) IMPLICATION (Malatesta, p. 184). (The thing is at
books.google.com/books?isbn=0852444990).

Now, from online study by Suber, too, we learn how we can TEST students
about this. Suber goes on to use * and # as variables for polyadic
connectives, which include some realisation of 'or' (or other).

At www.earlham.edu/~peters/courses/logsys/exercise.htm

"No monadic connective (operator) can express all truth-functions." "Many
triadic connectives can." "Any triadic connective is reducible to some set
of dyadic connectives." "Dyadic connectives can express all
truth-functions and cannot be further reduced. "Why this privilege or special status for
two-ness?" "There is NO privilege for twoness." "Just as n-adic
connectives (n>2) are translatable into dyadic connectives,
so are dyadic connectives translatable into n-adic (n>2)." "These are
theoretically equivalent."

"If we prefer the dyadic connectives, it is only in practice, for
elegance, economy, simplicity." (Grice's rationale in WoW:iv).

Let us invent a triadic connective that can express all truth-functions
and, let's prove that it can. Then let's show how it can be reduced to
(replaced by) some set of dyadic connectives.

Let, indeed,

*

be, in

*(p,q,r)

a triadic connective expressing

~(p v q v r)

i.e. "all the following are false". With this truth-function we can
express

~p

as

*(p.p,p)

We can also express

p /\ q

as

*(*(p,p,p),*(q,q,q),*(q,q,q)).

By Hunter's metatheorem (21.4)
if we can express both "~p" and "p v q", we can express ALL aleatory (if
not hapzard) truth-functions. Now let

#

in

#(p,q,r)

be a triadic connective expressing

~(p·q·r)

i.e. "not all the following are true". With THIS truth-function we can
express

~p

as

#(p,p,p).

And we can also express

pq

as

#(#(p,p,p),#(q,q,q),#(q,q,q)).

Again, by Hunter's metatheorem 21.3, if we can express both "~p", and, now
"p /\ q", we can express all (aleatory, if not hapzard) truth-functions.
From these two, it should be clear how to produce adequate
n-adic truth-functions for any n>2. Those like "*" will be generalized
dagger functions. Those like "#" will be generalized stroke functions.

Next: how this apply to Grice's reflections on 'alcohol-free'. Or not!