The Grecian Griceian: Galen and Grice on n-adic connectives

In "The primary logic: instruments for a dialogue between the two cultures" (Herefordshire, 1997), Michele Malatesta writes in the section on

Syncategoremata -- Polyadic connectives


DYADIC CONNECTIVE:

It rains iff it pours.

'if and only if'

TRIADIC CONNECTIVE:

It rains and it pours and it's cold.

"And ... and"

TETRADIC CONNECTIVE:

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

'Or.. .or.. .or' in (gi) is a tetradic connective. ...

"We do not know if the Stoics had inference schemata
in which polyadic connectives of logic product appeared. But
the truth table we draw from the Gellius quotation is
correct and corresponds to modern mathematical patterns."

Malatesta goes on to analyse Galen's Isagoge dialektike as the first treatise to distinguish three usages for 'or'.

---

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 goes on to study each as 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's example is complex enough.

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

----

"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, since it is
necessary that one of them must be true
and the others not." (epeidepter anagkaion
estin en men huparkhein to en autois oukh'
huparkhein ta alla. Galen, Inst. Log., V, 2."

Malatesta notes that scholars now regard
that the polyadic disjunctive (complete battle, teleia
makhe) and quasi-disjunctive (elippes makhe) are not
simple extensions of non-equivalence (Lukasiewicz's J)
and non-product (Lukasiewicz's D).

Malatesta, "Foundations of the Probability Calculus" (Metalogicon, 1989) and commentary on Galen by J. Mau (Berlin, 1960).

Also:

Stakelum, "Galen and the logic of the proposition" (Rome, 1940)

In particular, as 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 a note on Galen: "This passage illustrates the difficulty of trying to express, as
Galen wants, a complex logical situation without a proper
symbology."

---- (quoting from Kieffer).

Galen distinguishes the inferences with dyadic
from those with connectives which are triadic,
tetradic, etc.

But surely his complications can be resolved by
introducing the idea of polyadic (or n-adic, as I prefer)
IMPLICATION (as Malatesta does on p. 184).

books.google.com/books?isbn=0852444990...

From online study by Suber, too

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."

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

*

in

*(p,q,r)

be a triadic connective
expressing

~(p v q v r)

i.e. "all the following are false".

With this truth-function we can express

~p

thus:

*(p,p,p).

We can express

p /\ q

thus:

*(*(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 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).

We can express

pq

as

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

By Hunter's metatheorem 21.3,

if we can express both ~p and pq, we can express all 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.

"Invent a triadic connective that can express
all truth-functions and prove that it can. Then
show how it can be reduced to (replaced by) some
set of dyadic connectives."