List:
I discovered over the weekend that Fernando Zalamea's former student Arnold
Oostra has proposed some relatively simple adjustments that make
Existential Graphs (EGs) compatible with intuitionistic logic (
https://en.wikipedia.org/wiki/Intuitionistic_logic), rather than classical
logic. As Joan Moschovakis states, "Philosophically, intuitionism differs
from logicism by treating logic as a part of mathematics rather than as the
foundation of mathematics; from finitism by allowing constructive reasoning
about uncountable structures ... and from Platonism by viewing mathematical
objects as mental constructs with no independent ideal existence" (
https://plato.stanford.edu/entries/logic-intuitionistic/)--all
characteristics that likely would have appealed to Peirce. Moreover, as
Zalamea summarizes, "Oostra shows that the natural graphs akin to a logic
of continuity, or topological logic, are in fact the intuitionistic
existential graphs" (
https://riviste.unimi.it/index.php/Lebenswelt/article/view/11113/10508, p.
83).
The most obvious difference between the two systems is the absence in
intuitionistic logic of the law of excluded middle (LEM), and consequently
the invalidity of double negation elimination; i.e., the EG rule that
allows erasure of a double cut ("not not-A" becomes "A"). Peirce
consistently maintained that the *principle *of excluded middle--which is
similar, but not identical, to LEM--does not apply to anything that is *general
*or *continuous*, and therefore *indeterminate*; and he also was among the
first, perhaps the *very *first, to suggest an outline for developing a
three-valued logic accordingly.
CSP: Triadic Logic is that logic which, though not rejecting entirely the
Principle of Excluded Middle, nevertheless recognizes that every
proposition, S is P, is either true, or false, or else has a lower mode of
being such that it can neither be determinately P, nor determinately not-P,
but is at the limit between P and not P.
Of course it remains true, as far as the principle of contradiction is
concerned that the state of things represented by the proposition cannot be
V and F, *verum atque falsum* and must be VψF if by F is meant LψF (L
signifying the limit, i.e., that S is not capable of the determination P or
of the determination F). Thus, a blot is made on a sheet. Then every point
of the sheet is unblackened or is blackened. But there are points on the
boundary line; and those points are insusceptible of being unblackened or
of being blackened, since these predicates refer to the area about S and a
line has no area about any point of it ...
Thus the Triadic Logic does not *conflict* with Dyadic Logic; only, it
recognizes, what the latter does not ...
Triadic Logic is universally true. But Dyadic Logic is not absolutely
false, it is only *L*. (R 339:624[344r]; 1909 Feb 23)
CSP: I have long felt that it is a serious defect in existing logic that
it takes no heed of the *limit *between two realms. I do not say that the
Principle of Excluded Middle is downright *false*; but I *do *say that in
every field of thought whatsoever there is an intermediate ground
between *positive
assertion* and *positive negation* which is just as Real as they.
Mathematicians always recognize this, and seek for that limit as the
presumable lair of powerful concepts; while metaphysicans and oldfashioned
logicians,--the sheep and goat separators,--never recognize this. The
recognition does not involve any denial of existing logic but it involves a
great addition to it to recognize such a mode of logic that it takes no
heed of the *limit *between two realms. (NEM 3:851; 1909 Feb 26)
In the first excerpt, ψ approximates Peirce's symbol for "logical
aggregation"--a modification of the rotated division sign previously
proposed by William Stanley Jevons--which he defined within his entry for
"Logic (exact)" in Baldwin's Dictionary as "The operation of uniting two or
more terms or propositions, called aggregants, to produce an aggregate term
or proposition which is true of everything of which any aggregant is true,
and false of everything of which all the aggregants are false" (
http://www.jfsowa.com/peirce/baldwin.htm). In other words, "V and F" is
still ruled out by "the principle of contradiction," but "V or F" is
replaced by "V or L or F."
Unfortunately for those of us who are not fluent in Spanish--as the saying
goes, I know just enough to be dangerous--Oostra has published most of his
work to date in that language, in an ongoing series of online volumes that
he has been co-editing with Zalamea called *Cuadernos de Sistemática
Peirceana* [Notebooks of Peircean Systematics]. The first seven are
available online (
https://ucaldas.academia.edu/CuadernosSistem%C3%A1ticaPeirceana), and the
eighth is supposed to be coming out soon; its entire contents will be
material about EGs from an international symposium that occurred in 2016,
including a contribution from John Sowa (http://www.jfsowa.com/pubs/), so
perhaps there will be an English version of Oostra's complete exposition.
In the meantime, we can glean the basics from the many examples of IEGs in
Oostra's papers, along with his 2012 slide presentation in English, "A
Lattice of Intuitionistic Existential Graph Systems" (
https://matematicas.uniandes.edu.co/eventos/SLALM2012/slides/arnold_oostra.pdf).
It is intriguing to observe how much of this approach is grounded in
Peirce's own texts about EGs, which makes it mildly surprising that he did
not develop intuitionistic logic or IEGs himself. The following are what I
understand to be the steps to get from EGs to IEGs, translating Oostra's
terminology as "cut" (*corte*) for an outer enclosure (CP 4.556; 1906) and
"loop" (*lazo*) for an inner enclosure (CP 4.437; 1903).
1. Use a continuous scroll for implication, rather than nested cuts or
shading, such that the conditional is more fundamental than negation (CP
4.564; 1906). This is because in intuitionistic logic, unlike in classical
logic, "if A then B" is *not equivalent* to "not (A and not B)"; the
former entails the latter, but not the other way around. All of the usual
permissions are maintained--insertion in odd areas, erasure in even areas,
iteration and deiteration toward the inside--except that double cut erasure
is replaced by "scrolling," in which a scroll with nothing in its outer
close and a proposition in its inner close is interchangeable with the
unenclosed proposition by itself.
2. Define negation as a scroll with a proposition in its outer close and
the "pseudograph" in its inner close. The pseudograph is an absurdity by
virtue of "implying that every proposition is true" as either an empty cut
or "a black spot entirely filling the [inner] close," which "may be drawn
invisibly small" to leave only the outer close (CP 4.454-456; 1903).
However, Peirce suggested at least once that this "blot" should *not* be
completely omitted, lest the derivation be forgotten (CP 4.564n; c. 1906).
3. Represent disjunction with a "multiple scroll" that has more than one
loop within its cut (CP 4.457; 1903) and apply the usual permissions to
these loops--insertion in odd areas, erasure in even areas, iteration and
deiteration within the same cut. Accordingly, an evenly enclosed empty cut
can always be transformed into any proposition, since this is equivalent to
inserting a loop with the proposition in it and then scrolling.
Oostra provides several helpful clarifications in one of his papers--a
scroll *cannot *be iterated within itself; an additional loop *cannot *be
inserted within another loop, only within a cut; and although a loop can be
detached from its cut, thus becoming a nested cut, a nested cut *cannot *be
attached to its outer cut, thus becoming a loop. Again, a double cut *cannot
*be erased, but an unenclosed proposition can have a double cut added
around it by scrolling and then detaching the loop from the cut. Lines of
identity can be employed in IEGs to represent indefinite individuals as in
Beta EGs, while broken cuts and loops can be employed in IEGs to represent
modalities as in Gamma EGs. Converting IEGs back into EGs only requires
restoring double cut erasure.
I am always happy to receive feedback from anyone, including well-founded
criticism; but in this case, I would especially welcome substantive
comments from John Sowa, including any corrections if I have misstated
something. Despite our recent disagreements on various matters, I readily
acknowledge your considerable expertise in logic and especially EGs.
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Professional Engineer, Amateur Philosopher, Lutheran Layman
www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt
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