Helmut - if you read Peirce's cosmological outlines [6.203 and
1.412], he begins with 1ns, moves on to the instantiations of 2ns,
and then, into the developing habits of 3ns. So, the 'actualization'
of the modes in spatiotemporal existence is linear.
But - all three modes are potential and necessary, therefore, I
don't think that one can say that any one of them is 'more
fundamental'.
And I'm not sure how a singular description of a variable [A human
is a featherless biped] can be transformed into a sound deductive
argument [IF it is a featherless biped THEN it is human]. As you
point out, the connection of the attribute [featherless biped] might
not always apply to the variable of 'human'.
Edwina
On Fri 29/01/21 11:02 AM , Helmut Raulien [email protected] sent:
All, I think, the difference is not the meaning, but what it
is. Though the double negation´s meaning is the same as the
conclusion´s meaning, the double negation has the form of a
proposition, or a definition, which is secondness: "There is not a
featherless biped that is not a human" may be classificationally
instantated from exclusion of exclusion to definition: "A human is a
featherless biped". The conclusion "If it is a featherless biped then
it is a human" may be individually instantated to an argument; "It is
a human because it is a featherless biped". It is thirdness, a rule
or law. Which of both is more fundamental? Both mean the same, so if
one changes, the other changes too. But which is more likely to
change? The law "If it is a featherless biped then it is a human"
cannot change just so, by itself. But the situation, the secondness,
the truth of the proposition "There is not a featherless biped that
is not a human" can easily change, and make the law obsolete, e.g. if
an ape quits using its arms for walking, or if a dog sadly has two
legs amputated, or if a chicken, due to a mutation, is born without
feathers. So, is the double negation more fundamental, because it
breaks the law? People will say, that the ape is just stupid, the
chicken and the dog are just handicapped, exceptions only corrobate
the rule, and all people who claim that the law is no longer in
charge are heretics. A law (3ns) is more tenacious than actualities
(2ns), that first have to prove worthwile one by one to add up to a
certain measure, so is it more fundamental? Helmut Freitag, 29.
Januar 2021 um 00:49 Uhr
"Jon Alan Schmidt"
wrote: John, List, All: In this post, I will simply respond to
the numbered items below rather than quoting them. 1. I am not
aware of any evidence that Peirce ever explicitly denied that
illation is essential for deduction or rejected the use of the scroll
for that purpose in EGs. The absence of these specific terms in his
relatively sparse writings during the last 34 months of his life does
not outweigh their abundance in his voluminous output over the
preceding decades. Again, my hypothesis is that in June 1911 he
merely decided to simplify his presentation of EGs for the
uninitiated by omitting the derivation of negation (oval) from
consequence (scroll). 2. The psychological process of noticing a
difference or distinction in perception does seem to be more
primitive than the linguistic process of verbalizing the inference of
modus ponens, but as Peirce repeatedly affirmed, the logical relation
of negation is absolutely not more primitive than the logical
relation of consequence. Moreover, "We do not derive these notions
[universal elementary relations of logic] from observation, nor by
any sense of being opposed, but from our own reason" (CP 8.352, EP
2:485, 1908). 3. I have repeatedly acknowledged that in classical
logic, which is what Peirce obviously had in mind in 1911, a scroll
is indeed equivalent to a nest of two ovals. Although he anticipated
intuitionistic logic in several remarkable ways, including his
explicit statements in R 300 (1908) that analyzing a consequence as a
composite of two negations is erroneous, unfortunately he did not take
the additional steps that would have been necessary to formalize it.
Otherwise, we might today be calling it synechistic logic instead.
4. According to Peirce, "I have a complete theory of this process
[logical analysis], including its methodeutic, which I base upon my
existential graphs which is my chef d'oeuvre" (NEM 3:885, 1908).
Whether it was logical analysis or EGs that he considered to be his
masterpiece, it is clear that they are closely linked. In the context
of logical analysis, he repeatedly defines "more analytical" as
"breaking up inferences into the greatest possible number of steps"
(CP 4.373, 1902). In every single classical or intuitionistic proof
that involves negation, deriving it from a consequence with "the
essence of falsity" as its consequent is technically a necessary
additional step. As Bellucci and Pietarinen rightly put it, "Taking
the idea of negation as primary is philosophically inaccurate." 5.
Peirce was obviously not advocating that we add unnecessary steps,
and he generally sought to minimize the number of axioms. For
example, he eagerly embraced non-Euclidean geometry as demonstrating
that the parallel postulate is not essential for a consistent system.
I believe that he likewise would have endorsed intuitionistic logic as
demonstrating that excluded middle and its corollaries are not
essential for a consistent system, had he managed to work out the
details himself--or had he lived to see Brouwer and Heyting do so,
despite undoubtedly disagreeing with them about the philosophical
motivations. In that sense, non-Euclidean geometry is more analytical
than Euclidean geometry and intuitionistic logic is more analytical
than classical logic, because in each case the former requires an
additional postulate/axiom for certain proofs--effectively, an
additional step that is presupposed by the latter. 6. According to
his own persistent testimony, Peirce was not terribly interested in
improving the efficiency of proof procedures, and the expression
"dumping the scroll" is frankly both crude and misleading. This
exchange seems like a good summary of our different positions that
result from our different purposes, so hopefully we can leave it at
that. Regards, Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
www.LinkedIn.com/in/JonAlanSchmidt [1] - twitter.com/JonAlanSchmidt
[2] On Wed, Jan 27, 2021 at 11:38 PM John F. Sowa wrote:
Jon, List,
A few more points:
1. The quotations you cited are from a time when Peirce still
thought that a sign of illation was important for deduction. Note
that in R670, he says that the EGs have just three syntactic
features: a line of identity, a spot for a rheme and a shaded area
for negation. The scroll is "equivalent" to a nest of two negations.
It is not a primitive feature. In L231 and the later MSS, he did not
draw a scroll or use the word.
2. When I wrote (in slide 10) that an inference was required for
negation, I meant a "process of inference", not a "special sign for
inference". But I admit that I should have been more precise: A
negation results from an observation of a difference or distinction
between two perceptions or two aspects of a single perception. That
observation may be expressed in the form "A is not B". That process
is far more primitive than an application of modus ponens.
3. In R670, the word he actually used to compare a scroll to a nest
of two negations is "equivalent". Equivalence implies that one can
be substituted for the other in any context. Since I wasn't looking
at the MS at the moment, I said that a scroll is "nothing but" a nest
of two negations. Equivalence implies that point. It also implies
that a nest of two negations is "nothing but" a scroll. In any case,
he did not draw a scroll or mention the word in L231 or the later MSS.
4. The word 'analytical' means "pertaining to analysis". It's not
at all obvious what the phrase "more analytical" would mean.
Although Peirce stated his "permissions" in different ways over the
years, every proof from 1897 to the end took exactly the same number
of steps.
5. Notice the proof of the Praeclarum Theorema in egintro.pdf. That
proof took exactly 7 steps from a blank to the conclusion. In the
Principia Mathematica, Whitehead & Russell took 43 steps, starting
from 5 non-obvious axioms. That length does not make their method
"more analytical" . More appropriate adjectives would be
inefficient, inelegant, awkward, clumsy, not recommended... (And by
the way, one of the 5 axioms in the 1910 edition was redundant, But
nobody noticed that fact until 1926.
6. The most efficient proof procedures used today, do not depend on
a special sign of illation or the rule of modus ponens. In dumping
the scroll, Peirce was, as usual, ahead of his time.
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