Supp-supplement: A class is something very else than a group: Did marxism fail because of this? A class cannot rule, but a group can. To expect the working class, or the class of proletarians, to being able to rule, could not work, because it simply is not a group. The ruling group were the functionaries (functional composition), who put up a dictationship, ok, but never ever have represented the proletarian class (I am talking about the Soviet Union, just for an example). Well, what I want to say is: This whole topic is politically relevant, isnt it? Can Peirce help making the world a better place? Step down from the ivory tower?
Supplement: At the moment I am not very clear about the difference and the relatedness of mereology (composition) and classification (e.g. taxonomy). Though my temporary hunch is, that a class (classification) is, though not transformed into, abstracted as a group (composition) by an analyzing mind. And, because we dont always observe our own abstractions, we dont see the difference (between composition and classification), as it exists in reality whithout being analyzed by us. I suspect this difference to be huge (quite contradictional), and suspect our not seeing it to be a collusion based on insufficient self-observation (of missing to observe our observing processs). Just a hunch as I said. I am not clear about that all at the moment.
John, List,
thank you, John! This is very interesting to me, the fact that mereology, effect, and classification were treated mathematically, by Boole. I must read Boole (and Peirce and Peano).
Best,
Helmut
On 11/17/2017 5:05 PM, Helmut Raulien wrote:
> I think, the three kinds of implication or hierarchy are: Composition,
> power, and classification:
>
> Composition: "a contains b" or "b is a part of a", "if we have a, then
> we have b too".
> Power: "a can have an effect on b", "if a changes, then b is not safe of
> remaining unchanged either".
> Classification: "b is a kind of a" or "a implies b", "if a then b".
They are related. In fact, George Boole used the same operations
to represent them. We often think of Boolean algebra as a version
of propositional logic. But in his two books of 1847 and 1854, he
used the same algebra for all three. Peirce certainly knew that.
Boole had only three operators: '+' for OR; '×' for AND; '-' for NOT.
He had two "Boolean values" 0 and 1 with the following axioms:
0+0=0. 0+1=1. 1+0=1. 1+1=0.
0×0=0. 0×1=0. 1×0=0. 1×1=1.
-0=1. -1=0.
Note that Boole assumed an exclusive OR. Peirce replaced
that assumption with an inclusive OR: 1+1=1.
Peirce also introduced an if-then symbol, which he defined
as less-than-or-equal: if p implies q, the truth value of
p is always less than or equal to the truth value of q.
But instead of using the more common symbol '≤', Peirce chose to
write it as '-<'. He explained that writing -< can be done without
lifting pen from paper, but ≤ requires two separate strokes. It's
not that Peirce was lazy, but that he wanted to emphasize -< as a
single operator, not as a compound of two distinct operators.
The symbol -< could be defined in two equivalent ways:
p -< q is defined as -(p × -q)
p -< q is defined as (-p + q)
The above explanation of '-<' emphasizes implication of
propositions, which corresponds to "power or effect".
Boole also used his algebra for a simplified set theory:
p×q is intersection; p+q is disjoint union; and -p is
the complement. But Peirce would interpret p+q as union,
and p-<q as subset.
This theory is today called 'mereology' (from the Greek word
'meros' for part). In effect, Peirce's version of Boolean
algebra can represent composition (part-whole theory).
Boole also used his algebra for relating the terms of
a syllogism. In modern logic, those terms represent
monadic predicates, which can be used to specify the
classes of a classification. If p(x) and q(x) are
predicates that specify classes, Boole would use his
algebra to relate them.
With Peirce's notation, p(x) -< q(x) would say that
p(x) specifies a subclass of q(x).
Note that Peirce's symbol is more convenient that Peano's.
To say that every cat is an animal:
Peirce: For every x, cat(x) -< animal(x).
Peano: For every x, cat(x) ⊃ animal(x).
To say that the set Cats is a subset of the set Animals:
Peirce: Cats -< Animals.
Peano: Cats ⊂ Animals.
Peirce's symbol points in the same direction for both. But
Peano's notation is more confusing.
John
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> I think, the three kinds of implication or hierarchy are: Composition,
> power, and classification:
>
> Composition: "a contains b" or "b is a part of a", "if we have a, then
> we have b too".
> Power: "a can have an effect on b", "if a changes, then b is not safe of
> remaining unchanged either".
> Classification: "b is a kind of a" or "a implies b", "if a then b".
They are related. In fact, George Boole used the same operations
to represent them. We often think of Boolean algebra as a version
of propositional logic. But in his two books of 1847 and 1854, he
used the same algebra for all three. Peirce certainly knew that.
Boole had only three operators: '+' for OR; '×' for AND; '-' for NOT.
He had two "Boolean values" 0 and 1 with the following axioms:
0+0=0. 0+1=1. 1+0=1. 1+1=0.
0×0=0. 0×1=0. 1×0=0. 1×1=1.
-0=1. -1=0.
Note that Boole assumed an exclusive OR. Peirce replaced
that assumption with an inclusive OR: 1+1=1.
Peirce also introduced an if-then symbol, which he defined
as less-than-or-equal: if p implies q, the truth value of
p is always less than or equal to the truth value of q.
But instead of using the more common symbol '≤', Peirce chose to
write it as '-<'. He explained that writing -< can be done without
lifting pen from paper, but ≤ requires two separate strokes. It's
not that Peirce was lazy, but that he wanted to emphasize -< as a
single operator, not as a compound of two distinct operators.
The symbol -< could be defined in two equivalent ways:
p -< q is defined as -(p × -q)
p -< q is defined as (-p + q)
The above explanation of '-<' emphasizes implication of
propositions, which corresponds to "power or effect".
Boole also used his algebra for a simplified set theory:
p×q is intersection; p+q is disjoint union; and -p is
the complement. But Peirce would interpret p+q as union,
and p-<q as subset.
This theory is today called 'mereology' (from the Greek word
'meros' for part). In effect, Peirce's version of Boolean
algebra can represent composition (part-whole theory).
Boole also used his algebra for relating the terms of
a syllogism. In modern logic, those terms represent
monadic predicates, which can be used to specify the
classes of a classification. If p(x) and q(x) are
predicates that specify classes, Boole would use his
algebra to relate them.
With Peirce's notation, p(x) -< q(x) would say that
p(x) specifies a subclass of q(x).
Note that Peirce's symbol is more convenient that Peano's.
To say that every cat is an animal:
Peirce: For every x, cat(x) -< animal(x).
Peano: For every x, cat(x) ⊃ animal(x).
To say that the set Cats is a subset of the set Animals:
Peirce: Cats -< Animals.
Peano: Cats ⊂ Animals.
Peirce's symbol points in the same direction for both. But
Peano's notation is more confusing.
John
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