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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